Cuntz semigroups of ultraproduct C*-algebras

TL;DR

This paper establishes the categorical properties of Cuntz semigroups for ultraproducts of C*-algebras, showing they form a bicomplete category and aligning their structure with ultraproducts of individual semigroups, with applications to K-theory and simplicity.

Contribution

It proves the bicompleteness of the category of abstract Cuntz semigroups and demonstrates that scaled Cuntz semigroups of ultraproducts match the ultraproducts of individual semigroups, advancing the understanding of their structure.

Findings

The category of abstract Cuntz semigroups is bicomplete.

Scaled Cuntz semigroup of an ultraproduct equals the ultraproduct of scaled Cuntz semigroups.

Characterization of when ultraproducts are simple and criteria for order properties.

Abstract

We prove that the category of abstract Cuntz semigroups is bicomplete. As a consequence, the category admits products and ultraproducts. We further show that the scaled Cuntz semigroup of the (ultra)product of a family of C*-algebras agrees with the (ultra)product of the scaled Cuntz semigroups of the involved C*-algebras. As applications of our results, we compute the non-stable K-Theory of general (ultra)products of C*-algebras and we characterize when ultraproducts are simple. We also give criteria that determine order properties of these objects, such as almost unperforation.

Equations

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$$\operatorname{Cu}(\prod_{j}A_{j})\cong\big{\{}x\in\prod_{j}\operatorname{Cu}(A_{j})\mid x\leq\infty u\big{\}},$$

$$\mathrm{Cu}(A):=(A\otimes \mathcal{K}{)}_{+}/\sim .$$

$$\gamma (S):=\bar{S}/\sim .$$

$$\mathrm{Cu}\big{(}\gamma(S),T\big{)}\cong\mathrm{W}\big{(}S,\iota(T)\big{)},$$

$$\tau (S):=\mathcal{P}(S)/\sim .$$

$$\mathrm{Cu}\big{(}T,\tau(S)\big{)}\cong\mathcal{Q}\big{(}\iota(T),S\big{)},$$

$$F(i)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathrm{PoM}\text{-}\!\prod_{i\in I}S_{i}:=\big{\{}(s_{i})_{i\in I}:s_{i}\in S_{i}\text{ for all }i\in I\big{\}}\,.$$

$$S:=\left\{(s_i{)}_{i\in I}\in \mathrm{PoM}\text{-}\prod _{i\in I}F(i):F(f)(s_i)=s_j\text{ for all }f:i\to j\text{ in }I\right\}.$$

$$F(f)(s_i)=F(f)(\underset{n}{\sup }{s}_i^{(n)})=\underset{n}{\sup }F(f)(s_i^{(n)})=\underset{n}{\sup }{s}_j^{(n)}=s_j.$$

$$\sigma (t)=({\sigma }_i(t){)}_i=(\underset{n}{\sup }{\sigma }_i(t_n){)}_i=\underset{n}{\sup }({\sigma }_i(t_n){)}_i=\underset{n}{\sup }\sigma (t_n).\text{qed}$$

$$\mathrm{Cu}\text{-}\underleftarrow{\lim }F=\tau \left(\mathcal{Q}\text{-}\underleftarrow{\lim }F\right)=\tau \left(\mathrm{PoM}\text{-}\underleftarrow{\lim }F,{\ll }_{\mathrm{pw}}\right)$$

$$\prod _{i\in I}{S}_i=\tau \left(\mathcal{Q}\text{-}\prod _{i\in I}{S}_i\right)=\tau \left(\mathrm{PoM}\text{-}\prod _{i\in I}{S}_i,{\ll }_{\mathrm{pw}}\right).$$

$$\mathrm{Cu}\text{-}\underset{i\in I}{\underleftarrow{\lim }}{S}_i=\tau \left(\mathcal{Q}\text{-}\underset{i\in I}{\underleftarrow{\lim }}{S}_i\right).$$

$$\coprod _{j\in J}{X}_j\cong \underset{F⋐J}{\underrightarrow{\lim }}\coprod _{j\in F}{X}_j.$$

$$\coprod _{j\in J}{X}_j\cong \underset{F⋐J}{\underrightarrow{\lim }}\coprod _{j\in F}{X}_j\to \underset{F⋐J}{\underrightarrow{\lim }}\prod _{j\in F}{X}_j=\underset{j\in J}{⨁}{X}_j.$$

$${\coprod }_{j\in J}{X}_j\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$x=z+\phi (a)+\psi (b)\text{ and }y=z+\phi (b)+\psi (a).$$

$$x\sim a\prec b\sim y.$$

$$x=z+\phi (a)+\psi (b),\text{ and }y=z+\phi (b)+\psi (a).$$

$$z^{\prime \prime }\prec z^{\prime }\prec z,\text{ and }v\prec \phi (a),\text{ and }w\prec \psi (b),\text{ and }u\prec z^{\prime \prime }+v+w.$$

$$u\prec z^{\prime \prime }+v+w\prec z^{\prime }+\phi (a^{\prime })+\psi (b^{\prime })=x^{\prime }.$$

$$x=x_0{\sim }_0{x}_1{\sim }_0\dots {\sim }_0{x}_n=y.$$

$$u\precsim x^{\prime }{\precsim }_{0}x\sim y.$$

$$x=z+\phi (a)+\psi (b)\text{ and }y=z+\phi (b)+\psi (a).$$

$$h(x)=h(z)+h(\phi (a))+h(\psi (b))=h(z)+h(\psi (a))+h(\phi (b))=h(y).$$

$$x\sim a\prec b\sim y.$$

$$\tilde{h}([x])=h(x)=h(a)\prec h(b)=h(y)=\tilde{h}([y]).$$

$$(\mathrm{Cu}\text{-}\underset{i\in I}{\underrightarrow{\lim }}{S}_i)\otimes T\cong \mathrm{Cu}\text{-}\underset{i\in I}{\underrightarrow{\lim }}(S_i\otimes T).$$

Full text

As applications of our results, we compute the non-stable K-Theory of general (ultra)products of $C^{*}$-algebras and we characterize when ultraproducts are simple. We also give criteria that determine order properties of these objects, such as almost unperforation.

1. Introduction

Product and ultraproduct constructions are important tools in mathematics. For $C^{*}$-algebras, they are technical devices in which one can suitably encompass notions that have come to the forefront of the theory. Most notably, ultrapoducts of $C^{*}$-algebras appear in aspects of the Toms-Winter conjecture; see, for example, [BBS*+*19]. They also play an important role in detecting tensorial absorption by the Jiang-Su algebra for $C^{*}$-algebras with finite nuclear dimension; see [Kir06] (and also [RT17]).

The Cuntz semigroup is an invariant for $C^{*}$-algebras, first introduced in [Cun78] with the aim of proving that simple, stably finite, $C^{*}$-algebras admit quasitraces. Due to the fact that it contains a great deal of information about the $C^{*}$-algebra, its uses have been proved to be far-reaching in subsequent works. For example, the Cuntz semigroup has become key to distinguishing $C^{*}$-algebras that fail to satisfy regularity conditions and thus are not amenable to classification using the Elliott invariant (see, for example, [Rør03] and [Tom08]). Further, for the class of simple, separable, nuclear $C^{*}$-algebras that absorb the Jiang-Su algebra tensorially, it was proved in [ADPS14] that the Cuntz semigroup after tensoring with the circle is functorially equivalent to the Elliott invariant.

The categorical framework needed to analyse the Cuntz semigroup as an invariant was developed in the pioneering work by Coward, Elliott and Ivanescu in [CEI08]. They defined the category $\mathrm{Cu}$ of abstract Cuntz semigroups (also called $\mathrm{Cu}$-semigroups) and showed that the process of assigning to each $C^{*}$-algebra $A$ the (concrete) Cuntz semigroup of its stabilisation is a sequentially continuous functor. The properties of the category $\operatorname{Cu}$ and its relation with the category of $C^{*}$-algebras was further developed in [APT18], [APT17], and [APT20]. It is therefore interesting to seek efficient methods to compute the Cuntz semigroup of products and ultraproducts. This is one of the main objectives of this paper.

Buiding on previous work (as explained below), we summarize the main structural features of the category $\mathrm{Cu}$ as follows:

Theorem**.**

The category $\mathrm{Cu}$ of abstract Cuntz semigroups is a closed, symmetric, monoidal, bicomplete category.

That $\mathrm{Cu}$ is symmetric and monoidal was established in [APT18]; on the other hand, it was shown in [APT17] that $\mathrm{Cu}$ is closed. We prove in this paper that $\mathrm{Cu}$ is bicomplete, that is, complete and cocomplete (see Theorems 3.1 and 3.2 below). This means that $\mathrm{Cu}$ admits arbitrary small limits and colimits, and thus arbitrary products and inductive limits. (The latter was already proved in [APT18].)

In order to show that $\mathrm{Cu}$ is complete, we take advantage of the fact that it sits coreflectively in a bigger category, termed $\mathcal{Q}$ in [APT17]. It follows that the product of a family of $\mathrm{Cu}$-semigroups is not just their set-theoretic product, but rather a completion of the latter by means of the $\tau$-construction (see Subsection 2.3 for more details). Roughly speaking, the $\tau$-construction consists of considering certain equivalence classes of continuous paths indexed on the interval $(-\infty,0]$.

We further apply our results to $C^{*}$-algebras. To this end, we first study to what extent the Cuntz semigroup functor preserves these constructions. The fact that the Cuntz semigroup functor is stable by definition introduces a difficulty when dealing with products and ultraproducts of arbitrary infinite families of $C^{*}$-algebras, since these are not stable even if the algebras are. In order to circumvent this difficulty, we need a convenient substitute for stability. This is found in property (S), as introduced by Hjelmborg and Rørdam in [HR98].

Another important ingredient in our discussion is the notion of scale. For a $C^{*}$-algebra $A$, the natural scale of $\operatorname{Cu}(A)$ captures the information of equivalence classes coming from positive elements in $A$. This was first considered in [PTWW14]. We define in Section 4 scales in general abstract $\mathrm{Cu}$-semigroups and introduce the category $\mathrm{Cu}_{\mathrm{sc}}$, in which $\mathrm{Cu}$ sits reflexively. This is the natural categorical framework in which Cuntz semigroups of products and ultraproducts of arbitrary families of $C^{*}$-algebras can be computed.

Ultraproducts over an ultrafilter $\mathcal{U}$ are introduced categorically in Subsection 7.1 as inductive limits of products over sets in $\mathcal{U}$ following, for example, [Ekl77]. This becomes relevant because, as we show later, the scaled Cuntz semigroup functor preserves arbitrary inductive limits and products. We prove:

Theorem** (Theorems 4, 5 and 7.1).**

The category $\mathrm{Cu}_{\mathrm{sc}}$ is bicomplete, and the scaled Cuntz semigroup functor preserves arbitrary products and ultraproducts of $C^{*}$-algebras.

More precisely, given a family $(A_{j})_{j\in J}$ of $C^{*}$-algebras and an ultrafilter $\mathcal{U}$ over $J$, there is a natural commutative diagram

[TABLE]

where the horizontal maps are order-embeddings whose images are ideals. These maps induce order-isomorphisms between scaled Cuntz semigroups. In the unital setting, the said isomorphisms can be viewed as

[TABLE]

where $u$ is the image of the class of the unit via the map $\Phi$ and $\infty u=\sup_{n}nu$. A similar description is obtained for the ultraproduct.

Since the categorical picture of an ultraproduct is not the most commonly used in the category of $C^{*}$-algebras, we make the connection explicit in Subsection 7.2 to also view an ultraproduct of $C^{*}$-algebras as the quotient of their product modulo the ideal of strings that converge to zero along the ultrafilter. This is also a convenient way to compute its Cuntz semigroup, as a quotient of the Cuntz semigroup of the product of the algebras modulo a suitable ideal, and it becomes a useful way to deal with ultraproducts in concrete examples (see Subsection 7.2 and Subsection 7.2).

As applications of our results, we compute the monoid of Murray-von Neumann classes of projections of a product and ultraproduct of a family of stably finite $C^{*}$-algebras, recovering and completing previous work of Li ([Li05]). We also characterize in Subsection 8.2 when an ultraproduct of $C^{*}$-algebras over a countably incomplete ultrafilter is simple; this unifies previous results due to Kirchberg (see [Rør02]) and also results from [FHL*+*16].

Acknowledgements

Part of this research was conducted during stays at the CRM (Barcelona), and the ICMAT (Madrid). The authors would like to thank both institutions for financial support and for providing inspiring working conditions.

The authors would also like to thank Leonel Robert and Gábor Szabó for valuable conversations and feedback.

The two first named authors were partially supported by MINECO (grant No. MTM2017-83487-P), and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya (grant No. 2017-SGR-1725). The third named author was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under the SFB 878 (Groups, Geometry & Actions) and under Germany’s Excellence Strategy EXC 2044-390685587 (Mathematics Münster: Dynamics-Geometry-Structure).

We wish to thank the referee for numerous comments that have helped to improve the manuscript.

2. Preliminaries: Categorical Framework

2.1. The category $\mathrm{Cu}$ of abstract Cuntz semigroups

In this subsection, we recall the definition of the category $\mathrm{Cu}$ as introduced in [CEI08].

\thepgrCt**.**

A positively ordered monoid is a commutative monoid $M$, written additively with neutral element [math], equipped with a translation invariant partial order $\leq$ for which [math] is the smallest element. The category of positively ordered monoids will be denoted by $\mathrm{PoM}$, where the morphisms are maps that preserve addition, order, and the zero element. See [APT18, Appendix B.2] for details.

\thepgrCt**.**

Let $(X,\leq)$ be a partially ordered set. A binary relation $\prec$ on $X$ is called an auxiliary relation (see [GHK*+*03, Definition I-1.11, p.57]) if:

• (i)

If $x\prec y$ then $x\leq y$, for all $x,y\in X$.

• (ii)

If $w\leq x\prec y\leq z$ then $w\prec z$, for all $w,x,y,z\in X$.

If, further, $X$ is a monoid, then an auxiliary relation $\prec$ is said to be additive if it is compatible with addition and $0\prec x$ for every $x\in X$. Observe that an auxiliary relation is automatically transitive: if $x\prec y$ and $y\prec z$, then by condition (i) $x\leq y$ and applying condition (ii) to $x\leq y\prec z\leq z$, we obtain $x\prec z$.

Of particular importance for us is the (sequential version of the) compact containment relation, also called way-below relation. Given $x,y\in X$, we say that $x$ is compactly contained in $y$, or that $x$ is way-below $y$, in symbols $x\ll y$, provided that for every increasing sequence $(y_{n})_{n}$ such that the supremum $\sup_{n}y_{n}$ exists and satisfies $y\leq\sup_{n}y_{n}$, there exists $k\in\mathbb{N}$ such that $x\leq y_{k}$.

Definition \thedfnCt ([CEI08]).

A $\mathrm{Cu}$-semigroup, also called an abstract Cuntz semigroup, is a positively ordered monoid $S$ that satisfies the following axioms:

• (O1)

Every increasing sequence $(x_{n})_{n}$ in $S$ has a supremum $\sup_{n}x_{n}\in S$.

• (O2)

Every element of $S$ is the supremum of a $\ll$-increasing sequence.

• (O3)

If $x^{\prime}\ll x$ and $y^{\prime}\ll y$ then $x^{\prime}+y^{\prime}\ll x+y$, for all $x^{\prime},y^{\prime},x,y\in S$.

• (O4)

If $(x_{n})_{n}$ and $(y_{n})_{n}$ are increasing sequences in $S$, then $\sup_{n}(x_{n}+y_{n})=\sup_{n}x_{n}+\sup_{n}y_{n}$.

A $\mathrm{Cu}$-morphism is a $\mathrm{PoM}$-morphism that preserves the way-below relation and suprema of increasing sequences. Given $\mathrm{Cu}$-semigroups $S$ and $T$, the set of all $\mathrm{Cu}$-morphisms from $S$ to $T$ is denoted by $\mathrm{Cu}(S,T)$.

An ideal in a $\mathrm{Cu}$-semigroup $S$ is a submonoid $J\subseteq S$ that is downward hereditary (if $x,y\in S$ satisfy $x\leq y\in J$, then $x\in J$) and closed under passing to suprema of increasing sequences.

\thepgrCt**.**

Let $A$ be a $C^{*}$-algebra, and let $a,b$ be positive elements in the stabilization $A\otimes\mathcal{K}$. Recall that $a$ is Cuntz subequivalent to $b$, in symbols $a\precsim b$, provided that there is a sequence $(x_{n})_{n}$ in $A\otimes\mathcal{K}$ such that $x_{n}bx_{n}^{*}$ converges to $a$ in norm. In case that $a\precsim b$ and $b\precsim a$, then $a$ and $b$ are said to be Cuntz equivalent, and we write $a\sim b$. Then $\sim$ is an equivalence relation on $(A\otimes\mathcal{K})_{+}$. The Cuntz semigroup of $A$ is the quotient set

[TABLE]

The equivalence class of an element $a\in(A\otimes\mathcal{K})_{+}$ in $\operatorname{Cu}(A)$ is denoted by $[a]$. Addition on $\operatorname{Cu}(A)$ is defined by $[a]+[b]=[\left(\begin{smallmatrix}a&0\\ 0&b\end{smallmatrix}\right)]$, for $a,b\in(A\otimes\mathcal{K})_{+}$ (using an isomorphism $M_{2}(\mathcal{K})\cong\mathcal{K}$). The neutral element for this operation is the class of the zero element in $A\otimes\mathcal{K}$. Finally, we define a partial order on $\operatorname{Cu}(A)$ by setting $[a]\leq[b]$ provided $a\precsim b$. In this way $\operatorname{Cu}(A)$ becomes a positively ordered monoid.

It was proved in [CEI08] that $\operatorname{Cu}(A)$ is a $\mathrm{Cu}$-semigroup. Moreover, a ∗-homomorphism $\varphi\colon A\to B$ induces a $\mathrm{Cu}$-morphism $\operatorname{Cu}(\varphi)\colon\operatorname{Cu}(A)\to\operatorname{Cu}(B)$ by $\operatorname{Cu}(\varphi)([a])=[(\varphi\otimes\mathrm{id}_{\mathcal{K}})(a)]$, for $a\in(A\otimes\mathcal{K})_{+}$. This defines a functor $\operatorname{Cu}(\_\,)$, from the category $C^{*}$ of $C^{*}$-algebras with ∗-homomorphisms to the category $\mathrm{Cu}$.

It was also shown in [CEI08] that the category $\mathrm{Cu}$ admits sequential inductive limits and that the functor $\operatorname{Cu}(\_\,)$ is sequentially continuous, that is, it preserves sequential inductive limits. This was extended to inductive limits over arbitrary directed sets in [APT18, Corollary 3.2.9]. In Subsection 3.2 we show that $\mathrm{Cu}$ is even a cocomplete category, that is, it admits arbitrary (small) colimits.

We will often use the following result, which is called ‘Rørdam’s lemma’ (see [Rør92, Proposition 2.4] or [Thi17, Theorem 2.30]): Let $a,b\in A_{+}$. Then the following are equivalent:

1. (1)

$a\precsim b$. 2. (2)

For every $\varepsilon>0$ there exists $\delta>0$ such that $(a-\varepsilon)_{+}\precsim(b-\delta)_{+}$. 3. (3)

For every $\varepsilon>0$ there exist $\delta>0$ and $s\in A$ such that $(a-\varepsilon)_{+}=ss^{*}$ and $s^{*}s\in\operatorname{Her}((b-\delta)_{+})$.

2.2. The category $\mathrm{W}$

In this subsection, we recall the definition of the category $\mathrm{W}$ from [APT18], with a slight deviation; see Subsection 2.2.

Definition \thedfnCt.

A $\mathrm{W}$-semigroup is a commutative monoid $S$ together with a transitive binary relation $\prec$ satisfying $0\prec x$ for every $x\in S$, and such that the following conditions are satisfied:

1. (W1)

For each $x\in S$, there is a $\prec$-increasing sequence $(x_{n})_{n}$ with $x_{n}\prec x$ for each $n$, and such that for any $y\prec x$ there is $n\in\mathbb{N}$ satisfying $y\prec x_{n}$. 2. (W3)

If $x^{\prime}\prec x$ and $y^{\prime}\prec y$, then $x^{\prime}+y^{\prime}\prec x+y$, for all $x^{\prime},y^{\prime},x,y\in S$. 3. (W4)

If $x,y,z\in S$ satisfy $x\prec y+z$, then there are $y^{\prime},z^{\prime}\in S$ such that $y^{\prime}\prec y$, $z^{\prime}\prec z$, and $x\prec y^{\prime}+z^{\prime}$.

Given $\mathrm{W}$-semigroups $S$ and $T$, a $\mathrm{W}$-morphism is a map $f\colon S\to T$ that preserves addition, the relation $\prec$, the zero element, and that is continuous in the sense that for every $x\in S$ and $y\in T$ satisfying $y\prec f(x)$ there exists $x^{\prime}\in S$ such that $x^{\prime}\prec x$ and $y\prec f(x^{\prime})$. The set of all $\mathrm{W}$-morphisms from $S$ to $T$ is denoted by $\mathrm{W}(S,T)$.

Remark \thermkCt.

In [APT18, Definition 2.1.1], we gave a slightly different definition of a $\mathrm{W}$-semigroup. To clarify the change from Subsection 2.2, let us denote the category defined in [APT18] by $\mathrm{W}^{+}$. Thus, a $\mathrm{W}^{+}$-semigroup is a positively ordered monoid $S$ together with an auxiliary relation $\prec$ satisfying $0\prec x$ for every $x\in S$, satisfying (W1), (W3) and (W4) as in Subsection 2.2, and satisfying (W2) which requires that each $x$ in $S$ is the supremum of $x^{\prec}:=\{x^{\prime}:x^{\prime}\prec x\}$. Further, a $\mathrm{W}^{+}$-morphism is an order-preserving monoid homomorphism that satisfies the continuity condition from Subsection 2.2.

If $S$ is a $\mathrm{W}^{+}$-semigroup, then by forgetting the partial order on $S$ we obtain a $\mathrm{W}$-semigroup. Further, every $\mathrm{W}^{+}$-morphism naturally induces a $\mathrm{W}$-morphism. This assignment defines a ‘forgetful’ functor $F\colon\mathrm{W}^{+}\to\mathrm{W}$.

Conversely, let $(S,\prec)$ be a $\mathrm{W}$-semigroup. We define a binary relation on $S$ by setting $x\leq y$ if and only if $x^{\prec}\subseteq y^{\prec}$. Then $\leq$ is a translation-invariant pre-order. Now, for $x,y\in S$, we set $x\prec_{+}y$ if and only if there exists $y^{\prime}\in S$ such that $x\leq y^{\prime}$ and $y^{\prime}\prec y$. Given $w,x,y,z\in S$, we have that $x\prec y$ implies $x\prec_{+}y$, also that $x\prec_{+}y$ implies $x\leq y$, and finally we have that $w\leq x\prec_{+}y\leq z$ implies $w\prec_{+}z$. Let us check the latter. If $x\prec_{+}y$, then there is $y^{\prime}\in S$ with $x\leq y^{\prime}\prec y$, and it follows that $w\leq y^{\prime}\prec y$. Since $y^{\prime}\prec y$ and $y\leq z$, we have $y^{\prime}\prec z$, and therefore $w\prec_{+}z$. It is easy to check that $(S,\prec_{+})$ satisfies (W1), (W3), and (W4).

Given $x\in S$, let us verify $x=\sup x^{\prec_{+}}$. Clearly, $x$ is an upper bound for $x^{\prec_{+}}$. To show that $x$ is the smallest upper bound, let $y\in S$ be another upper bound for $x^{\prec_{+}}$. Given $w\in S$ with $w\prec x$, we can apply (W1) to find $w^{\prime}\in S$ such that $w\prec w^{\prime}\prec x$, and in particular $w^{\prime}\in x^{\prec_{+}}$. By assumption we have $w^{\prime}\leq y$ and, since $w\prec w^{\prime}$, we obtain $w\prec y$. This implies $x\leq y$.

Thus, $(S,\leq,\prec_{+})$ is a $\mathrm{W}^{+}$-semigroup if and only if the pre-order $\leq$ is a partial order. By antisymmetrization applied to the preordered set $(S,\leq)$, one can naturally associate a partially ordered set $(S^{\prime},\leq^{\prime})$, and one can show that $\prec_{+}$ induces an auxiliary relation $\prec_{+}^{\prime}$ on $S^{\prime}$ in such a way that $(S^{\prime},\leq^{\prime},\prec_{+}^{\prime})$ is a $\mathrm{W}^{+}$-semigroup. This defines a functor $G\colon\mathrm{W}\to\mathrm{W}^{+}$ such that $G\circ F\cong\mathrm{id}$. However, $F\circ G\ncong\mathrm{id}$. For example, take $S=\mathbb{N}$, with $x\prec y$ if and only if $x=0$. Then $F(G(S))\cong\{0\}$, which is not isomorphic to $S$.

In the proofs below we work exclusively in the category $\mathrm{W}$ as given by Subsection 2.2. Thus, we do not require that $\prec$ induces a partial order. Moreover, by dropping this requirement, we may forget about the (partial) order altogether. Furher, we will see below that many results from [APT18] remain valid for the more relaxed definition of the category $\mathrm{W}$ used here.

\thepgrCt**.**

Given a $\mathrm{Cu}$-semigroup $S$, the monoid $S$ together with the relation $\ll$ is a $\mathrm{W}$-semigroup. Moreover, given two $\mathrm{Cu}$-semigroups $S$ and $T$, a map $\varphi\colon S\to T$ is a $\mathrm{Cu}$-morphism if and only if $\varphi$, considered as a map from the $\mathrm{W}$-semigroup $(S,\ll)$ to the $\mathrm{W}$-semigroup $(T,\ll)$, is a $\mathrm{W}$-morphism. Thus, we obtain a functor $\iota\colon\mathrm{Cu}\to\mathrm{W}$ that embeds $\mathrm{Cu}$ as a full subcategory of $\mathrm{W}$.

\thepgrCt**.**

Given a $\mathrm{W}$-semigroup $S$, let $\bar{S}$ denote the set of all $\prec$-increasing sequences in $S$. We equip $\bar{S}$ with componentwise addition, giving it the structure of a commutative monoid. For sequences $\textbf{x}:=(x_{n})_{n}$ and $\textbf{y}:=(y_{n})_{n}$ in $\bar{S}$, we write $\textbf{x}\subseteq\textbf{y}$ if for every $k\in\mathbb{N}$, there is $l\in\mathbb{N}$ such that $x_{k}\prec y_{l}$. Then $\subseteq$ is a preorder on $\bar{S}$. Set $\textbf{x}\sim\textbf{y}$ if and only if $\textbf{x}\subseteq\textbf{y}$ and $\textbf{y}\subseteq\textbf{x}$. Then $\sim$ is an equivalence relation on $\bar{S}$. Define

[TABLE]

Given $\textbf{x}\in\bar{S}$, we use use $[\textbf{x}]$ to denote its class in $\gamma(S)$. The relation $\subseteq$ induces a partial order on $\gamma(S)$, by setting $[\textbf{x}]\leq[\textbf{y}]$ if and only if $\textbf{x}\subseteq\textbf{y}$.

The proof of [APT18, Proposition 3.1.6] shows that $\gamma(S)$ is a $\mathrm{Cu}$-semigroup, which we call the $\mathrm{Cu}$-completion of $S$. Further, given a $\mathrm{W}$-morphism $\varphi\colon S\to T$, we define $\gamma(\varphi)\colon\gamma(S)\to\gamma(T)$ by $\gamma(\varphi)([(x_{n})_{n}]):=[(\varphi(x_{n}))_{n}]$, for $(x_{n})_{n}\in\bar{S}$. One can show that $\gamma(\varphi)$ is a $\mathrm{Cu}$-morphism. This defines a covariant functor $\gamma\colon\mathrm{W}\to\mathrm{Cu}$.

Let $S$ be a $\mathrm{W}$-semigroup. We define $\alpha_{S}\colon S\to\gamma(S)$ by $\alpha_{S}(x):=[(x_{n})_{n}]$, for $x\in S$, where $(x_{n})_{n}$ is a $\prec$-increasing sequence obtained by applying (W1) to $x$. One verifies that $\alpha_{S}$ is a well-defined $\mathrm{W}$-morphism. Adapting the proof of [APT18, Theorem 3.1.10], we obtain the following result.

Recall that a full subcategory is said to be reflective if the inclusion functor has a left adjoint.

Theorem \thethmCt.

The category $\mathrm{Cu}$ is a full, reflective subcategory of $\mathrm{W}$. The functor $\gamma\colon\mathrm{W}\to\mathrm{Cu}$ is a left adjoint to the inclusion functor $\iota\colon\mathrm{Cu}\to\mathrm{W}$. Moreover, given a $\mathrm{W}$-semigroup $S$, the map $\alpha_{S}\colon S\to\gamma(S)$ is a universal $\mathrm{W}$-morphism: For every $\mathrm{Cu}$-semigroup $T$, there is a natural bijection

[TABLE]

implemented by sending a $\mathrm{Cu}$-morphism $\varphi\colon\gamma(S)\to T$ to $\varphi\circ\alpha_{S}$.

2.3. The category $\mathcal{Q}$

In this subsection, we recall the definition of the category $\mathcal{Q}$ and of the $\tau$-construction, as introduced in [APT17]. Since we will apply the $\tau$-construction only to $\mathcal{Q}$-semigroups, we use a simplified definition of path that is different from that used in [APT17]. We remark that the $\tau$-construction can be applied to a much more general class of semigroups equipped with an auxiliary relation by using a more general concept of paths; see [APT17, Definition 3.3].

Definition \thedfnCt.

A $\mathcal{Q}$-semigroup is a positively ordered monoid satisfying axioms (O1) and (O4) from Subsection 2.1, together with an additive auxiliary relation $\prec$. A $\mathcal{Q}$-morphism is a $\mathrm{PoM}$-morphism that preserves the auxiliary relation and suprema of increasing sequences. Given $\mathcal{Q}$-semigroups $S$ and $T$, the set of all $\mathcal{Q}$-morphisms from $S$ to $T$ is denoted by $\mathcal{Q}(S,T)$.

\thepgrCt**.**

Given a $\mathrm{Cu}$-semigroup $S$, then $S$ together with the relation $\ll$ is a $\mathcal{Q}$-semigroup. Moreover, given two $\mathrm{Cu}$-semigroups $S$ and $T$, a map $\varphi\colon S\to T$ is a $\mathrm{Cu}$-morphism if and only if it is a $\mathcal{Q}$-morphism when considered as a map from $(S,\ll)$ to $(T,\ll)$. We obtain a functor $\iota\colon\mathrm{Cu}\to\mathcal{Q}$ that sends a $\mathrm{Cu}$-semigroup $S$ to the $\mathcal{Q}$-semigroup $(S,\ll)$, and embeds $\mathrm{Cu}$ as a full subcategory of $\mathcal{Q}$.

\thepgrCt**.**

Let $S=(S,\prec)$ be a $\mathcal{Q}$-semigroup. A path in $S$ is an order-preserving map $f\colon(-\infty,0]\to S$ such that $f(t)=\sup_{t^{\prime}<t}f(t^{\prime})$ for all $t\in(-\infty,0]$, and such that $f(t^{\prime})\prec f(t)$ whenever $t^{\prime}<t$. We denote the set of paths in $S$ by $\mathcal{P}(S)$.

For $f,g\in\mathcal{P}(S)$ we define $f+g$ by $(f+g)(t)=f(t)+g(t)$. Together with the constant zero path, this gives $\mathcal{P}(S)$ the structure of a commutative monoid.

Given $f,g\in\mathcal{P}(S)$, we write $f\precsim g$ if, for every $t<0$, there is $t^{\prime}<0$ such that $f(t)\prec g(t^{\prime})$. Set $f\sim g$ if $f\precsim g$ and $g\precsim f$. It follows from [APT17, Lemma 3.4] that the relation $\precsim$ is reflexive, transitive, and compatible with the addition of paths. We set

[TABLE]

Given $f\in\mathcal{P}(S)$, its equivalence class in $\tau(S)$ is denoted by $[f]$. Equip $\tau(S)$ with an addition and order by setting $[f]+[g]:=[f+g]$ and $[f]\leq[g]$ provided $f\precsim g$. It is proved in [APT17, Theorem 3.15] that $\tau(S)$ is a $\mathrm{Cu}$-semigroup.

We define $\varphi_{S}\colon\tau(S)\to S$ by $\varphi_{S}([f]):=f(0)$, for $f\in\mathcal{P}(S)$. One verifies that $\varphi_{S}$ is a well-defined $\mathcal{Q}$-morphism. Given a path $f\in\mathcal{P}(S)$, we think of $f(0)$ as the endpoint of $f$. Accordingly, we call $\varphi_{S}$ the endpoint map.

The construction just outlined will be referred to as the $\tau$-construction. Given a $\mathcal{Q}$-morphism $\varphi\colon S\to T$, we define $\tau(\varphi)\colon\tau(S)\to\tau(T)$ by $\tau(\varphi)([f]):=[\varphi\circ f]$, for $f\in\mathcal{P}(S)$. One can show that $\tau(\varphi)$ is a $\mathrm{Cu}$-morphism. This defines a covariant functor $\tau\colon\mathcal{Q}\to\mathrm{Cu}$.

Theorem \thethmCt ([APT17, Theorem 4.12]).

The category $\mathrm{Cu}$ is a full, coreflective subcategory of $\mathcal{Q}$. The functor $\tau\colon\mathcal{Q}\to\mathrm{Cu}$ is a right adjoint to the inclusion functor $\iota\colon\mathrm{Cu}\to\mathcal{Q}$. Moreover, given a $\mathcal{Q}$-semigroup $S$, the endpoint map $\varphi_{S}\colon\tau(S)\to S$ is a universal $\mathcal{Q}$-morphism: For every $\mathrm{Cu}$-semigroup $T$, there is a natural bijection

[TABLE]

implemented by sending a $\mathrm{Cu}$-morphism $\psi\colon T\to\tau(S)$ to $\varphi_{S}\circ\psi$.

3. Bicompleteness of $\mathrm{Cu}$

In this section we show that the category $\mathrm{Cu}$ is bicomplete, that is, complete and cocomplete. To achieve this, we use that $\mathrm{Cu}$ is a full, coreflective subcategory of $\mathcal{Q}$ and also a full, reflective subcategory of $\mathrm{W}$; see Theorems 2.3 and 2.2.

We begin by briefly recalling the notions of small (co)limits and (co)completeness in arbitrary categories; see, for example, [Bor94, Section 2.6].

\thepgrCtSmall limits and colimits.

Let $I$ be a small category, and let $F\colon I\to\mathcal{C}$ be a functor to a category $\mathcal{C}$. Recall that a cone to $F$ is a pair $(L,\varphi)$, where $L$ is an object in $\mathcal{C}$ and $\varphi=(\varphi_{i})_{i\in I}$ is a collection of $\mathcal{C}$-morphisms $\varphi_{i}\colon L\to F(i)$ with the property that, whenever $f\colon i\to j$ is a morphism in $I$, we have $F(f)\circ\varphi_{i}=\varphi_{j}$. This means that the left diagram below commutes.

By definition, a limit of $F$ is a cone $(L,\varphi)$ that is universal in the sense that every cone $(L^{\prime},\varphi^{\prime})$ factors through $(L,\varphi)$, that is, there exists a unique $\mathcal{C}$-morphism $\alpha\colon L^{\prime}\to L$ such that $\varphi_{i}^{\prime}=\varphi_{i}\circ\alpha$ for every $i$. This means that the right diagram below commutes.

[TABLE]

If a limit of $F$ exists then it is unique up to natural isomorphism and we shall denote it by $(\mathcal{C}\text{-}\!\varprojlim F,\pi)$ or just by $\mathcal{C}\text{-}\!\varprojlim F$.

The notion of cocone to a functor $F\colon I\to\mathcal{C}$ is defined dually, as a pair $(M,\psi)$ with $M$ an object of $\mathcal{C}$ and $\psi=(\psi_{i})_{i\in I}$ a collection of $\mathcal{C}$-morphisms $\psi_{i}\colon F(i)\to M$ with dual compatibility relations. A colimit of $F$ is then defined as a universal cocone. If a colimit of $F$ exists then it is unique up to natural isomorphism and we shall denote it by $(\mathcal{C}\text{-}\varinjlim F,\sigma)$, or simply by $\mathcal{C}\text{-}\varinjlim F$.

The category $\mathcal{C}$ is termed complete (cocomplete) if all small limits (colimits) exist, that is, if every functor from a small category to $\mathcal{C}$ has a limit (colimit).

\thepgrCt**.**

Let $F\colon I\to\mathcal{C}$ be a functor from a small category $I$ to a category $\mathcal{C}$. Recall that, if $I$ has no morphisms (except the identities at each object), then $F$ corresponds to a selection of a family $(X_{i})_{i\in I}$ of objects in $\mathcal{C}$. In this case, a limit of $F$ is called a product of the family $(X_{i})_{i\in I}$ and is denoted by $\prod_{i\in I}X_{i}$. Dually we define a coproduct which we denote by $\coprod_{\in I}X_{i}$.

In case $I$ is an upward directed, partially ordered set, then the functor $F$ corresponds to a family $(X_{i})_{i\in I}$ of objects in $\mathcal{C}$ together with compatible $\mathcal{C}$-morphisms $\varphi_{j,i}\colon X_{i}\to X_{j}$ whenever $i\leq j$. A colimit of $F$ is called an inductive limit or direct limit. Inverse limits are defined similarly.

Finally, if $I=\{0,1\}$ is the category with two objects and two morphisms $0\to 1$, then $F\colon I\to\mathcal{C}$ corresponds to two $\mathcal{C}$-morphisms $\varphi,\psi\colon X\to Y$. A colimit of $F$ is then called a coequalizer. Thus, a coequalizer for $(\varphi,\psi)$ is a universal pair $(C,\eta)$, where $C$ is an object of $\mathcal{C}$ and $\eta\colon Y\to C$ is a $\mathcal{C}$-morphism such that $\eta\circ\varphi=\eta\circ\psi$.

3.1. Completeness

In this subsection we prove that $\mathrm{Cu}$ is a complete category; see Subsection 3.1. In particular, $\mathrm{Cu}$ has products over infinite index sets. We will prove in the sequel that the scaled Cuntz semigroup functor preserves (ultra)products of $C^{*}$-algebras; see Section 5 and Subsection 7.1.

\thepgrCt**.**

Let $I$ be a set. Let $(S_{i})_{i\in I}$ be a family of positively ordered monoids. Define

[TABLE]

Equipped with componentwise order and addition, $\mathrm{PoM}\text{-}\!\prod_{i\in I}S_{i}$ is a positively ordered monoid. Given $i\in I$, we let $\pi_{i}\colon\mathrm{PoM}\text{-}\!\prod_{i}S_{i}\to S_{i}$ be the projection map given by \pi_{i}\big{(}(s_{j})_{j\in I}\big{)}:=s_{i}. It is easy to verify that $\left(\mathrm{PoM}\text{-}\!\prod_{i\in I}S_{i},(\pi_{i})_{i}\right)$ is the product of the family $(S_{i})_{i}$ in $\mathrm{PoM}$.

Next, let $I$ be a small category and let $F\colon I\to\mathrm{PoM}$ be a functor. Set

[TABLE]

It is straightforward to verify that $0\in S$ and that $S$ is closed under addition in $\mathrm{PoM}\text{-}\!\prod_{i}F(i)$. Thus, $S$ inherits the structure of a positively ordered monoid.

For each $i\in I$, the projection map $\pi_{i}\colon\mathrm{PoM}\text{-}\!\prod_{j}F(j)\to F(i)$ restricts to a $\mathrm{PoM}$-morphism $\pi_{i}\colon S\to F(i)$. It is straightforward to verify that $(S,\pi)$, where $\pi=(\pi_{i})_{i\in I}$, is the limit of $F$ in $\mathrm{PoM}$. (Observe that $\mathrm{PoM}\text{-}\!\varprojlim F$ agrees, as a set, with the limit of the functor $F$ taking as target category the category of sets.)

Lemma 3.1**.**

Let $I$ be a small category, let $F\colon I\to\mathrm{PoM}$ be a functor, and let $(S,\pi)$ be the limit of $F$ in $\mathrm{PoM}$. Assume that $F(i)$ satisfies (O1) and (O4), for each $i\in I$, and assume that $F(f)\colon F(i)\to F(j)$ preserves suprema of increasing sequences for each morphism $f\colon i\to j$ in $I$. Then $S$ satisfies (O1) and (O4), and for each $i\in I$ the projection map $\pi_{i}\colon S\to F(i)$ preserves suprema of increasing sequences.

Proof.

We may assume that $S$ is defined as in (3.1). To verify (O1) for $S$, let $(s^{(n)})_{n\in\mathbb{N}}$ be an increasing sequence in $S$. For each $n$, we have $s^{(n)}=(s_{i}^{(n)})_{i\in I}$. Since the order in $S$ is defined componentwise and $F(i)$ satisfies (O1) for each $i$, we may set $s_{i}:=\sup_{n}s_{i}^{(n)}$. To verify that $s:=(s_{i})_{i}$ belongs to $S$, let $f\colon i\to j$ be a morphism in $I$. Since $F(f)\colon F(i)\to F(j)$ preserves suprema of increasing sequences, we deduce that

[TABLE]

Thus, $s$ belongs to $S$, and it is easy to verify that $s$ is the supremum of $(s^{(n)})_{n}$ in $S$. This shows that $S$ satisfies (O1).

The first part of the proof shows immediately that the projection maps $\pi_{i}$ preserve suprema of increasing sequences. Moreover, since addition in $S$ is also taken componentwise, we obtain that $S$ satisfies (O4). ∎

Definition \thedfnCt.

Let $I$ be a small category, and let $F\colon I\to\mathcal{Q}$ be a functor. Let $(S,\pi)$ be the limit of $F$ in $\mathrm{PoM}$ with $S$ defined as in (3.1). Define a binary relation $\prec_{\mathrm{pw}}$ on $S$ by setting $s\prec_{\mathrm{pw}}t$ for $s=(s_{i})_{i\in I}$ and $t=(t_{i})_{i\in I}$ in $S$ if and only if $s_{i}\prec t_{i}$ in $F(i)$ for each $i\in I$.

Proposition \theprpCt.

The category $\mathcal{Q}$ is complete. More precisely, given a small category $I$, and a functor $F\colon I\to\mathcal{Q}$, let $(S,(\pi_{i})_{i})$ be the limit of $F$ in $\mathrm{PoM}$, with $S$ defined as in (3.1). Then $(S,\prec_{\mathrm{pw}})$ is a $\mathcal{Q}$-semigroup and $\pi_{i}\colon S\to F(i)$ is a $\mathcal{Q}$-morphism for each $i\in I$. Moreover, $(S,\prec_{\mathrm{pw}})$ with $(\pi_{i})_{i}$ is a limit of $F$ in $\mathcal{Q}$.

Proof.

By Lemma 3.1, $S$ satisfies (O1) and (O4). It is straightforward to check that $\prec_{\mathrm{pw}}$ defines an additive auxiliary relation on $S$. Thus, $(S,\prec_{\mathrm{pw}})$ is a $\mathcal{Q}$-semigroup. Given $i\in I$, the projection map $\pi_{i}\colon S\to F(i)$ clearly preserves the auxiliary relation. By Lemma 3.1, $\pi_{i}$ is also a $\mathrm{PoM}$-morphism that preserves suprema of increasing sequences, and consequently $\pi_{i}$ is a $\mathcal{Q}$-morphism. To simplify, we denote the $\mathcal{Q}$-semigroup $(S,\prec_{\mathrm{pw}})$ by $S$. It follows that $(S,\pi)$ is a cone to $F$.

To show that $(S,\pi)$ is a limit of $F$ in $\mathcal{Q}$, let $(T,\sigma)$ be a cone to $F$. By the universal property of the limit in $\mathrm{PoM}$, there is a (unique) $\mathrm{PoM}$-morphism $\sigma\colon T\to S$ such that $\sigma_{i}=\pi_{i}\circ\sigma$ for all $i\in I$. Note that, $\sigma(t)=(\sigma_{i}(t))_{i\in I}$, for $t\in T$. We have to show that $\sigma$ is a $\mathcal{Q}$-morphism.

To verify that $\sigma$ preserves the auxiliary relation, let $t^{\prime},t\in T$ satisfy $t^{\prime}\prec t$. Since $\sigma_{i}$ preserves the auxiliary relation, we obtain that $\sigma_{i}(t^{\prime})\prec\sigma_{i}(t)$, for each $i\in I$. By definition, we then have $\sigma(t^{\prime})\prec_{\mathrm{pw}}\sigma(t)$ in $S$, as desired.

To show that $\sigma$ preserves suprema of increasing sequences, let $(t_{n})_{n}$ be an increasing sequence in $T$. Set $t=\sup_{n}t_{n}$. Since $\sigma_{i}$ preserves suprema of increasing sequences, we have $\sigma_{i}(t)=\sup_{n}\sigma_{i}(t_{n})$ for each $i\in I$. We deduce that

[TABLE]

Remark \thermkCt.

It follows from Subsection 3.1 that $\mathcal{Q}$ has products. The product in $\mathcal{Q}$ of a family $(S_{i})_{i\in I}$ of $\mathcal{Q}$-semigroups is precisely \mathcal{Q}\text{-}\!\prod_{i}S_{i}=\big{(}\mathrm{PoM}\text{-}\!\prod_{i}S_{i},\prec_{\mathrm{pw}}\big{)}

Theorem \thethmCt.

The category $\mathrm{Cu}$ is complete. More precisely, given a small category $I$ and a functor $F\colon I\to\mathrm{Cu}$, let $(S,(\pi_{i})_{i})$ be the limit of $F$ in $\mathcal{Q}$. Let $\tau(S)$ be the $\tau$-completion of $S$. Setting $S_{i}:=\tau(F(i))$ and identifying it with $F(i)$, we set $\psi_{i}:=\tau(\pi_{i})\colon\tau(S)\to\tau(S_{i})\cong S_{i}$. Then $\tau(S)$ together with $(\psi_{i})_{i}$ is the limit of $F$ in $\mathrm{Cu}$, that is:

[TABLE]

Proof.

By Subsection 2.3, $\mathrm{Cu}$ is a coreflective subcategory of $\mathcal{Q}$, which in turn is complete by Subsection 3.1. It follows that $\mathrm{Cu}$ is complete, since completeness passes to coreflective subcategories; see, for example, [Bor94, Proposition 3.5.3]. ∎

Corollary \thecorCt.

The category $\mathrm{Cu}$ has arbitrary products, arbitrary inverse limits, and finite pullbacks. In particular:

1. (1)

Let $(S_{i})_{i\in I}$ be a family of $\mathrm{Cu}$-semigroups. Then

[TABLE] 2. (2)

Let $((S_{i})_{i\in I},(\varphi_{ij})_{i,j\in I,i\geq j})$ be an inverse system of $\mathrm{Cu}$-semigroups. Then

[TABLE]

The proof of the following result is straightforward, using that the $\tau$-construction applied to $\mathrm{Cu}$-semigroups is vacuous; see [APT17, Proposition 4.10].

Proposition \theprpCt.

Let $(S_{j})_{j\in J}$ be a finite family of $\mathrm{Cu}$-semigroups. Then $\prod_{j\in J}S_{j}$ is naturally isomorphic to $\mathrm{PoM}\text{-}\!\prod_{j}S_{j}$, the (set-theoretic) product of the $S_{j}$ equipped with componentwise order and addition.

3.2. Cocompleteness

In this subsection we prove that $\mathrm{Cu}$ is a cocomplete category; see Subsection 3.2. In particular, it has coproducts and pushouts.

\thepgrCt**.**

Let $\mathcal{C}$ be a category that has finite coproducts and inductive limits. Let $(X_{j})_{j\in J}$ be a family of objects in $\mathcal{C}$. Denote $F\Subset J$ to mean that $F$ is a finite subset of $J$. The collection of such subsets is upward directed with respect to inclusion, and we have

[TABLE]

Thus, $\mathcal{C}$ has coproducts; see, for example, [Mac71, Theorem IX.1.1, p.212].

Now assume that $\mathcal{C}$ also has finite products and a zero object. Using universality we obtain, for $F\Subset J$, a natural morphism $\coprod_{j\in F}X_{j}\to\prod_{j\in F}X_{j}$. Given $F\subseteq G\Subset J$, the existence of a zero object allows us to construct a canonical morphism $\prod_{j\in F}X_{j}\to\prod_{j\in G}X_{j}$. In this way, one obtains an inductive system. Its inductive limit is termed the direct sum of the family $(X_{j})_{j\in J}$ and is denoted by $\bigoplus_{j\in J}X_{j}$. It follows from our discussion that there is a natural map

[TABLE]

In the particular case that finite products and finite coproducts are isomorphic, it follows that this morphism is in fact an isomorphism and hence we may identify the coproduct of a family of objects in the category with their direct sum. Therefore, it should not be confusing that we use the symbol ‘$\bigoplus$’ to denote both. As we shall see below, this is the case for the category $\mathrm{Cu}$.

Next, suppose that $\mathcal{C}$ has arbitrary products. Since there are maps $\bigoplus_{j\in J}X_{j}\to X_{i}$ for each $i\in J$, we obtain by the universal property of the product a natural map $\bigoplus_{j\in J}X_{j}\to\prod_{j\in J}X_{j}$ and a commutative diagram

[TABLE]

Now, let $(S_{j})_{j\in J}$ be a finite family of $\mathrm{W}$-semigroups. Let $\bigoplus_{j}S_{j}$, together with maps $\psi_{i}\colon S_{i}\to\bigoplus_{j\in J}S_{j}$ for $i\in J$, be the coproduct of the underlying abelian monoids. Note that $\bigoplus_{j}S_{j}$ is a $\mathrm{W}$-semigroup when endowed with the componentwise auxiliary relation. Moreover, this turns each $\psi_{i}$ into a $\mathrm{W}$-morphism. It follows easily that this is a coproduct of $(S_{j})_{j\in J}$ in the category $\mathrm{W}$. It is straightforward to verify that it also agrees with the finite product of $(S_{j})_{j\in J}$ in $\mathrm{W}$. Thus, according to the above discussion, we shall denote this coproduct by $\mathrm{W}\text{-}\!\bigoplus_{j}S_{j}$.

Lemma 3.2**.**

The category $\mathrm{W}$ has coproducts.

Proof.

By Subsection 3.2, $\mathrm{W}$ has finite coproducts. An argument analogous to [APT18, Theorem 2.1.10, p.15] shows that $\mathrm{W}$ has inductive limits. This implies that $\mathrm{W}$ has coproducts, as discussed in Subsection 3.2. ∎

Lemma 3.3**.**

Let $S$, $T$ be $\mathrm{W}$-semigroups, and let $\varphi,\psi\colon S\to T$ be $\mathrm{W}$-morphisms. For $x,y\in T$, set $x\sim_{0}y$ provided there are $z\in T$, and $a,b\in S$ such that

[TABLE]

Let $\sim$ be the transitive closure of the relation $\sim_{0}$. Then $\sim$ is a congruence on $T$.

For $x,y\in T$, set $x\precsim_{0}y$ provided there are $a,b\in T$ such that

[TABLE]

Let $\precsim$ be the transitive closure of $\precsim_{0}$. Then $\precsim$ is a transitive, additive relation.

Given $u,x,y\in T$ satisfying $u\precsim x\sim y$ there exists $y^{\prime}\in T$ such that $u\precsim y^{\prime}\prec y$.

Proof.

It is straightforward to check that $\sim_{0}$ is additive. It follows that $\sim$ is a congruence, and that $\precsim$ is additive and transitive.

Claim 1: Let $u,x,y\in T$ satisfy $u\prec x\sim_{0}y$. Then there exists $y^{\prime}\in T$ such that $u\precsim y^{\prime}\prec y$. To prove the claim, choose $z\in T$ and $a,b\in S$ such that

[TABLE]

Apply (W4) to choose $z^{\prime},z^{\prime\prime},v,w\in T$ such that

[TABLE]

Now, as $\varphi$ and $\psi$ are continuous, there are $a^{\prime},b^{\prime}\in S$ such that $a^{\prime}\prec a$, $b^{\prime}\prec b$, $v\prec\varphi(a^{\prime})$, and $w\prec\psi(b^{\prime})$. Set $x^{\prime}:=z^{\prime}+\varphi(b^{\prime})+\psi(c^{\prime})$, and $y^{\prime}:=z^{\prime}+\varphi(c^{\prime})+\psi(b^{\prime})$. Then

[TABLE]

We have $x^{\prime}\sim_{0}y^{\prime}$, and therefore $u\precsim y^{\prime}$. Further, we have $y^{\prime}\prec y$ since both $\varphi$ and $\psi$ preserve the auxiliary relation.

Claim 2: Let $u,x,y\in T$ satisfy $u\prec x\sim y$. Then there exists $y^{\prime}\in T$ such that $u\precsim y^{\prime}\prec y$. To prove the claim, we use that $\sim$ is the transitive closure of $\sim_{0}$ to choose $n\in\mathbb{N}$ and elements $x_{0},\dots,x_{n}\in T$ such that

[TABLE]

Applying claim 1 to $u\prec x_{0}\sim_{0}x_{1}$, we obtain $x_{1}^{\prime}\in T$ with $x_{0}\precsim x_{1}^{\prime}\prec x_{1}$. Applying claim 1 repeatedly, we obtain $x_{k}^{\prime}$ for $k=1,\ldots,n$ with $x_{k-1}^{\prime}\precsim x_{k}^{\prime}\prec x_{k}$. Then $u\precsim x_{n}^{\prime}\prec x_{n}$, which shows that $y^{\prime}:=x_{n}^{\prime}$ has the desired properties.

Claim 3: Let $u,x,y\in T$ satisfy $u\precsim_{0}x\sim y$. Then there exists $y^{\prime}\in T$ such that $u\precsim y^{\prime}\prec y$. To prove the claim, choose $a,b\in T$ with $u\sim a\prec b\sim x$. Then $a\prec b\sim y$. Applying claim 2, we obtain $y^{\prime}$ such that $a\precsim y^{\prime}\prec y$. It follows that $u\precsim y^{\prime}\prec y$, as desired.

Finally, to prove the last statement of the lemma, let $u,x,y\in T$ satisfy $u\precsim x\sim y$. Since $\precsim$ is the transitive closure of $\precsim_{0}$, we can choose $x^{\prime}\in T$ such that

[TABLE]

By claim 3, we obtain $y^{\prime}\in T$ with $x^{\prime}\precsim y^{\prime}\prec y$. Then $u\precsim y^{\prime}\prec y$, as desired. ∎

Lemma 3.4**.**

The category $\mathrm{W}$ has coequalizers.

Proof.

Let $S$ and $T$ be $\mathrm{W}$-semigroups, and let $\varphi,\psi\colon S\to T$ be $\mathrm{W}$-morphisms. For ease of notation, we shall denote the auxiliary relations of both $S$ and $T$ by $\prec$. We have to construct a $\mathrm{W}$-semigroup $L$ and a universal $\mathrm{W}$-morphism $\eta\colon T\to L$ such that $\eta\circ\varphi=\eta\circ\psi$.

Let $\sim$ be the congruence on $T$ defined as in Lemma 3.3. Notice that, with this definition, we have $\varphi(d)\sim\psi(d)$ for any $d\in S$. Set $L:=T/\sim$, and let $\eta\colon T\to L$ denote the quotient map. Given $x\in T$, we let $[x]$ denote the class of $x$ in $L$. Thus, we have $\eta(x)=[x]$, for $x\in T$.

Let $\precsim$ be the transitive relation on $T$ defined as in Lemma 3.3. This induces a binary relation $\prec$ on $L$ by setting $[x]\prec[y]$ if and only if $x\precsim y$, for $x,y\in T$. It follows from Lemma 3.3 that $\prec$ is a well-defined, transitive relation on $L$.

We claim that $(L,\prec)$ is a $\mathrm{W}$-semigroup. In order to establish this claim, we only need to verify axioms (W1) and (W4), as (W3) is trivially satisfied.

To verify (W1), let $x\in T$. Using that $T$ satisfies (W1), we can choose a $\prec$-increasing sequence $(x_{n})_{n}$ that is cofinal in $x^{\prec}:=\{x^{\prime}\in T:x^{\prime}\prec x\}$. It is clear that $x_{n}\prec x_{n+1}$ implies $[x_{n}]\prec[x_{n+1}]$. Thus, $([x_{n}])_{n}$ is a $\prec$-increasing sequence in $[x]^{\prec}$. To show that this sequence is cofinal in $[x]^{\prec}$, let $a\in T$ satisfy $[a]\prec[x]$. By definition, we have $a\precsim x$. By Lemma 3.3, there exists $x^{\prime}\in T$ with $a\precsim x^{\prime}\prec x$. Choose $n$ such that $x^{\prime}\prec x_{n}$. Then $[a]\prec[x_{n}]$, as desired.

To verify (W4), let $a,x,y\in T$ satisfy $[a]\prec[x]+[y]=[x+y]$. By definition, we have $a\precsim x+y$. By Lemma 3.3, there exists $u\in T$ with $a\precsim u\prec x+y$. Using that $T$ satisfies (W4), we can choose $x^{\prime},y^{\prime}\in T$ such that $u\prec x^{\prime}+y^{\prime}$, $x^{\prime}\prec x$, and $y^{\prime}\prec y$. Then the element $[u]$ in $L$ satisfies $[a]\prec[u]$, $[u]\prec[x^{\prime}]+[y^{\prime}]$, $[x^{\prime}]\prec[x]$, and $[y^{\prime}]\prec[y]$, as desired.

We have verified that $L$ is a $\mathrm{W}$-semigroup. Moreover, the map $\eta\colon T\to L$ is a monoid morphism and preserves the auxiliary relation. To verify that $\eta$ is continuous, let $a,b\in T$ satisfy $[a]\prec\eta(b)$. Then $a\precsim b$. By Lemma 3.3, we can choose $b^{\prime}\in T$ such that $a\precsim b^{\prime}\prec b$. Then $b^{\prime}$ has the desired properties. It is clear by the construction that $\eta\circ\varphi=\eta\circ\psi$.

We now claim that $(L,\eta)$ is a coequalizer of the pair $(\varphi,\psi)$. Thus, suppose that $R$ is a $\mathrm{W}$-semigroup and $h\colon T\to R$ is a $\mathrm{W}$-morphism such that $h\circ\varphi=h\circ\psi$.

Define $\tilde{h}\colon L\to R$ by $\tilde{h}([x]):=h(x)$, for $x\in T$. If we prove that $\tilde{h}$ is $\mathrm{W}$-morphism, then clearly it will be be the unique $\mathrm{W}$-morphism satisfying $\tilde{h}\circ\eta=h$.

In order to see that $\tilde{h}$ is well defined, let $x,y\in T$ satisfy $[x]=[y]$, that is, $x\sim y$. We need to show that $h(x)=h(y)$. Since $\sim$ is the transitive closure of $\sim_{0}$, we may assume that $x\sim_{0}y$. By definition, we can choose $a,b\in S$ and $z\in T$ such that

[TABLE]

Using at the second step that $h\circ\varphi=h\circ\psi$, we obtain

[TABLE]

Using that $h$ is additive, it follows that $\tilde{h}$ is additive. To show that $\tilde{h}$ preserves the auxiliary relation, let $x,y\in T$ satisfy $[x]\prec[y]$. Then, $x\precsim y$. Since $\precsim$ is the transitive closure of $\precsim_{0}$, we may assume that $x\precsim_{0}y$. By definition, we can choose $a,b\in T$ such that

[TABLE]

Using at the third step that $h$ preserves the auxiliary relation, we deduce

[TABLE]

Finally, to show that $\tilde{h}$ is continuous, let $x\in T$ and $z\in R$ satisfy $z\prec\tilde{h}([x])$. This means that $z\prec h(x)$, and since $h$ is continuous, there is $x^{\prime}\in T$ with $x^{\prime}\prec y$ and $z\leq h(x^{\prime})$. We have $[x^{\prime}]\prec[x]$ in $L$, and clearly $z\leq\tilde{h}([x^{\prime}])$ in $R$, which shows that $[x^{\prime}]$ has the desired properties. ∎

Proposition \theprpCt.

The category $\mathrm{W}$ is cocomplete.

Proof.

By Lemmas 3.2 and 3.4, $\mathrm{W}$ has small coproducts and coequalizers. This implies that $\mathrm{W}$ is cocomplete; see for example [Mac71, Corollary V.2.2, p.113] or (the dual statement of) [Bor94, Theorem 2.8.1]. ∎

Theorem \thethmCt.

The category $\mathrm{Cu}$ is cocomplete. More precisely, if $I$ is a small category and $F\colon I\to\mathrm{Cu}$ is a functor, then $\mathrm{Cu}\text{-}\!\varinjlim F=\gamma(\mathrm{W}\text{-}\!\varinjlim F)$.

Proof.

By Subsection 3.2, $\mathrm{W}$ is cocomplete. By Subsection 2.2, $\mathrm{Cu}$ is a full, reflective subcategory of $\mathrm{W}$, with reflector given by $\gamma\colon\mathrm{W}\to\mathrm{Cu}$. It follows that $\mathrm{Cu}$ is cocomplete, with the colimit of a functor $F$ given by the reflection of the colimit of $F$ in $\mathrm{W}$; see [Bor94, Proposition 3.5.3]. ∎

Corollary \thecorCt.

The category $\mathrm{Cu}$ has coproducts and pushouts. In particular, the coproduct of a family $(S_{j})_{j\in J}$ of $\mathrm{Cu}$-semigroups is \bigoplus_{j}S_{j}=\gamma\big{(}\mathrm{W}\text{-}\!\bigoplus_{j}S_{j}\big{)}.

The following result will be used in Section 6. Its proof, as the one of Subsection 3.1, is also straightforward, using that the $\gamma$-completion applied to $\mathrm{Cu}$-semigroups is vacuous.

Proposition \theprpCt.

Let $(S_{j})_{j\in J}$ be a finite family of $\mathrm{Cu}$-semigroups. Then $\bigoplus_{j}S_{j}$ is naturally isomorphic to $\prod_{j\in J}S_{j}$.

\thepgrCt**.**

It was shown in [APT17] that $\mathrm{Cu}$ is a closed, symmetric, monoidal category. The tensor product construction in the category $\mathrm{Cu}$ was introduced and developed in [APT18, Chapter 6]. It was proved in [APT18, Proposition 6.4.1] that the bifunctor $\_\,\otimes\_\,\colon\mathrm{Cu}\times\mathrm{Cu}\to\mathrm{Cu}$ is continuous in both variables. This result is recovered below. On the other hand, [APT17, Theorem 5.11] shows that, for any $\mathrm{Cu}$-semigroup $T$, the functor $\_\,\otimes_{\mathrm{Cu}}T$ has a right adjoint, denoted by $\llbracket T,\_\,\rrbracket$. The bifunctor $\llbracket\_\,,\_\,\rrbracket\colon\mathrm{Cu}\times\mathrm{Cu}\to\mathrm{Cu}$ is referred to as the internal-hom bifunctor.

Theorem \thethmCt.

Let $T$ be a $\mathrm{Cu}$-semigroup, let $((S_{i})_{i\in I},(\varphi_{j,i})_{i\leq j})$ be an inductive system in $\mathrm{Cu}$, and let $((R_{i})_{i\in I},(\psi_{j,i})_{i\leq j})$ be an inverse system in $\mathrm{Cu}$. Then:

1. (1)

We have an inductive system $((S_{i}\otimes T)_{i\in I},(\varphi_{j,i}\otimes\mathrm{id}_{T})_{i\leq j})$ in $\mathrm{Cu}$, and

[TABLE] 2. (2)

We have an inverse system $((\llbracket T,R_{i}\rrbracket)_{i\in I},((\psi_{j,i})_{*})_{i\leq j})$ in $\mathrm{Cu}$, and

[TABLE] 3. (3)

We have an inverse system $((\llbracket S_{i},T\rrbracket)_{i\in I},((\varphi_{j,i}^{*})_{i\leq j})$ in $\mathrm{Cu}$, and

[TABLE]

Proof.

The functors $F_{T}:=\_\,\otimes T$ and $G_{T}:=\llbracket T,\_\,\rrbracket$ form an adjoint pair by [APT17, Theorem 5.10]. Then (1) and (2) follow from general category theory.

Let us prove statement (3). Using the notation in [APT17, Paragraph 5.5], we have that $((\llbracket S_{i},T\rrbracket)_{i\in I},(\varphi_{i,j}^{*})_{i,j\in I})$ is an inverse system of $\mathrm{Cu}$-semigroups. Let $R$ be any other $\mathrm{Cu}$-semigroup. Since $\tau\colon\mathcal{Q}\to\mathrm{Cu}$ is a coreflection (Subsection 2.3), we obtain natural bijections:

[TABLE]

Given an inverse system of $\mathcal{Q}$-semigroups $(Y_{i})_{i}$, the set $\mathcal{Q}(R,\mathcal{Q}\text{-}\!\varprojlim_{i}Y_{i})$ is naturally bijective to the (set-theoretic) inverse limit $\varprojlim_{i}\mathcal{Q}(R,Y_{i})$. Similarly, given an inductive system of $\mathrm{W}$-semigroups $(X_{i})_{i}$, the set $\mathrm{W}(\mathrm{W}\text{-}\!\varinjlim X_{i},T)$ is naturally bijective to $\varprojlim\mathrm{W}(X_{i},T)$. Using these observations, and the adjunction between tensor product and internal-hom, the result follows. ∎

4. Scaled Cuntz semigroups

In this section, we introduce the category $\mathrm{Cu}_{\mathrm{sc}}$ of scaled $\mathrm{Cu}$-semigroups (see [PTWW14]), which we show to be complete and cocomplete.

Definition \thedfnCt.

Let $S$ be a $\mathrm{Cu}$-semigroup. A scale for $S$ is a downward hereditary subset $\Sigma\subseteq S$ that is closed under passing to suprema of increasing sequences in $S$, and that generates $S$ as an ideal. We call $(S,\Sigma)$ a scaled $\mathrm{Cu}$-semigroup.

Given scaled $\mathrm{Cu}$-semigroups $(S,\Sigma)$ and $(T,\Theta)$, a scaled $\mathrm{Cu}$-morphism is a $\mathrm{Cu}$-morphism $\alpha\colon S\to T$ satisfying $\alpha(\Sigma)\subseteq\Theta$. We let $\mathrm{Cu}_{\mathrm{sc}}$ denote the category of scaled $\mathrm{Cu}$-semigroups and scaled $\mathrm{Cu}$-morphisms.

\thepgrCt**.**

Let $A$ be a $C^{*}$-algebra. Then

[TABLE]

is a scale for $\operatorname{Cu}(A)$. We call $\mathrm{Cu}_{\mathrm{sc}}(A):=(\operatorname{Cu}(A),\Sigma_{A})$ the scaled Cuntz semigroup of $A$. Given a ∗-homomorphism $\varphi\colon A\to B$ between $C^{*}$-algebras, the induced $\mathrm{Cu}$-morphism $\operatorname{Cu}(\varphi)\colon\operatorname{Cu}(A)\to\operatorname{Cu}(B)$ (see Subsection 2.1) maps $\Sigma_{A}$ into $\Sigma_{B}$. Hence, we obtain a functor $C^{*}\to\mathrm{Cu}_{\mathrm{sc}}$. In Theorems 4, 5 and 7.1, we show that this functor preserves inductive limits and (ultra)products.

The following result is straightforward to prove. We omit the details.

Proposition \theprpCt.

Let $A$ be a $C^{*}$-algebra, let $J\subseteq A$ be an ideal, and let $\pi\colon A\to A/J$ denote the quotient map. Then

[TABLE]

\thepgrCt**.**

Given a $\mathrm{Cu}$-semigroup $S$, the whole of $S$ is a scale for $S$, and we say that $(S,S)$ is trivially scaled. A map between $\mathrm{Cu}$-semigroups is a $\mathrm{Cu}$-morphism if and only if it is a scaled $\mathrm{Cu}$-morphism when both $\mathrm{Cu}$-semigroups are equipped with trivial scales. Hence, $\mathrm{Cu}$ is isomorphic to the full subcategory of trivially scaled $\mathrm{Cu}$-semigroups in $\mathrm{Cu}_{\mathrm{sc}}$. In the other direction, we have the forgetful functor $\mathcal{F}\colon\mathrm{Cu}_{\mathrm{sc}}\to\mathrm{Cu}$, that maps $(S,\Sigma)$ to $S$. For every scaled $\mathrm{Cu}$-semigroup $(S,\Sigma)$ and every $\mathrm{Cu}$-semigroup $T$, there is a natural bijection

[TABLE]

showing that $\mathcal{F}$ is a left adjoint to the inclusion $\mathrm{Cu}\to\mathrm{Cu}_{\mathrm{sc}}$, and thus $\mathrm{Cu}$ is a full, reflective subcategory of $\mathrm{Cu}_{\mathrm{sc}}$.

\thepgrCt**.**

Let $I$ be a small category, and let $F\colon I\to\mathrm{Cu}_{\mathrm{sc}}$, $i\mapsto F(i)=(S_{i},\Sigma_{i})$ be a functor. Consider the underlying functor $I\to\mathrm{PoM}$, $i\mapsto S_{i}$, and let $(S,(\pi_{i})_{i})$ be the limit of $F$ in $\mathrm{PoM}$ (see Subsection 3.1) with $S$ defined as in (3.1). Set $\Sigma_{0}:=S\cap\prod_{i}\Sigma_{i}$. Note that

[TABLE]

Then $\Sigma_{0}$ is a downward hereditary subset of $S$ satisfying $\pi_{i}(\Sigma_{0})\subseteq\Sigma_{i}$ for each $i$.

The limit of the underlying functor $\mathcal{F}\circ F\colon I\to\mathrm{Cu}$, $i\mapsto S_{i}$, is given as $\tau(S,\ll_{\mathrm{pw}})$ together with maps $\psi_{i}:=\tau(\pi_{i})\colon\tau(S,\ll_{\mathrm{pw}})\to\tau(S_{i},\ll)\cong S_{i}$; see Subsection 3.1. Set

[TABLE]

Then $\Sigma$ is a downward hereditary subset of $\tau(S,\ll_{\mathrm{pw}})$ that is closed under passing to suprema of increasing sequences. Let $\langle\Sigma\rangle$ denote the ideal of $\tau(S,\ll_{\mathrm{pw}})$ generated by $\Sigma$. Then $(\langle\Sigma\rangle,\Sigma)$ is a scaled $\mathrm{Cu}$-semigroup. Moreover, for each $i\in I$ we have $\psi_{i}(\Sigma)\subseteq\Sigma_{i}$, which shows that $\psi_{i}\colon(\langle\Sigma\rangle,\Sigma)\to(S_{i},\Sigma_{i})$ is a scaled $\mathrm{Cu}$-morphism. One can show that this defines a limit for $F$ in $\mathrm{Cu}_{\mathrm{sc}}$. We omit the details.

Theorem \thethmCt.

The category $\mathrm{Cu}_{\mathrm{sc}}$ is bicomplete.

Proof.

It remains to show that $\mathrm{Cu}_{\mathrm{sc}}$ is cocomplete. Let $I$ be a small category, and let $F\colon I\to\mathrm{Cu}_{\mathrm{sc}}$, $i\mapsto F(i)=(S_{i},\Sigma_{i})$ be a functor. By Subsection 3.2, $\mathrm{Cu}$ is cocomplete, and so there is a $\mathrm{Cu}$-semigroup $S$ and $\mathrm{Cu}$-morphisms $\sigma_{i}\colon S_{i}\to S$ that are compatible with $F(f)\colon S_{i}\to S_{j}$ for each $f\colon i\to j$ in $I$, and such that $(S,(\sigma_{i})_{i})$ is universal with these properties. Let $\Sigma$ be the smallest subset of $S$ that contains $\bigcup_{i}\sigma_{i}(\Sigma_{i})$, and that is downward hereditary and closed under suprema of increasing sequences in $S$, that is,

[TABLE]

Then $(S,\Sigma)$ is a scaled $\mathrm{Cu}$-semigroup, and each $\sigma_{i}$ is a scaled $\mathrm{Cu}$-morphism. To show that this gives a limit for $F$ in $\mathrm{Cu}_{\mathrm{sc}}$, let $(T,\Theta)$ be a scaled $\mathrm{Cu}$-semigroup together with compatible scaled $\mathrm{Cu}$-morphisms $\tau_{i}\colon(S_{i},\Sigma_{i})\to(T,\Theta)$. Using the universal property of $(S,(\sigma_{i})_{i})$, we obtain a (unique) $\mathrm{Cu}$-morphism $\beta\colon S\to T$ such that $\tau_{i}=\beta\circ\sigma_{i}\colon S_{i}\to T$. Since each $\tau_{i}$ is a scaled $\mathrm{Cu}$-morphism, we have

[TABLE]

which implies that $\beta(\Sigma)\subseteq\Theta$, and thus $\beta$ is a scaled $\mathrm{Cu}$-morphism. ∎

Theorem \thethmCt.

The scaled Cuntz semigroup functor preserves inductive limits: Given an inductive system $((A_{j})_{j\in J},(\varphi_{k,j})_{j\leq k})$ of $C^{*}$-algebras, we have

[TABLE]

Proof.

Set $A:=\varinjlim_{j}A_{j}$ and let $\varphi_{\infty,j}\colon A_{j}\to A$ denote the maps to the inductive limit. They induce scaled $\mathrm{Cu}$-morphisms $\psi_{\infty,j}\colon\mathrm{Cu}_{\mathrm{sc}}(A_{j})\to\mathrm{Cu}_{\mathrm{sc}}(A)$, which in turn induce a scaled $\mathrm{Cu}$-morphism

[TABLE]

It follows from [APT18, Corollary 3.2.9, p.29] that $\psi$ is an order-isomorphism. Let $\Sigma$ denote the scale of $\varinjlim_{j}\mathrm{Cu}_{\mathrm{sc}}(A_{j})$. We have $\psi(\Sigma)\subseteq\Sigma_{A}$, and it remains to show that $\psi$ maps $\Sigma$ onto $\Sigma_{A}$. Let $x\in\Sigma_{A}$, and let $x^{\prime}\ll x$. It is enough to verify $x^{\prime}\in\psi(\Sigma)$. By definition of $\Sigma_{A}$, there exists $a\in A_{+}$ with $x^{\prime}\ll[a]$. Thus, there is $\varepsilon>0$ with $x^{\prime}\ll[(a-\varepsilon)_{+}]$. Choose $j$ and $b\in(A_{j})_{+}$ such that $\|a-\varphi_{\infty,j}(b)\|<\varepsilon$. Set $y:=[b]\in\Sigma_{A_{j}}$, and let $\tilde{y}$ be the image of $y$ in $\varinjlim_{j}\operatorname{Cu}(A_{j})$. Then $\tilde{y}\in\Sigma$, and

[TABLE]

It follows that $\psi^{-1}(x^{\prime})\in\Sigma$, and thus $x^{\prime}\in\psi(\Sigma)$. ∎

5. Cuntz semigroups of products

In this section, we show that the scaled Cuntz semigroup functor preserves products; see Section 5.

\thepgrCt**.**

Let $(A_{j})_{j\in J}$ be a family of $C^{*}$-algebras. For each $i\in J$, the natural projection $\pi_{i}\colon\prod_{j}A_{j}\to A_{i}$ induces a $\mathrm{Cu}$-morphism $\tilde{\pi}_{i}\colon\operatorname{Cu}(\prod_{j}A_{j})\to\operatorname{Cu}(A_{i})$. Then, by the universal property of the product, there is a unique $\mathrm{Cu}$-morphism

[TABLE]

such that $\tilde{\pi}_{i}=\sigma_{i}\circ\Phi$ for all $i\in I$, where $\sigma_{i}\colon\prod_{j}\operatorname{Cu}(A_{j})\to\operatorname{Cu}(A_{j})$ denotes the natural $\mathrm{Cu}$-morphism associated to the product in the category $\mathrm{Cu}$.

Let us describe the map $\Phi$ more concretely. Given a positive element $\underline{a}=(a_{j})_{j\in J}$ in $\prod_{j}A_{j}$, and $t\in(-\infty,0]$, set

[TABLE]

Then $t\mapsto\varphi_{t}(\underline{a})$ is a path in $(\mathrm{PoM}\text{-}\!\prod_{j}\operatorname{Cu}(A_{j}),\ll_{\mathrm{pw}})$, and we have

[TABLE]

Next, we need to deal with the stabilization that occurs in the construction of $\operatorname{Cu}(\prod_{j}A_{j})$. We note that even if each $A_{j}$ is stable, the product $\prod_{j}A_{j}$ is not stable (unless $J$ is finite); see Subsection 8.2. Nevertheless, we will see that $\prod_{j}A_{j}$ satisfies property (S) of Hjelmborg-Rørdam, which is enough for our purposes.

Definition \thedfnCt (Hjelmborg, Rørdam, [HR98]).

A $C^{*}$-algebra $A$ is said to have property (S) if for every $a\in A_{+}$ and every $\varepsilon>0$ there exist $b\in A_{+}$ and $x\in A$ such that $a=x^{*}x$, $b=xx^{*}$ and $\|ab\|\leq\varepsilon$.

Theorem \thethmCt ([HR98]).

Let $A$ be a $C^{*}$-algebra. If $A$ is stable, then it has property (S). The converse holds if $A$ is $\sigma$-unital (in particular, if $A$ is separable).

Lemma 5.1**.**

Property (S) passes to products.

Proof.

Let $(A_{j})_{j\in J}$ be a family of $C^{*}$-algebras with property (S). To verify that $A:=\prod_{j\in J}A_{j}$ has property (S), let $a=(a_{j})_{j}\in A_{+}$ and $\varepsilon>0$. Given $j\in J$, use that $A_{j}$ satisfies (S), to choose $b_{j}\in(A_{j})_{+}$ and $x_{j}\in A_{j}$ such that

[TABLE]

Set $b:=(b_{j})_{j}$ and $x:=(x_{j})_{j}$. For each $j$, we have $\|x_{j}\|=\|a_{j}\|^{1/2}\leq\|a\|^{1/2}$ and therefore $\|x\|=\sup_{i}\|x_{i}\|<\infty$. Similarly, we have $\|b\|<\infty$. Hence, $b$ and $x$ belong to $A$, and we have $a=x^{*}x$, $b=xx^{*}$ and $\|ab\|\leq\varepsilon$, as desired. ∎

We remark that it is straightforward to verify that property (S) passes to quotients, and hence to ultraproducts. The following result is well-known (although we couldn’t locate it in the literature). It can be proved using Blackadar’s technique of constructing separable sub-$C^{*}$-algebras with prescribed properties ([Bla06, Section II.8.5, p.176ff]), which is the $C^{*}$-algebraic version of the downward Löwenheim-Skolem theorem from model theory.

Lemma 5.2**.**

Let $A$ be a $C^{*}$-algebra with property (S), and let $A_{0}\subseteq A$ be a separable sub-$C^{*}$-algebra. Then there exists a separable sub-$C^{*}$-algebra $B\subseteq A$ with property (S) (and hence $B$ is stable by Section 5), and such that $A_{0}\subseteq B$.

Lemma 5.3**.**

Let $A$ be a $C^{*}$-algebra with property (S), and let $p\in\mathcal{K}$ be a rank one projection. Then the map $\iota\colon A\to A\otimes\mathcal{K}$, given by $\iota(a)=a\otimes p$, induces a natural bijection between Cuntz equivalence classes in $A_{+}$ and $(A\otimes\mathcal{K})_{+}$. Moreover, $a,b\in A_{+}$ satisfy $\iota(a)\precsim\iota(b)$ in $A\otimes\mathcal{K}$ if and only if $a\precsim b$ in $A$.

Proof.

In general, if $D\subseteq B$ is a hereditary sub-$C^{*}$-algebra, and given $x,y\in D_{+}$, then $x$ is Cuntz (sub)equivalent to $y$ in $D$ if and only if $x$ is Cuntz (sub)equivalent to $y$ in $B$; see e.g. [Thi17, Proposition 2.18]. Since $\iota$ identifies $A$ with a hereditary sub-$C^{*}$-algebra of $A\otimes\mathcal{K}$, it follows that the map $\tilde{\iota}\colon(A_{+})_{/\sim}\to\operatorname{Cu}(A)$, given by $\tilde{\iota}([a])=[a\otimes p]$, is a well-defined order-embedding.

To show that $\tilde{\iota}$ is surjective, let $a\in(A\otimes\mathcal{K})_{+}$. Choose a separable sub-$C^{*}$-algebra $A_{0}\subseteq A$ such that $a$ belongs to $A_{0}\otimes\mathcal{K}\subseteq A\otimes\mathcal{K}$. (For instance, let $(e_{kl})_{k,l\geq 1}$ denote matrix units for $\mathcal{K}$ such that $p=e_{11}$, and let $A_{0}$ be generated by $\iota^{-1}(e_{1k}ae_{l1})$ for $k,l\geq 1$.) Apply Lemma 5.2 to obtain a separable, stable sub-$C^{*}$-algebra $B\subseteq A$ with $A_{0}\subseteq B$. Note that $a$ belongs to $B\otimes\mathcal{K}$. Since $B$ is stable, there exists a unitary in $M(B\otimes\mathcal{K})$ such that $uxu^{*}\in B\otimes p$ for each $x\in B\otimes\mathcal{K}$. Then, $uau^{*}$ belongs to $B\otimes p$, and hence to the image of $\iota$. Further, $uau^{*}$ is Cuntz equivalent to $a$ in $M(B\otimes\mathcal{K})$, and consequently in the hereditary sub-$C^{*}$-algebra $B\otimes\mathcal{K}$, and therefore also in $A\otimes\mathcal{K}$. ∎

The following technical lemmas will be used to compute the Cuntz semigroups of product $C^{*}$-algebras.

Lemma 5.4**.**

Let $(A_{j})_{j\in J}$ be a family of $C^{*}$-algebras, and let $\underline{a}=(a_{j})_{j\in J}$ and $\underline{b}=(b_{j})_{j\in J}$ be positive elements in $\prod_{j}A_{j}$. Then the following are equivalent:

1. (1)

We have $\underline{a}\precsim\underline{b}$ in $\prod_{j}A_{j}$. 2. (2)

For every $\varepsilon>0$ there exists $\delta>0$ such that $(a_{j}-\varepsilon)_{+}\precsim(b_{j}-\delta)_{+}$ in $A_{j}$ for every $j\in J$. 3. (3)

We have $\Phi([\underline{a}])\leq\Phi([\underline{b}])$ in $\prod_{j}\operatorname{Cu}(A_{j})$.

Proof.

As in Section 5, set

[TABLE]

for each $t\in(-\infty,0]$. Then

[TABLE]

By definition of the order in $\prod_{j}\operatorname{Cu}(A_{j})=\tau(\mathrm{PoM}\text{-}\!\prod_{j}\operatorname{Cu}(A_{j}),\ll_{\mathrm{pw}})$, we have $\Phi([\underline{a}])\leq\Phi([\underline{b}])$ if and only if for every $t<0$ there exists $t^{\prime}<0$ such that $\varphi_{t}(\underline{a})\ll_{\mathrm{pw}}\varphi_{t^{\prime}}(\underline{b})$. Equivalently, for every $\varepsilon>0$ there exists $\delta>0$ such that $\varphi_{-\varepsilon}(\underline{a})\ll_{\mathrm{pw}}\varphi_{-\delta}(\underline{b})$. We deduce that (2) and (3) are equivalent.

It follows directly from Rørdam’s Lemma (Subsection 2.1) that (1) implies (2). Conversely, assume (2), and let us verify (1) with Rørdam’s Lemma. Let $\varepsilon>0$. We need to find $\underline{s}$ in $\prod_{j}A_{j}$ such that $\underline{s}\underline{s}^{*}=(\underline{a}-\varepsilon)_{+}$ and $\underline{s}^{*}\underline{s}\in\operatorname{Her}(\underline{b})$. By assumption, for each $j\in J$ there is $\delta>0$ such that $(a_{j}-\tfrac{\varepsilon}{2})_{+}\precsim(b_{j}-\delta)_{+}$ in $A_{j}$. Apply Rørdam’s Lemma for $(a_{j}-\tfrac{\varepsilon}{2})_{+}\precsim(b_{j}-\delta)_{+}$ and $\tfrac{\varepsilon}{2}$ to obtain $s_{j}\in A_{j}$ such that

[TABLE]

Observe that $\|s_{j}\|\leq\|a_{j}\|^{1/2}$ for all $j\in J$, and hence $(\|s_{j}\|)_{j}$ is bounded. Set $\underline{s}:=(s_{j})_{j\in J}$. Then $\underline{s}$ is an element in $\prod_{j}A_{j}$ satisfying $\underline{s}\underline{s}^{*}=(\underline{a}-\epsilon)_{+}$. Let $f_{\delta}\colon\mathbb{R}\to[0,1]$ be a continuous function satisfying $f(t)=0$ for $t\leq 0$ and $f(t)=1$ for $t\geq\delta$. Then $f_{\delta}(b_{j})s_{j}^{*}s_{j}f_{\delta}(b_{j})=s_{j}^{*}s_{j}$ for each $j$, and therefore $f_{\delta}(\underline{b})\underline{s}^{*}\underline{s}f_{\delta}(\underline{b})=\underline{s}^{*}\underline{s}$. This implies that $\underline{s}^{*}\underline{s}\in\operatorname{Her}(\underline{b})$, as desired. ∎

Lemma 5.5**.**

Let $A$ be a $C^{*}$-algebra, let $a\in A_{+}$, and let $x\in\operatorname{Cu}(A)$ with $x\ll[a]$. Then there exists a positive, contractive $b\in A$ such that $x\ll[(b-\tfrac{1}{2})_{+}]$ and $[b]=[a]$.

Proof.

Choose $\delta>0$ such that $x\ll[(a-\delta)_{+}]$. We may assume that $\delta<1/2$. Let $h\colon\mathbb{R}\to[0,1]$ be defined by

[TABLE]

Set $b:=h(a)$. Then $[a]=[b]$. Consider the functions $\mathbb{R}\to\mathbb{R}$, given by $t\mapsto(h(t)-\tfrac{1}{2})_{+}$ and $t\mapsto(t-\delta)_{+}$. They have the same support (namely $(\delta,\infty)$), which implies that $(h(a)-1/2)_{+}$ and $(a-\delta)_{+}$ are Cuntz equivalent (first in the commutative $C^{*}$-algebra $C^{*}(a)\cong C(\sigma(a))$, and thus also in $A$). Hence,

[TABLE]

Proposition \theprpCt.

Let $(A_{j})_{j\in J}$ be a family of stable $C^{*}$-algebras. Then, the map $\Phi\colon\operatorname{Cu}(\prod_{j}A_{j})\to\prod_{j}\operatorname{Cu}(A_{j})$ from Section 5 is an isomorphism.

Proof.

Set $A:=\prod_{j}A_{j}$. By Lemma 5.1, $A$ has property (S). Hence, by Lemma 5.3 there is a natural identification of $\operatorname{Cu}(A)$ with the Cuntz equivalence classes in $A_{+}$. Thus, it remains to show that $\Phi$ is an order-isomorphism when restricted to Cuntz classes of elements in $A_{+}$.

It follows from Lemma 5.4 that $\Phi$ is an order-embedding. It remains to show that $\Phi$ is surjective. Since $\Phi$ is an order-embedding, and since $\operatorname{Cu}(A)$ has suprema of increasing sequences that moreover are preserved by $\Phi$, it is enough to verify that the image of $\Phi$ is order-dense.

Let $x,y\in\prod_{j}\operatorname{Cu}(A_{j})$ satisfy $x\ll y$. We will find $\underline{b}\in A_{+}$ such that $x\ll\Phi([\underline{b}])\ll y$. Choose $\ll_{\mathrm{pw}}$-increasing paths $(\textbf{x}_{t})_{t\in(-\infty,0]}$ and $(\textbf{y}_{t})_{t\in(-\infty,0]}$ in $\mathrm{PoM}\text{-}\!\prod_{j}\operatorname{Cu}(A_{j})$ representing $x$ and $y$, respectively. It follows from [APT17, Lemma 3.16] that there exists $t<0$ such that $\textbf{x}_{0}\ll_{\mathrm{pw}}\textbf{y}_{t}$.

Choose elements $x_{0,j}\in\operatorname{Cu}(A_{j})$ such that $\textbf{x}_{0}=(x_{0,j})_{j}$, and choose positive elements $a_{j}\in A_{j}$ such that $\textbf{y}_{t}=([a_{j}])_{j}$. Then, for each $j\in J$, we have $x_{0,j}\ll[a_{j}]$. Applying Lemma 5.5, we obtain a positive, contractive $b_{j}\in A_{j}$ such that

[TABLE]

Set $\underline{b}:=(b_{j})_{j}$, which is a positive, contractive element in $\prod_{j}A_{j}$. We have

[TABLE]

and therefore

[TABLE]

This shows that $\underline{b}$ has the claimed properties. ∎

Remark \thermkCt.

More generally, Section 5 (and consequently Subsection 7.1) holds for every family of $C^{*}$-algebras that satisfy property (S). However, Section 5 shows that it does not hold for every family of $C^{*}$-algebras.

The proof of the following result is standard. We therefore omit it.

Lemma 5.6**.**

Let $A\subseteq B$ be a hereditary sub-$C^{*}$-algebra. Then the inclusion $\iota\colon A\to B$ induces an order-embedding $\operatorname{Cu}(\iota)\colon\operatorname{Cu}(A)\to\operatorname{Cu}(B)$ that identifies $\operatorname{Cu}(A)$ with an ideal of $\operatorname{Cu}(B)$.

\thepgrCt**.**

Let $(S_{j},\Sigma_{j})_{j\in J}$ be a family of scaled $\mathrm{Cu}$-semigroups. Let $(S,\Sigma)$ be their product in $\mathrm{Cu}_{\mathrm{sc}}$, which we also call their scaled product.

Let us describe $(S,\Sigma)$ more concretely. The set-theoretic product $\prod_{j}\Sigma_{j}$ is a downward hereditary subset of $\mathrm{PoM}\text{-}\!\prod_{j}S_{j}$. Then

[TABLE]

and $S$ is the ideal of $\prod_{j}S_{j}$ generated by $\Sigma$. Given $[(\textbf{x}_{t})_{t\leq 0}]\in\prod_{j}S_{j}$, we have $[(\textbf{x}_{t})_{t\leq 0}]\in S$ if and only if for every $t<0$ there exist $\sigma^{(1)},\ldots,\sigma^{(N)}\in\prod_{j}\Sigma_{j}$ such that $\textbf{x}_{t}\ll_{\mathrm{pw}}\sigma^{(1)}+\ldots+\sigma^{(N)}$.

Theorem \thethmCt.

The scaled Cuntz semigroup functor preserves products.

More concretely, let $(A_{j})_{j\in J}$ be a family of $C^{*}$-algebras, and let $(S,\Sigma)$ be the scaled product of $(\mathrm{Cu}_{\mathrm{sc}}(A_{j}))_{j}$ with $\Sigma\subseteq S\subseteq\prod_{j}\operatorname{Cu}(A_{j})$ as in Section 5. Then, the map $\Phi\colon\operatorname{Cu}(\prod_{j}A_{j})\to\prod_{j}\operatorname{Cu}(A_{j})$ from Section 5 is an order-embedding with image $S$ that moreover identifies the scale of $\operatorname{Cu}(\prod_{j\in J}A_{j})$ with $\Sigma$:

[TABLE]

Proof.

Let $p\in\mathcal{K}$ be a rank-one projection. For each $j$, let $\iota_{j}\colon A_{j}\to A_{j}\otimes\mathcal{K}$ be given by $\iota_{j}(a):=a\otimes p$. This induces a natural map $\iota\colon\prod_{j}A_{j}\to\prod_{j}(A_{j}\otimes\mathcal{K})$, which identifies $\prod_{j}A_{j}$ with a hereditary sub-$C^{*}$-algebra of $\prod_{j}(A_{j}\otimes\mathcal{K})$. By Lemma 5.6, $\operatorname{Cu}(\iota)$ is an order-embedding whose image is an ideal.

Let $\Phi^{\prime}$ be the natural map from Section 5 for the product $\prod_{j}(A_{j}\otimes\mathcal{K})$. For each $j$, we have a $\mathrm{Cu}$-morphism $\operatorname{Cu}(\iota_{j})\colon\operatorname{Cu}(A_{j})\to\operatorname{Cu}(A_{j}\otimes\mathcal{K})$, which induces a $\mathrm{Cu}$-morphism $\alpha\colon\prod_{j}\operatorname{Cu}(A_{j})\to\prod_{j}\operatorname{Cu}(A_{j}\otimes\mathcal{K})$. Then

[TABLE]

which means that the following diagram commutes:

[TABLE]

By Section 5, $\Phi^{\prime}$ is an order-isomorphism. Further, each $\operatorname{Cu}(\iota_{j})$ is an isomorphism, and consequently so is $\alpha$. It follows that $\Phi$ is an order-embedding whose image is an ideal. It remains to show that $\Phi$ maps $\Sigma_{\prod_{j}A_{j}}$ onto $\Sigma$.

First, let $x\in\Sigma_{\prod_{j}A_{j}}$. By definition, for every $x^{\prime}\ll x$, there exists a positive $\underline{a}\in\prod_{j}A_{j}$ with $x^{\prime}\leq[\underline{a}]$. Since $\Phi$ preserves suprema of increasing sequences, and since $\Sigma$ is downward hereditary, it is enough to verify that $\Phi([\underline{a}])\in\Sigma$ for every positive $\underline{a}\in\prod_{j}A_{j}$. But this follows directly from the description of $\Phi$ in Section 5.

Conversely, using that $\Phi$ is an order-embedding, and that $\Sigma_{\prod_{j}A_{j}}$ is closed under suprema of increasing sequences that moreover are preserved by $\Phi$, it is enough to verify that the image of $\Phi(\Sigma_{\prod_{j}A_{j}})$ is order-dense in $\Sigma$. Thus, let $y^{\prime},y\in\Sigma$ satisfy $y^{\prime}\ll y$. Choose an $\ll_{\mathrm{pw}}$-increasing path $(\textbf{y}_{t})_{t\in(-\infty,0]}$ such that $y=[(\textbf{y}_{t})_{t\leq 0}]$, and choose $y_{t,j}\in\operatorname{Cu}(A_{j})$ such that $\textbf{y}_{t}=(y_{t,j})_{j}$. Then choose $\varepsilon>0$ such that $y^{\prime}\leq[(\textbf{y}_{t-3\varepsilon})_{t\leq 0}]$. By definition of $\Sigma$, we have $\textbf{y}_{t}\in\mathrm{PoM}\text{-}\!\prod_{j}\Sigma_{A_{j}}$ for each $t<0$. In particular, $\textbf{y}_{-\varepsilon}\in\mathrm{PoM}\text{-}\!\prod_{j}\Sigma_{A_{j}}$. For each $j\in J$, we have $y_{-2\varepsilon,j}\ll y_{-\varepsilon,j}\in\Sigma_{A_{j}}$. By definition of $\Sigma_{A_{j}}$, we obtain $a_{j}\in(A_{j})_{+}$ such that $y_{-2\varepsilon,j}\leq[a_{j}]$.

Applying Lemma 5.5, we obtain a positive, contractive $b_{j}\in A_{j}$ such that

[TABLE]

Then $\underline{b}:=(b_{j})_{j}$ is a positive element in $\prod_{j}A_{j}$. Hence, the Cuntz class of $\underline{b}$ belongs to $\Sigma_{\prod_{j}A_{j}}$. As in the proof of Section 5, we obtain $y^{\prime}\leq\Phi([\underline{b}])\leq y$. ∎

In the case of finitely many $C^{*}$-algebras, the map $\Phi$ from Section 5 is easily seen to be surjective, whence we obtain the following (well-known) result. We note that the product of finitely many $\mathrm{Cu}$-semigroups is simply their set-theoretic product, equipped with componentwise order and addition; see Subsection 3.1.

Corollary \thecorCt.

Let $(A_{j})_{j\in J}$ be a finite family of $C^{*}$-algebras. Then the map from Section 5 induces a natural isomorphism $\operatorname{Cu}(\prod_{j\in J}A_{j})\cong\prod_{j\in J}\operatorname{Cu}(A_{j})$.

Example \theexaCt.

Set $A_{j}:=\mathbb{C}$ for each $j\in\mathbb{N}$. Set $A:=\prod_{j}A_{j}\cong\ell^{\infty}(\mathbb{N})$. We have $\operatorname{Cu}(A_{j})\cong\overline{\mathbb{N}}$ for each $j$, and the product $\prod_{j}\overline{\mathbb{N}}$ is defined as the equivalence classes of $\ll_{\mathrm{pw}}$-increasing paths $(-\infty,0]\to\mathrm{PoM}\text{-}\!\prod_{j}\overline{\mathbb{N}}$. It follows that $\prod_{j}\overline{\mathbb{N}}$ may be identified with equivalence classes of componentwise increasing paths $(-\infty,0)\to\mathrm{PoM}\text{-}\!\prod_{j}\mathbb{N}$. In particular, compact elements in $\prod_{j}\overline{\mathbb{N}}$ naturally corresponds to functions $\mathbb{N}\to\mathbb{N}$. (In Subsection 8.1 we will generalize this to give a description of compact elements in arbitrary products.)

Let us see that the natural order-embedding $\Phi\colon\operatorname{Cu}(A)\to\prod_{j}\overline{\mathbb{N}}$ is not surjective. Recall that an element $x$ in a $\mathrm{Cu}$-semigroup $S$ is said to be compact if $x\ll x$. Note that $\Phi$ maps the Cuntz class of the unit of $A$ to the compact element in $\prod_{j}\overline{\mathbb{N}}$ that correspond to the function $f\colon\mathbb{N}\to\mathbb{N}$ with $f(j)=1$ for all $j$. Consider the compact element $x$ in $\prod_{j}\overline{\mathbb{N}}$ that corresponds to the function $g\colon\mathbb{N}\to\mathbb{N}$ with $g(j)=j$ for all $j$. Since $g\nleq nf$ for every $n\in\mathbb{N}$, it follows that $x\nleq\infty\Phi([1_{A}])$. In particular, $\Phi([1_{A}])$ is not full, and $\operatorname{Cu}(\prod_{j}A_{j})$ is isomorphic to a proper ideal of $\prod_{j}\operatorname{Cu}(A_{j})$.

It is natural to ask to what extent the Cuntz semigroup functor preserves arbitrary limits. We are thankful to Eusebio Gardella for pointing out this example.

Example \theexaCt.

Let $\Gamma$ be a countable, discrete, exact group satisfying the approximation property. (For instance, any countable, discrete, amenable group.) In [Suz17, Theorem A], Suzuki constructs a decreasing sequence of $C^{*}$-algebras $(A_{n})_{n\in\mathbb{N}}$, each of which is isomorphic to $\mathcal{O}_{2}$, whose intersection is isomorphic to the reduced group $C^{*}$-algebra $C_{r}^{*}(\Gamma)$.

It follows that $C_{r}^{*}(\Gamma)\cong\varprojlim A_{n}$. For each $n\in\mathbb{N}$, we have $\operatorname{Cu}(A_{n})\cong\{0,\infty\}$. Since the maps in the inverse system given by the algebras $A_{n}$ are defined by the inclusions $A_{n}\subseteq A_{n-1}$, at the level of the Cuntz semigroup they induce the identity maps. It follows that $\mathrm{Cu}\text{-}\!\varprojlim\limits\operatorname{Cu}(A_{n})\cong\{0,\infty\}$. However, $\operatorname{Cu}(C_{r}^{*}(\Gamma))\not\cong\{0,\infty\}$ since, for example, $C_{r}^{*}(\Gamma)$ has a normalized trace.

Remark \thermkCt.

One can consider a product over an infinite index set $J$ as the inverse limit of the subproducts of finite subsets of $J$. By Section 5, the Cuntz semigroup functor preserves products of stable $C^{*}$-algebras, and therefore it does preserve this particular inverse limit. One difference between Section 5 and a product of $C^{*}$-algebras is that, in the latter, all connecting maps are surjective. Therefore, the following question is pertinent:

Question \theqstCt.

Does the Cuntz semigroup functor preserve inverse limits with surjective connecting morphisms? That is, do we have

[TABLE]

for every (sequential) inverse system $(A_{n})_{n\in\mathbb{N}}$ of (stable) $C^{*}$-algebras where each map $A_{n+1}\to A_{n}$ is surjective?

6. Cuntz semigroups of direct sums and sequence algebras

In this section we show that the Cuntz semigroup functor preserves direct sums. It follows that the scaled Cuntz semigroup functor preserves sequence algebras; see Section 6.

In $\mathrm{Cu}$, product and coproduct of finite families are naturally isomorphic; see Propositions 3.1 and 3.2. It follows that for any family $(S_{j})_{j\in J}$ of $\mathrm{Cu}$-semigroups, the direct sum $\bigoplus_{j\in J}S_{j}$ is naturally isomorphic to the coproduct $\coprod_{j\in J}S_{j}$; see Subsection 3.2. The analogous statement holds in $\mathrm{Cu}_{\mathrm{sc}}$. In the category of $C^{*}$-algebras, it is not true that finite products and coproducts coincide (see Section 6). Nevertheless, there is a zero object and the categorical construction of the direct sum yields the usual definition of direct sums for $C^{*}$-algebras.

Theorem \thethmCt.

Let $(A_{j})_{j\in J}$ be a family of $C^{*}$-algebras. Then there is are natural isomorphisms

[TABLE]

Proof.

We use that direct sums are defined as inductive limits of finite products. Using that the Cuntz semigroup functor preserves inductive limits ([APT18, Corollary 3.2.9, p.29]) and finite products (Section 5), we obtain the first isomorphism. The second isomorphism follows analogously, using Section 4. ∎

Example \theexaCt.

The Cuntz semigroup functor does not preserve coproducts. Given two $C^{*}$-algebras $A$ and $B$, their coproduct in the category of $C^{*}$-algebras is the (full) free product $A\ast B$. Consider for example $A=B=\mathbb{C}$. It is known that

[TABLE]

see [Bla06, Example IV.1.4.2, p.330]. We have $\operatorname{Cu}(\mathbb{C})\cong\overline{\mathbb{N}}$, and the coproduct of $\overline{\mathbb{N}}$ and $\overline{\mathbb{N}}$ in the category $\mathrm{Cu}$ is simply $\overline{\mathbb{N}}\times\overline{\mathbb{N}}$ with coordinatewise order and addition. On the other hand, it is easy to see that $\operatorname{Cu}(\mathbb{C}\ast\mathbb{C})\ncong\overline{\mathbb{N}}\times\overline{\mathbb{N}}$. (Using [APS11, Corollary 3.5], one can in fact explicitly compute $\operatorname{Cu}(\mathbb{C}\ast\mathbb{C})$.)

Notation \thentnCt.

Let $(S_{j})_{j\in J}$ be a family of $\mathrm{Cu}$-semigroups. Given $\textbf{x}=(x_{j})_{j\in J}$ in $\mathrm{PoM}\text{-}\!\prod_{j\in J}S_{j}$, we define its support as $\operatorname{supp}(\textbf{x}):=\{j\in J:{x}_{j}\neq 0\}$. Set

[TABLE]

Given a $\mathrm{Cu}$-semigroup $S$, we write $\mathrm{c}_{0}(S)$ for \mathrm{c}_{0}\big{(}(S)_{n\in\mathbb{N}}\big{)}.

Proposition \theprpCt.

Let $(S_{j})_{j\in J}$ be a family of $\mathrm{Cu}$-semigroups. Then \mathrm{c}_{0}\big{(}(S_{j})_{j}\big{)} is an ideal in $\prod_{j}S_{j}$, and the canonical map $\bigoplus_{j}S_{j}\to\prod_{j}S_{j}$ is an order-embedding whose image is \mathrm{c}_{0}\big{(}(S_{j})_{j}\big{)}.

Proof.

For each $F\Subset J$, we let $\varphi_{J,F}\colon\prod_{j\in F}S_{j}\to\prod_{j\in J}S_{j}$ be the natural morphism. Since these maps are compatible with the inductive system defining $\bigoplus_{j}S_{j}$, they induce a natural morphism $\varphi\colon\bigoplus_{j}S_{j}=\varinjlim_{F\Subset J}\prod_{j\in F}S_{j}\to\prod_{j\in J}S_{j}$. Using that each $\varphi_{J,F}$ is an order-embedding, it follows that $\varphi$ is an order-embedding as well. To simplify, we write $\mathrm{c}_{0}$ for \mathrm{c}_{0}\big{(}(S_{j})_{j}\big{)}. It is easy to verify that $\mathrm{c}_{0}$ is an ideal in $\prod_{j}S_{j}$. Since the image of each $\varphi_{J,F}$ is contained in $\mathrm{c}_{0}$, so is the image of $\varphi$.

To show that the image of $\varphi$ is $\mathrm{c}_{0}$, let $x\in\mathrm{c}_{0}$. Choose $\textbf{x}_{t}=(x_{t,j})_{j}\in\mathrm{PoM}\text{-}\!\prod_{j}S_{j}$ such that $x=[(\textbf{x}_{t})_{t\leq 0}]$. For $n\geq 1$, set $F_{n}:=\operatorname{supp}(\textbf{x}_{-1/n})$. Since $x\in\mathrm{c}_{0}$, we obtain that $F_{n}$ is finite. For $t\leq 0$ set

[TABLE]

Then $(\textbf{x}^{(n)}_{t})_{t\leq 0}$ is a path, and we let $x^{(n)}$ denote its class in $\prod_{j\in F_{n}}S_{j}$. By construction, we have $\varphi_{J,F}(x^{(n)})=[(\textbf{x}_{t-1/n})_{t\leq 0}]$. The images of $x^{(n)}$ in $\bigoplus_{j}S_{j}$ form an increasing sequence, and we let $y$ denote their supremum. Then

[TABLE]

The following result is straightforward to prove. We omit the details.

Proposition \theprpCt.

Let $(S_{j},\Sigma_{j})_{j\in J}$ be a family of scaled $\mathrm{Cu}$-semigroups, and let $(S,\Sigma)$ be their scaled product with $\Sigma\subseteq S\subseteq\prod_{j}S_{j}$ as in Section 5. Then \mathrm{c}_{0}\big{(}(S_{j})_{j}\big{)}\subseteq S, and we have a canonical isomorphism:

[TABLE]

Recall that the sequence algebra of a $C^{*}$-algebra $A$ is defined as

[TABLE]

By combining Theorems 6 and 5, Section 6, [CRS10, Theorem 5] we can now describe the Cuntz semigroup of the sequence algebra:

Corollary \thecorCt.

Let $A$ be a $C^{*}$-algebra. Let $(S,\Sigma)$ be the scaled product of the family $(\operatorname{Cu}(A))_{n\in\mathbb{N}}$, with $\Sigma\subseteq S\subseteq\prod_{n}\operatorname{Cu}(A)$ as in Section 5. Then

[TABLE]

7. Cuntz semigroups of ultraproducts

In this section, we show that the scaled Cuntz semigroup functor preserves ultraproducts; see Subsection 7.1. In particular, the Cuntz semigroup functor preserves ultraproducts of stable $C^{*}$-algebras; see Subsection 7.1. We take the opportunity to offer a more categorical view on ultraproducts, which suits both $C^{*}$-algebras and $\mathrm{Cu}$-semigroups.

7.1. Categorical ultraproducts

We first recall the basic theory of categorical ultraproducts.

\thepgrCt**.**

Let $\mathcal{C}$ be a category that has (small) products and inductive limits, let $J$ be a set, let $\mathcal{U}$ be an ultrafilter on $J$, and let $(X_{j})_{j\in J}$ be a family of objects in $\mathcal{C}$. Let $G\subseteq F\subseteq J$. By the universal property of the product we obtain a morphism

[TABLE]

such that $\pi_{j,F}=\pi_{j,G}\circ\varphi_{G,F}$ for each $j\in G$.

We order the elements of $\mathcal{U}$ by reversed inclusion. Then $\mathcal{U}$ is upward directed, and we obtain an inductive system indexed over $\mathcal{U}$, with objects $\prod_{j\in F}X_{j}$ for $F\in\mathcal{U}$, and with morphism $\varphi_{G,F}$ for $F,G\in\mathcal{U}$ with $F\supseteq G$. The inductive limit of this system is called the (categorical) ultraproduct of $(X_{j})_{j\in J}$ along $\mathcal{U}$:

[TABLE]

We let $\pi_{\mathcal{U}}\colon\prod_{j\in J}X_{j}\to\prod_{\mathcal{U}}X_{j}$ denote the natural morphism to the inductive limit.

The ultraproduct of a constant family of objects is called ultrapower. Given an object $X$, we denote its ultrapower by $X_{\mathcal{U}}$, that is, $X_{\mathcal{U}}:=\prod_{\mathcal{U}}X$.

The ultraproduct construction is functorial in the following sense: Given morphisms $\alpha_{j}\colon X_{j}\to Y_{j}$, we obtain a natural morphism $\alpha_{\mathcal{U}}\colon\prod_{\mathcal{U}}X_{j}\to\prod_{\mathcal{U}}Y_{j}$ as follows: For every $F\in\mathcal{U}$, the universal property of products implies that the morphisms $(\alpha_{j})_{j\in F}$ induce a natural morphism $\alpha_{F}\colon\prod_{j\in F}X_{j}\to\prod_{j\in F}Y_{j}$. These morphisms then induce a morphism between the ultraproducts, defined as respective inductive limits. Similarly, a morphism $X\to Y$ induces a morphism $X_{\mathcal{U}}\to Y_{\mathcal{U}}$.

\thepgrCt**.**

Let $\mathcal{C}$ and $\mathcal{D}$ be categories that have products and inductive limits (and hence categorical ultraproducts), and let $F\colon\mathcal{C}\to\mathcal{D}$ be a functor that preserves inductive limits. Let $\mathcal{U}$ be an ultrafilter on a set $J$, and let $(X_{j})_{j\in J}$ be a family of objects in $\mathcal{C}$. Then there is a natural morphism

[TABLE]

If the functor $F$ also preserves products, then $\Phi_{\mathcal{U}}$ is a natural isomorphism.

\thepgrCt**.**

Let $\mathcal{U}$ be an ultrafilter on a set $J$, and let $(A_{j})_{j\in J}$ be a family of $C^{*}$-algebras. The category of $C^{*}$-algebras, and the categories $\mathrm{Cu}$ and $\mathrm{Cu}_{\mathrm{sc}}$ of (scaled) $\mathrm{Cu}$-semigroups are complete and cocomplete, and thus each of the categories admit categorical ultraproducts in the sense of Subsection 7.1. By [APT18, Corollary 3.2.9, p.29] and Section 4, the (scaled) Cuntz semigroup functor preserves arbitrary inductive limits. (It was shown in [CEI08] that the Cuntz semigroup functor preserves sequential inductive limits; for the application to ultraproducts it is however crucial that the functor also preserves inductive limits over non-sequential directed sets.) As explained in Subsection 7.1, we obtain natural (scaled) $\mathrm{Cu}$-morphisms

[TABLE]

Applying Section 5 and Section 5, we obtain the following results:

Proposition \theprpCt.

Let $\mathcal{U}$ be an ultrafilter on a set $J$, and let $(A_{j})_{j\in J}$ be a family of stable $C^{*}$-algebras. Then, the map $\Phi_{\mathcal{U}}\colon\operatorname{Cu}(\prod_{\mathcal{U}}A_{j})\to\prod_{\mathcal{U}}\operatorname{Cu}(A_{j})$ from Subsection 7.1 is an isomorphism.

Theorem \thethmCt.

The scaled Cuntz semigroup functor preserves ultraproducts: Given an ultrafilter $\mathcal{U}$ on a set $J$, and a family $(A_{j})_{j\in J}$ of $C^{*}$-algebras, the map $\Phi_{\mathcal{U},\mathrm{sc}}$ from Subsection 7.1 is an isomorphism:

[TABLE]

7.2. A different picture of ultraproducts

In many ‘everyday’ categories, the (categorical) ultraproduct admits a more concrete description, which is the one used in practice. We will first recall this for the category of $C^{*}$-algebras, and then obtain a similar result for the category $\mathrm{Cu}$.

\thepgrCt**.**

Let $\mathcal{U}$ be an ultrafilter on a set $J$, and let $(A_{j})_{j\in J}$ be a family of $C^{*}$-algebras. Set $A:=\prod_{j\in J}A_{j}$. We define the support of an element $\underline{a}=(a_{j})_{j}\in A$ as $\operatorname{supp}(\underline{a}):=\{j\in J:a_{j}\neq 0\}$. Given $F\in\mathcal{U}$, we set

[TABLE]

Then $I_{F}$ is a (closed, two-sided) ideal in $A$ that is naturally isomorphic to $\prod_{j\in F^{c}}A_{j}$. (We use the convention that the product over the empty set is the zero $C^{*}$-algebra.) Moreover, the canonical map $\varphi_{F,J}\colon A\to\prod_{j\in F}A_{j}$ is a surjective ∗-homomorphism with kernel $I_{F}$. Thus, we obtain a natural isomorphism $\prod_{j\in F}A_{j}\cong A/I_{F}$.

Given $F\supseteq G$, we have $I_{F}\subseteq I_{G}$. Under the isomorphisms $\prod_{j\in F}A_{j}\cong A/I_{F}$ and $\prod_{j\in G}A_{j}\cong A/I_{G}$, the connecting morphism $\varphi_{G,F}$ (defined as in Subsection 7.1) corresponds to the natural quotient map $A/I_{F}\to A/I_{G}$. Let $I$ be the ideal of $A$ generated by the $I_{F}$, that is,

[TABLE]

The inductive limit of the system $A/I_{F}$ is isomorphic to $A/I$. It follows that the ultraproduct $\prod_{\mathcal{U}}A_{j}$ is naturally isomorphic to $A/I$, as shown in the following commutative diagram:

[TABLE]

It remains to describe the ideal $I$. Given $(a_{j})_{j}\in A=\prod_{j}A_{j}$, we have a bounded function $J\to\mathbb{R}$, $j\mapsto\|a_{j}\|$, which allows us to consider $\lim_{j\to\mathcal{U}}\|a_{j}\|$. For $a=(a_{j})_{j}\in A$ we have $\lim_{j\to\mathcal{U}}\|a_{j}\|=0$ if and only if $\{j\in J:\|a_{j}\|<\varepsilon\}\in\mathcal{U}$ for every $\varepsilon>0$. It follows that

[TABLE]

This shows that the ‘category theoretic’ ultraproduct of $C^{*}$-algebras agrees with the notion of ultraproduct of $C^{*}$-algebras considered in (continuous) model theory; see for example [GH01]. The ideal $I$ is also denoted by $c_{\mathcal{U}}((A_{j})_{j})$.

For every positive element $a_{j}\in A_{j}$ and $\varepsilon>0$ we have $\|a_{j}\|\leq\varepsilon$ if and only if $(a_{j}-\varepsilon)_{+}=0$. Thus,

[TABLE]

\thepgrCt**.**

Let $\mathcal{U}$ be an ultrafilter on a set $J$, and let $(S_{j})_{j\in J}$ be a family of $\mathrm{Cu}$-semigroups. Set $P:=\mathrm{PoM}\text{-}\!\prod_{j}S_{j}$, the set-theoretic product of the $S_{j}$. The product $S:=\prod_{j\in J}S_{j}$ in $\mathrm{Cu}$ is given by equivalence classes of $\ll_{\mathrm{pw}}$-increasing paths $(-\infty,0]\to P$. Given $F\in\mathcal{U}$, we set

[TABLE]

Then $I_{F}$ is an ideal of $S$ that is canonically isomorphic to $\prod_{j\in F^{c}}S_{j}$. As for $C^{*}$-algebras, the map $\varphi_{F,J}\colon S\to\prod_{j\in F}S_{j}$ induces an isomorphism $\prod_{j\in F}S_{j}\cong S/I_{F}$. Let $I$ be the ideal of $S$ generated by the $I_{F}$, that is,

[TABLE]

We obtain natural isomorphisms

[TABLE]

Let $\pi_{\mathcal{U}}\colon\prod_{j\in J}S_{j}\to\prod_{\mathcal{U}}S_{j}$ denote the quotient map with kernel $I$. The next result describes the ideal $I$.

Lemma 7.1**.**

We retain the notation from Subsection 7.2. Let $(\textbf{x}_{t})_{t\leq 0}$ be a path in $(\mathrm{PoM}\text{-}\!\prod_{j}S_{j},\ll_{\mathrm{pw}})$. Then the following are equivalent:

1. (1)

We have $[(\textbf{x}_{t})_{t\leq 0}]\in I$. 2. (2)

For every $\varepsilon>0$, there exists $F\in\mathcal{U}$ such that the $\varepsilon$-cut-down $[(\textbf{x}_{t-\varepsilon})_{t\leq 0}]$ belongs to $I_{F}$. 3. (3)

For every $t<0$, we have $\operatorname{supp}(\textbf{x}_{t})\notin\mathcal{U}$.

Proof.

(1)$\Rightarrow$(2): Suppose $[(\textbf{x}_{t})_{t\leq 0}]\in I$. Then there is an increasing sequence $(x_{n})_{n}$ in $\bigcup_{F\in\mathcal{U}}I_{F}$ such that $[(\textbf{x}_{t})_{t\leq 0}]=\sup x_{n}$. Let $\varepsilon>0$. Using that $[(\textbf{x}_{t-\varepsilon})_{t\leq 0}]\ll[(\textbf{x}_{t})_{t\leq 0}]$, we obtain $n$ such that $[(\textbf{x}_{t-\varepsilon})_{t\leq 0}]\leq x_{n}$. Since $x_{n}\in\bigcup_{F\in\mathcal{U}}I_{F}$ and since each $I_{F}$ is an ideal of $S$, there exists $F\in\mathcal{U}$ such that $[(\textbf{x}_{t-\varepsilon})_{t\leq 0}]$ belongs to $I_{F}$.

(2)$\Rightarrow$(3): Let $t<0$. Set $\varepsilon:=-t$. By assumption, there exists $F\in\mathcal{U}$ such that $[(\textbf{x}_{s-\varepsilon})_{s\leq 0}]$ belongs to $I_{F}$. By definition, this means $\operatorname{supp}(\textbf{x}_{0-\varepsilon})\cap F=\emptyset$, and thus $\operatorname{supp}(\textbf{x}_{t})\notin\mathcal{U}$.

(3)$\Rightarrow$(1): Suppose that for every $t<0$, we have $\operatorname{supp}(\textbf{x}_{t})\notin\mathcal{U}$. Now, write

[TABLE]

and put $F_{n}=\operatorname{supp}(\textbf{x}_{-\frac{1}{n}})^{c}$. It is clear now that $F_{n}\in\mathcal{U}$ and that $[(\textbf{x}_{t-\frac{1}{n}})_{t\leq 0}]\in I_{F_{n}}$. Therefore $[(\textbf{x}_{t})_{t\leq 0}]\in I$, as desired. ∎

Notation \thentnCt.

Let $\mathcal{U}$ be an ultrafilter on a set $J$, and let $(S_{j})_{j\in J}$ be a family of $\mathrm{Cu}$-semigroups. Set

[TABLE]

Proposition \theprpCt.

We retain the notation from Subsection 7.2. Then \mathrm{c}_{\mathcal{U}}\big{(}(S_{j})_{j}\big{)} is an ideal in $\prod_{j}S_{j}$, and we have a natural isomorphism

[TABLE]

Theorem \thethmCt.

Let $\mathcal{U}$ be an ultrafilter on a set $J$, and let $(A_{j})_{j\in J}$ be a family of $C^{*}$-algebras. Let $(S,\Sigma)$ be the scaled product of $(\mathrm{Cu}_{\mathrm{sc}}(A_{j}))_{j}$ with $\Sigma\subseteq S\subseteq\prod_{j}\operatorname{Cu}(A_{j})$ as in Section 5. Then we have a commutative diagram, where each row is a short exact sequence, where the upper vertical maps are isomorphisms, and where the lower vertical maps are inclusions as ideals:

[TABLE]

In particular, the map $\Phi_{\mathcal{U}}$ from Subsection 7.1 is an order-embedding that identifies $\operatorname{Cu}(\prod_{\mathcal{U}}A_{j})$ with the image of $S$ under the map $\pi_{\mathcal{U}}$ from Subsection 7.2. Further, $\Phi_{\mathcal{U}}$ identifies the scale of $\operatorname{Cu}(\prod_{\mathcal{U}}A_{j})$ with $\pi_{\mathcal{U}}(\Sigma)$.

Proof.

In general, if $J$ is an ideal in a $C^{*}$-algebra $A$, then the inclusion $J\to A$ induces a map that identifies $\operatorname{Cu}(J)$ with an ideal of $\operatorname{Cu}(A)$, and such that $\operatorname{Cu}(A)/\operatorname{Cu}(J)$ is naturally isomorphism to $\operatorname{Cu}(A/J)$; see [CRS10, Theorem 4.1], and note that the assumption that the ideal be $\sigma$-unital can be removed; see also [APT18, Section 5.1, p.37ff]. This shows that the top row in the diagram is exact.

By Section 5, $\Phi$ is an order-embedding with image $S$. To simplify, we denote $\mathrm{c}_{\mathcal{U}}((A_{j})_{j})$ by $\mathrm{c}_{\mathcal{U}}^{A}$, and we denote \mathrm{c}_{\mathcal{U}}\big{(}(\operatorname{Cu}(A_{j}))_{j\in J}\big{)} by $\mathrm{c}_{\mathcal{U}}^{S}$.

Claim: Let $\underline{a}\in(\prod_{j}A_{j})_{+}$. Then $\Phi([\underline{a}])\in\mathrm{c}_{\mathcal{U}}^{S}$ if and only if $\underline{a}\in\mathrm{c}_{\mathcal{U}}^{A}$.

To prove the claim, let $a_{j}$ such that $\underline{a}=(a_{j})_{j}$, and set $\textbf{x}_{t}:=([(a_{j}+t)_{+}])_{j}$ for $t\leq 0$. Then $\Phi([\underline{a}])=[(\textbf{x}_{t})_{t\leq 0}]$; see Section 5. As noted at the end of Subsection 7.2, we have $\underline{a}\in c_{\mathcal{U}}^{A}$ if and only if $\operatorname{supp}(((a_{j}-\varepsilon)_{+})_{j\in J})\notin\mathcal{U}$ for every $\varepsilon>0$, which in turn is equivalent to $\operatorname{supp}(\textbf{x}_{t})\notin\mathcal{U}$ for every $t<0$. Now, the claim follows from Lemma 7.1.

It follows that for $x\in\Sigma_{\prod_{j}A_{j}}$ we have $x\in\operatorname{Cu}(\mathrm{c}_{\mathcal{U}}((A_{j})_{j}))$ if and only if $\Phi(x)\in\mathrm{c}_{\mathcal{U}}^{S}$. Since $\Phi(\Sigma_{\prod_{j}A_{j}})=\Sigma$, it follows that $\Phi(\operatorname{Cu}(\mathrm{c}_{\mathcal{U}}^{A})\cap\Sigma_{\prod_{j}A_{j}})=\mathrm{c}_{\mathcal{U}}^{S}\cap\Sigma$. Using Section 4 at the third step, we obtain

[TABLE]

where $\langle\_\,\rangle$ denotes the generated ideal. It follows that the upper vertical maps in the diagram are isomorphisms.

Lastly, to show the statement about the scales, let $\psi\colon\operatorname{Cu}(\prod_{j}A_{j})\to\operatorname{Cu}(\prod_{\mathcal{U}}A_{j})$ be the $\mathrm{Cu}$-morphism induced by the quotient ∗-homomorphism. Using Section 4 at the last step, we get

[TABLE]

Example \theexaCt.

Let $J$ be an infinite set, and let $\mathcal{U}$ be an ultrafilter on $J$. We identify the Stone Čech compactification $\beta(J)$ of $J$ with the set of ultrafilters on $J$, and we equip it with the Stone topology, which has a basis of clopen sets given by

[TABLE]

for each $A\subseteq J$. Let $\mathrm{cpct}_{\sigma}(\beta(J))$ denote the family of $\sigma$-compact, open subsets of $\beta(J)$. Note that $\mathrm{cpct}_{\sigma}(\beta(I))$ is a positively ordered monoid, where addition of sets is given by their union, and order is defined by inclusion. Then we have isomorphisms of $\mathrm{Cu}$-semigroups:

[TABLE]

Proof.

Given $A,B\subseteq J$, we have $\mathcal{U}_{A\cup B}=\mathcal{U}_{A}\cup\mathcal{U}_{B}$, and $\mathcal{U}_{A}\subseteq\mathcal{U}_{B}$ if and only if $A\subseteq B$. We first compute $\prod_{J}\{0,\infty\}$. Let $\mathcal{P}(J)$ denote the powerset of $J$ considered as a positively ordered monoid with addition given by union and order given by inclusion. It is clear that $\mathrm{PoM}\text{-}\!\prod_{j}\{0,\infty\}\cong\mathcal{P}(J)$. Under this identification, the auxiliary relation $\ll_{\mathrm{pw}}$ becomes inclusion and it is easy to see that in this case the $\tau$-construction agrees with the (sequential) round ideal completion $\gamma$. Hence,

[TABLE]

The elements in $\gamma(\mathcal{P}(J),\subseteq)$ are equivalence classes of increasing sequences $\mathbf{A}=(A_{n})_{n\in\mathbb{N}}$ of subsets of $J$, under the antisymmetrization of the relation $(A_{n})_{n}\precsim(B_{n})_{n}$ if for every $n\in\mathbb{N}$ there is $m\in\mathbb{N}$ such that $A_{n}\subseteq B_{m}$. We use $[\mathbf{A}]$ to denote the class of $\mathbf{A}$, and recall that addition is given by $[(A_{n})_{n}]+[(B_{n})_{n}]=[(A_{n}\cup B_{n})_{n}]$. Define $\Phi\colon\gamma(\mathcal{P}(J),\subseteq)\to\mathrm{cpct}_{\sigma}(\beta(J))$ by

[TABLE]

for every increasing sequence $\mathbf{A}=(A_{n})_{n\in\mathbb{N}}$. Using the properties of the clopen sets $\mathcal{U}_{A}$ mentioned above, it follows that $\Phi$ is well-defined, order-preserving and additive.

To see that $\Phi$ is surjective, let $U\subseteq\beta(J)$ be a $\sigma$-compact, open subset. Choose an increasing sequence $(K_{n})_{n}$ of compact subsets such that $U=\bigcup_{n}K_{n}$. Since the $\mathcal{U}_{A}$ form a basis of the topology, and the $K_{n}$ are compact, we can find sets $D_{n}$ such that $K_{n}\subseteq\mathcal{U}_{D_{n}}\subseteq U$. Therefore, setting $A_{n}=\bigcup_{i=1}^{n}D_{i}$ to make them increasing, we have $U=\bigcup_{n}\mathcal{U}_{A_{n}}$ and $U=\Phi([\textbf{A}])$ where $\textbf{A}=(A_{n})_{n}$.

Now suppose that $\Phi([\mathbf{A}])\leq\Phi([\mathbf{B}])$. This implies that for each $n$, $\mathcal{U}_{A_{n}}\subseteq\bigcup_{m\in\mathbb{N}}\mathcal{U}_{B_{m}}$. Since $\mathcal{U}_{A_{n}}$ is compact and the sequence $(B_{m})_{m}$ is increasing, there is $m\in\mathbb{N}$ such that $\mathcal{U}_{A_{n}}\subseteq\mathcal{U}_{B_{m}}$ which implies $A_{n}\subseteq B_{m}$. Hence $\textbf{A}\precsim\textbf{B}$ proving that $\Phi$ is an order-embedding, and thus $\prod_{j}\{0,\infty\}\cong\mathrm{cpct}_{\sigma}(\beta(J))$.

We now compute the ultraproduct. Let us write $c_{\mathcal{U}}$ for $c_{\mathcal{U}}((\{0,\infty\})_{j})$. It follows from Subsection 7.2 that $c_{\mathcal{U}}$ corresponds to equivalence classes of $\mathbf{A}=(A_{n})_{n}$ with all $A_{n}\not\in\mathcal{U}$. Let $[\mathbf{A}],[\mathbf{B}]\in\gamma(\mathcal{P}(J),\subseteq)$ with $[\mathbf{B}]\notin c_{\mathcal{U}}$. Then there is $n_{0}$ such that $B_{n_{0}}\in\mathcal{U}$. For each $n$, we have

[TABLE]

Set $\textbf{C}:=(A_{n}\cap B_{n_{0}}^{c})_{n}$. Then $\textbf{A}\precsim\textbf{B}+\textbf{C}$. Since $(A_{n}\cap B_{n_{0}}^{c})\notin\mathcal{U}$, we have $\textbf{C}\in c_{\mathcal{U}}$.

By Subsection 7.2, we have $\prod_{\mathcal{U}}\{0,\infty\}\cong\prod_{j}\{0,\infty\}/c_{\mathcal{U}}$. It follows from the above argument that $\prod_{\mathcal{U}}\{0,\infty\}$ has only two elements, $\{0,\infty\}$, as desired. ∎

8. Applications

8.1. Compact elements and K-Theory

Recall that, if $S$ is a $\mathrm{Cu}$-semigroup, then an element $x\in S$ is compact provided $x\ll x$. If $S$ is only a $\mathcal{Q}$-semigroup, we shall also use this terminology for an element $x\in S$ such that $x\prec x$. The subsemigroup of compact elements of $S$ will be denoted by $S_{c}$. This semigroup is a positively ordered monoid, regardless of whether $S$ is a $\mathrm{Cu}$-semigroup or a $\mathcal{Q}$-semigroup. We next describe the compact elements of an ultraproduct of $\mathrm{Cu}$-semigroups, for which we use the picture of categorical ultraproducts; see Subsection 7.1. This will be used later to describe the Murray-von Neumann semigroup of an ultraproduct of unital $C^{*}$-algebras, thus recovering [Li05, Theorem 4.1] (see also [GH01, Proposition 2.1]).

Lemma 8.1**.**

Let $S$ be a $\mathcal{Q}$-semigroup. Then, the endpoint map (Subsection 2.3) induces a $\mathrm{PoM}$-isomorphism $\tau(S)_{c}\cong S_{c}$.

Proof.

Since the endpoint map $\varphi_{S}\colon\tau(S)\to S$ is a $\mathcal{Q}$-morphism, we see that its restriction to $\tau(S)_{c}$ is a $\mathrm{PoM}$-morphism that maps compact elements to compact elements. Given $a\in S_{c}$, the map $f_{a}\colon(-\infty,0]\to S$ given by $f_{a}(t)=a$ for every $t$ is clearly a path with $[f_{a}]\in\tau(S)_{c}$ and $\varphi_{S}([f_{a}])=a$. Thus, $\varphi_{S}$ induces an additive, order-preserving bijection $\tau(S)_{c}\to S_{c}$.

Now, let $[f],[g]\in\tau(S)_{c}$ satisfy $\varphi_{S}([f])\leq\varphi_{S}([g])$. By [APT17, Lemma 3.16] applied to $[g]\ll[g]$, there is $t_{0}<0$ such that $g(t)\prec g(t_{0})$ for all $t<0$. Therefore $\varphi_{S}([g])=\sup_{t<0}g(t)\leq g(t_{0})$ and thus, for every $t<0$, we have $f(t)\prec g(t_{0})$. This implies $[f]\leq[g]$ and therefore $\varphi_{S}$ induces an order-embedding. ∎

Corollary \thecorCt.

Let $(S_{j})_{j\in J}$ be a family of $\mathrm{Cu}$-semigroups. Then

[TABLE]

Proof.

We have $\prod_{j}S_{j}=\tau(\mathcal{Q}\text{-}\!\prod_{j}S_{j})$, by Subsection 3.1. Therefore, Lemma 8.1 implies that $(\prod_{j}S_{j})_{c}\cong(\mathcal{Q}\text{-}\!\prod_{j}S_{j})_{c}$. Since the auxiliary relation in $\mathcal{Q}\text{-}\!\prod_{J}S_{j}$ is $\ll_{\mathrm{pw}}$, it follows that $(\mathcal{Q}\text{-}\!\prod_{j}S_{j})_{c}$ is isomorphic to $\mathrm{PoM}\text{-}\!\prod_{j}(S_{j})_{c}$. ∎

Proposition \theprpCt.

Let $\mathcal{U}$ be an ultrafilter on a set $J$, and let $(S_{j})_{j\in J}$ be a family of $\mathrm{Cu}$-semigroups. Then there is a natural $\mathrm{PoM}$-isomorphism

[TABLE]

In particular, for every compact $x\in\prod_{\mathcal{U}}S_{j}$ there exists a compact $y\in\prod_{j}S_{j}$ with $x=\pi_{\mathcal{U}}(y)$, where $\pi_{\mathcal{U}}\colon\prod_{j\in J}S_{j}\to\prod_{\mathcal{U}}S_{j}$ is the natural quotient map.

Proof.

We have $\prod_{\mathcal{U}}S_{j}=\varinjlim_{F\in\mathcal{U}}\prod_{j\in F}S_{j}$, and thus $(\prod_{\mathcal{U}}S_{j})_{c}\cong\varinjlim_{F\in\mathcal{U}}(\prod_{j\in F}S_{j})_{c}$ with the restriction of the same connecting maps. Now, by Subsection 8.1, for each $F\in\mathcal{U}$ the end point map induces an isomorphism $(\prod_{j\in F}S_{j})_{c}\cong\prod_{j\in F}(S_{j})_{c}$, which is easily seen to be compatible with the connecting maps. Therefore

[TABLE]

For a $C^{*}$-algebra $A$, we denote as customary by $V(A)$ the semigroup of Murray-von Neumann equivalence classes of projections in $A\otimes\mathcal{K}$. By [BC09], if $A$ is stably finite, then $V(A)$ can be identified with the semigroup of compact elements in $\operatorname{Cu}(A)$. The following result for ultraproducts is implicit in [Li05, Theorem 4.1].

Corollary \thecorCt.

Let $\mathcal{U}$ be an ultrafilter on a set $J$, and let $(A_{j})_{j\in J}$ be a family of stably finite, unital $C^{*}$-algebras. Put $u_{j}=[1_{A_{j}}]\in V(A_{j})$ and $u=(u_{j})_{j\in J}$. Then:

1. (1)

V(\prod_{j\in J}A_{j})\cong\big{\{}x\in\prod_{j\in J}V(A_{j})\colon x\leq nu\text{ for some }n\in\mathbb{N}\big{\}}. 2. (2)

V(\prod_{\mathcal{U}}A_{j})\cong\big{\{}x\in\prod_{\mathcal{U}}V(A_{j})\colon x\leq n\pi_{\mathcal{U}}(u)\text{ for some }n\in\mathbb{N}\big{\}}.

Proof.

First observe that both $\prod_{i}A_{i}$ and $\prod_{\mathcal{U}}A_{i}$ are stably finite $C^{*}$-algebras. Therefore $V(\prod_{j}A_{j})$ is identified with $\operatorname{Cu}(\prod_{j}A_{j})_{c}$. By Section 5, we have

[TABLE]

and thus the desired conclusions follow from Subsection 8.1. Statement (2) follows analogously from Subsection 7.1 and Subsection 8.1. ∎

8.2. Simplicity

\thepgrCt**.**

Recall that an ultrafilter $\mathcal{U}$ is called countably complete if it is closed under taking intersections of countably many elements. If $\mathcal{U}$ it not countably complete, then it is called countably incomplete. Every free ultrafilter on a countable set is countably incomplete.

It is well-known that the ultrapower of a separable $C^{*}$-algebra $A$ over a countably complete (free) ultrafilter is canonically isomorphic to $A$ itself; see for example [GH01, Proposition 6.1]. Thus, with respect to a separable $C^{*}$-algebra, a countably complete, free ultrafilter behaves like a principal ultrafilter. Let now $A$ be a simple, stable, separable, infinite-dimensional $C^{*}$-algebra, and let $\mathcal{U}$ be a free ultrafilter. If $\mathcal{U}$ is countably incomplete, then $\prod_{\mathcal{U}}A$ is neither stable (see Subsection 8.2), nor separable, and usually not simple (see Subsection 8.2 for a characterization of simplicity of ultrapowers). It follows that the behaviour of ultraproducts and ultrapowers is immensely different for countably complete and countably incomplete ultrafilters.

We remark that it is not obvious that countably complete, free utrafilters exist. In fact, the existence of a countably complete, free ultrafilter is equivalent to the existence of a measurable cardinal, which puts this question outside of our usual axiomatic setting of ZFC and into the realm of set theory.

Remark \thermkCt.

In general, stability does not pass to ultraproducts or ultrapowers. More precisely, let $A$ be a (stable) $C^{*}$-algebra, let $I$ be a set and let $\mathcal{U}$ be a free ultrafilter on $I$. A result of Kania, [Kan15, Theorem 1.1], shows that a $C^{*}$-algebra that is a Grothendieck space is not isomorphic to the tensor product of two infinite-dimensional $C^{*}$-algebras. Moreover, he showed that $\prod_{\mathcal{U}}A$ is a Grothendieck space if $\mathcal{U}$ is countably incomplete. Thus, if $\mathcal{U}$ is countably incomplete and $A$ is infinite-dimensional, then $\prod_{\mathcal{U}}A$ is not stable. If $I$ is countable, this also follows from results of Farah and Hart, [FH13], and Ghasemi [Gha15].

Definition \thedfnCt.

We say that a scaled $\mathrm{Cu}$-semigroup $(S,\Sigma)$ has property $(C_{n})$ if for all $x,y\in\Sigma$ with $y\neq 0$, we have $x\leq ny$.

Lemma 8.2**.**

Let $\mathcal{U}$ be a free ultrafilter on a set $J$, and let $(S_{j},\Sigma_{j})_{j\in J}$ be a family of scaled $\mathrm{Cu}$-semigroups. Let $n\in\mathbb{N}$ and $F\in\mathcal{U}$ such that $(S_{j},\Sigma_{j})$ has $(C_{n})$ for all $j\in F$. Then $\prod_{\mathcal{U}}(S_{j},\Sigma_{j})$ has $(C_{n})$.

Proof.

Let $x,y\in\prod_{\mathcal{U}}\Sigma_{j}$ with $y\neq 0$. We need to show that $x\leq ny$. Choose $\tilde{x},\tilde{y}\in\prod_{j}(S_{j},\Sigma_{j})$ with $\pi_{\mathcal{U}}(\tilde{x})=x$ and $\pi_{\mathcal{U}}(\tilde{y})=y$. Then, choose $x_{t,j},y_{t,j}\in S_{j}$ such that $\tilde{x}=[((x_{t,j})_{j})_{t}]$ and $\tilde{y}_{t}=[((y_{t,j})_{j})_{t}]$. Since $y\neq 0$, we have $\tilde{y}\notin c_{\mathcal{U}}$ (see Subsection 7.2). Hence, we can fix $s<0$ such that $G:=\operatorname{supp}((y_{s,j})_{j})$ belongs to $\mathcal{U}$.

For $t\leq 0$ and $j\in J$ set

[TABLE]

Then $z:=[((z_{t,j})_{j})_{t}]$ belongs to $c_{\mathcal{U}}\cap\prod_{j}(S_{j},\Sigma_{j})$. We claim that $\tilde{x}\leq n\tilde{y}+z$. To prove this, let $t<0$. Choose $\tilde{s}$ with $t,s<\tilde{s}<0$ (with $s$ as chosen above).

Case 1: Let $j\in F\cap G$. Then $S_{j}$ has $(C_{n})$, and we have $x_{\tilde{s},j},y_{s,j}\in\Sigma_{j}$ with $y_{s,j}\neq 0$. We obtain that $x_{\tilde{s},j}\leq ny_{s,j}$, and thus

[TABLE]

Case 2: Let $j\notin F\cap G$. Then

[TABLE]

Thus, $(x_{t,j})_{j}\ll_{\mathrm{pw}}n(y_{\tilde{s},j})_{j}+(z_{\tilde{s},j})_{j}$. Hence, $\tilde{x}\leq n\tilde{y}+z$ in $\prod_{j}S_{j}$, and thus $x\leq ny$ in $\prod_{\mathcal{U}}(S_{j},\Sigma_{j})$. ∎

Recall that a $\mathrm{Cu}$-semigroup $S$ is said to satisfy (O6) if for all $x^{\prime},x,y,z\in S$ satisfying $x^{\prime}\ll x\leq y+z$, there exist $e,f\in S$ with $e\leq x,y$ and $f\leq x,z$, and such that $x^{\prime}\leq e+f$. By [Rob13, Proposition 5.1.1], Cuntz semigroups of $C^{*}$-algebras satisfy (O6).

Lemma 8.3**.**

Let $(S,\Sigma)$ be a scaled $\mathrm{Cu}$-semigroup satisfying (O6). Assume that $(S,\Sigma)$ has $(C_{n})$ for some $n\geq 1$. Then $S$ is simple.

Proof.

Pick $n\geq 1$ such that $(S,\Sigma)$ has $(C_{n})$. Let $x,y\in S$ with $y\neq 0$. We need to show that $x\leq\infty y$.

Claim: There exists $z\in\Sigma$ with $0\neq z\leq y$. To show this, choose $y^{\prime\prime},y^{\prime}\in S$ with $0\neq y^{\prime\prime}\ll y^{\prime}\ll y$. Using that $\Sigma$ generates $S$ as an ideal, we can then choose $y_{1},\ldots,y_{K}\in\Sigma$ such that

[TABLE]

Choose $y_{1}^{\prime},\dots,y_{K}^{\prime}$ such that $y^{\prime\prime}\ll y_{K}^{\prime}\ll\dots\ll y_{1}^{\prime}\ll y^{\prime}$. Applying (O6) for $y_{1}^{\prime}\ll y^{\prime}\leq y_{1}+(y_{2}+\ldots+y_{K})$, we obtain $e_{1}$ such that $e_{1}\leq y_{1},y^{\prime}$ (in particular, $e_{1}\in\Sigma$) and

[TABLE]

In the next step, we apply (O6) for $y_{2}^{\prime}\ll y_{1}^{\prime}\leq y_{2}+(e_{1}+y_{3}+\ldots+y_{K})$ to obtain $e_{2}\in\Sigma$ such that

[TABLE]

Iteratively, we obtain $e_{1},\ldots,e_{K}\in\Sigma$ such that

[TABLE]

Since $0\neq y^{\prime\prime}\leq y_{K}^{\prime}$, there exists $k\in\{1,\ldots,K\}$ with $e_{k}\neq 0$. Then $z:=e_{k}$ has the desired properties to verify the claim.

It is enough to verify that $x^{\prime}\leq\infty y$ for every $x^{\prime}\ll x$. Thus, let $x^{\prime}\ll x$. Using again that $\Sigma$ generates $S$ as an ideal, we obtain $x_{1},\ldots,x_{L}\in\Sigma$ such that

[TABLE]

Since $(S,\Sigma)$ has $(C_{n})$, we have $x_{l}\leq nz$ for every $l=1,\ldots,L$. Hence,

[TABLE]

The next examples shows that Lemma 8.3 does not hold without assuming (O6).

Example \theexaCt.

Set $S=\{0,x,y,\infty\}$ with addition given by

[TABLE]

and equipped with the algebraic order (for two elements $s,t$ we set $s\leq t$ if there exists $u$ with $s+u=t$). Note that $x$ and $y$ are incomparable. Then $S$ is a $\mathrm{Cu}$-semigroup. Further, $\Sigma:=\{0,x\}$ is a scale such that $(S,\Sigma)$ satisfies $(C_{1})$. We have $\infty\nleq\infty y$, which shows that $S$ is not simple. Note that $S$ does not satisfy (O6): We have $y\ll y\leq x+x$, but no elements $e,f$ satisfying $y\leq e+f$ and $e,f\leq x,y$.

Lemma 8.4**.**

Let $\mathcal{U}$ be a countably incomplete, free ultrafilter on a set $J$, and let $(S_{j},\Sigma_{j})_{j\in J}$ be a family of scaled $\mathrm{Cu}$-semigroups such that $\prod_{\mathcal{U}}(S_{j},\Sigma_{j})$ is simple. Then there exist $n\in\mathbb{N}$ and $F\in\mathcal{U}$ such that $S_{j}$ has $(C_{n})$ for all $j\in F$.

Proof.

We argue by contradiction. Thus, assume that for all $n\in\mathbb{N}$, and for all $F\in\mathcal{U}$, there is $j\in F$ such that $S_{j}$ does not have $(C_{n})$. Let us write $c_{\mathcal{U}}$ for $c_{\mathcal{U}}((S_{j})_{j})$. By Subsection 7.2, we have $\prod_{\mathcal{U}}S_{j}\cong(\prod_{j}S_{j})/c_{\mathcal{U}}$, and we let $\pi_{\mathcal{U}}\colon\prod_{j}S_{j}\to\prod_{\mathcal{U}}S_{j}$ denote the quotient map.

Since $\mathcal{U}$ is countably incomplete, we may find a decreasing sequence $(F_{n})_{n}$ of elements in $\mathcal{U}$ such that $\bigcap_{n}F_{n}=\emptyset$. By our assumption the sets

[TABLE]

also belong to $\mathcal{U}$, and they also form a decreasing sequence with empty intersection.

Replacing $F_{n}$ by $G_{n}$ we may assume that for every $j\in F_{n}$, the semigroup $S_{j}$ does not have $(C_{n})$. For each $n\geq 1$, and each $j\in F_{n}\setminus F_{n+1}$, choose elements $x_{j}$, $y_{j}\in\Sigma_{j}$ with $y_{j}\neq 0$ and $x_{j}\not\leq ny_{j}$.

Further, choose paths $(x_{t,j})_{t\leq 0}$ in $S_{j}$ with $x_{j}=x_{0,j}$ such that $x_{t,j}\not\leq ny_{j}$ for all $t\leq 0$. Likewise, choose paths $(y_{t,j})_{t\leq 0}$ in $S_{j}$ with $y_{j}=y_{0,j}$ and such that $y_{t,j}\neq 0$ for all $t\leq 0$. For $t\leq 0$ set

[TABLE]

Set $x=[(\textbf{x}_{t})_{t\leq 0}]$ and $y=[(\textbf{y}_{t})_{t\leq 0}]$ in $\prod_{j}S_{j}$. By construction, we have $x,y\in\prod_{j}(S_{j},\Sigma_{j})$. Since $\operatorname{supp}(\textbf{y}_{-1})=F_{1}\in\mathcal{U}$, we have $y\notin c_{\mathcal{U}}$, and therefore $\pi_{\mathcal{U}}(y)\neq 0$. Using that $\prod_{\mathcal{U}}(S_{j},\Sigma_{j})$ is simple, it follows that $\pi_{\mathcal{U}}(x)\leq\infty\pi_{\mathcal{U}}(y)$.

Let $\varepsilon>0$, and let $x_{\varepsilon}:=[(\textbf{x}_{t-\varepsilon})_{t\leq 0}]$ be the $\varepsilon$-cut-down of $x$. We have $x_{\varepsilon}\ll x$ in $\prod_{j}(S_{j},\Sigma_{j})$, and therefore

[TABLE]

in $\prod_{\mathcal{U}}(S_{j},\Sigma_{j})$. We obtain $m\in\mathbb{N}$ such that $\pi_{\mathcal{U}}(x_{\varepsilon})\leq m\pi_{\mathcal{U}}(y)$, and thus there is $z\in c_{\mathcal{U}}$ such that $x_{\varepsilon}\leq my+z$. Choose $z_{t,j}\in S_{j}$ such that $z=[((z_{t,j})_{j\in J})_{t\leq 0}]$.

Fix any $t<0$. Then there is $s<0$ such that $\textbf{x}_{t-\varepsilon}\ll_{\mathrm{pw}}m\textbf{y}_{s}+\textbf{z}_{s}$. Set $T:=(\operatorname{supp}(\textbf{z}_{s}))^{c}$. Since $z\in c_{\mathcal{U}}$, we have $T\in\mathcal{U}$, and thus there is $N\geq m$ with $T\cap(F_{N}\setminus F_{N+1})\neq\emptyset$. Choose $j$ in $T\cap(F_{N}\setminus F_{N+1})$. Then $z_{s,j}=0$, and thus

[TABLE]

in contradiction with our choice. ∎

Theorem \thethmCt.

Let $\mathcal{U}$ be a countably incomplete, free ultrafilter on a set $J$, and let $(S_{j},\Sigma_{j})_{j\in J}$ be a family of scaled $\mathrm{Cu}$-semigroups that satisfy (O6). Then the following are equivalent:

1. (1)

$\prod_{\mathcal{U}}(S_{j},\Sigma_{j})$* is simple.* 2. (2)

$\prod_{\mathcal{U}}(S_{j},\Sigma_{j})$* has $(C_{n})$ for some $n\geq 1$.* 3. (3)

There exist $n\in\mathbb{N}$ and $F\in\mathcal{U}$ such that $S_{j}$ has $(C_{n})$ for all $j\in F$.

In particular, if $(S,\Sigma)$ is a scaled $\mathrm{Cu}$-semigroup satisfying (O6), then the ultrapower $\prod_{\mathcal{U}}(S,\Sigma)$ is simple if and only if $(S,\Sigma)$ has $(C_{n})$ for some $n$.

Proof.

By Lemma 8.4, (1) implies (3), and by Lemma 8.2 (3) implies (2). It is straightforward to verify that (O6) passes to products and quotients in $\mathrm{Cu}$. Hence, $\prod_{\mathcal{U}}(S_{j},\Sigma_{j})$ satisfies (O6), whence we can apply Lemma 8.3 to deduce that (2) implies (1). ∎

Let $n\in\mathbb{N}$. Adapting [KR02, Definition 4.3] from $C^{*}$-algebras to $\mathrm{Cu}$-semigroups, let us say that a $\mathrm{Cu}$-semigroup $S$ is pi-$n$ if $2nx=nx$ for every $x\in S$. Further, we say that $S$ is weakly purely infinite if it is pi-$n$ for some $n$.

Note that a $\mathrm{Cu}$-semigroup $S$ is simple and pi-$n$ if and only if for all $x,y\in S$ with $y\neq 0$ we have $x\leq ny$. Thus, $S$ is simple and pi-$n$ if and only if the trivially scaled $\mathrm{Cu}$-semigroup $(S,S)$ has $(C_{n})$.

Corollary \thecorCt.

Let $\mathcal{U}$ be a countably incomplete, free ultrafilter on a set $J$, and let $(S_{j})_{j\in J}$ be a family of $\mathrm{Cu}$-semigroups. Then the following are equivalent:

1. (1)

$\prod_{\mathcal{U}}S_{j}$* is simple.* 2. (2)

$\prod_{\mathcal{U}}S_{j}$* is simple and weakly purely infinite.* 3. (3)

There exist $n\in\mathbb{N}$ and $F\in\mathcal{U}$ such that $S_{j}$ is simple and pi-$n$* for all $j\in F$.*

In particular, the ultrapower $\prod_{\mathcal{U}}S$ of a $\mathrm{Cu}$-semigroup $S$ is simple if and only if $S$ is simple and weakly purely infinite.

Proof.

It is clear that (2) implies (1). As in the proof of Subsection 8.2, we obtain that (1) implies (3), and that (3) implies (2). (Note that Lemmas 8.4 and 8.2 do not require (O6).) ∎

Lemma 8.5**.**

Let $A$ be a $C^{*}$-algebra, and let $n\geq 1$. Then $(\operatorname{Cu}(A),\Sigma_{A})$ satisfies $(C_{n})$ if and only if $A\cong M_{k}(\mathbb{C})$ for $k\leq n$, or $A$ is simple and purely infinite.

Proof.

The backward implication is easy to show. Thus, assume that $(\operatorname{Cu}(A),\Sigma_{A})$ satisfies $(C_{n})$. Since Cuntz semigroups of $C^{*}$-algebras satisfy (O6), it follows from Lemma 8.3 that $A$ is simple. Then $A$ is either elementary (that is, $A$ contains a minimal nonzero projection), or not. In the first case, we obtain $\operatorname{Cu}(A)\cong\overline{\mathbb{N}}$. It is easy to verify that $(\overline{\mathbb{N}},\Sigma)$ has $(C_{n})$ if and only if $\Sigma\subseteq\{0,\ldots,n\}$. It follows that $A$ contains only projections of rank at most $n$, and thus $A\cong M_{k}(\mathbb{C})$ for some $k\leq n$.

In the other case, it follows from the Glimm Halving Lemma that for every nonzero $x\in\operatorname{Cu}(A)$ there exists a nonzero $y$ with $2y\leq x$. Let $x\in\operatorname{Cu}(A)$ be nonzero. From the Claim in the proof of Lemma 8.3 we obtain a nonzero $y\in\Sigma_{A}$ with $y\leq x$. Applying the Glimm Halving Lemma several times, we obtain a nonzero $z$ with $2nz\leq y$. By $(C_{n})$, we also have $y\leq nz$, and thus $2y\leq 2nz\leq y$. Therefore $y=\infty$, and then $x=\infty$. It follows that $\operatorname{Cu}(A)\cong\{0,\infty\}$, which implies that $A$ is simple and purely infinite. ∎

Using Subsection 8.2, we obtain a characterization for when an ultraproduct of $C^{*}$-algebras is simple. In the case that we have a purely infinite simple $C^{*}$-algebra $A$, and a free ultrafilter $\mathcal{U}$ over $\mathbb{N}$, it was proved by Kirchberg (see [Rør02, Proposition 6.2.6]) that the ultrapower $A_{\mathcal{U}}$ is again purely infinite simple. The result that an ultraproduct of purely infinite, simple $C^{*}$-algebras is again purely infinite simple can be obtained as a consequence of [FHL*+*16, Theorem 2.5.1].

Theorem \thethmCt.

Let $\mathcal{U}$ be a countably incomplete, free ultrafilter on a set $J$, and let $(A_{j})_{j\in J}$ be a family of $C^{*}$-algebras. Then $\prod_{\mathcal{U}}A_{j}$ is simple if and only if either $A_{j}$ is simple and purely infinite for almost all $j$ in $J$, or else there is $n\in\mathbb{N}$ such that $A_{j}\cong M_{n}(\mathbb{C})$ for almost all $j$ in $J$.

In particular, the ultrapower $\prod_{\mathcal{U}}A$ of a $C^{*}$-algebra $A$ is simple if and only if $A$ is either simple and purely infinite or $A\cong M_{n}(\mathbb{C})$ for some $n\in\mathbb{N}$.

Proof.

If $A_{j}\cong M_{n}(\mathbb{C})$ for almost all $j\in J$, then $\prod_{\mathcal{U}}A_{j}\cong M_{n}(\mathbb{C})$, which is simple. Note that a $C^{*}$-algebra $A$ is simple and purely infinite if and only if $\operatorname{Cu}(A)\cong\{0,\infty\}$, with the trivial scale. Thus, if $A_{j}$ is simple and purely infinite for almost all $j\in J$, then using Section 5 and Subsection 7.2 we obtain

[TABLE]

which implies that $\prod_{\mathcal{U}}A_{j}$ is simple (and purely infinite).

To show the converse implication, assume that $\prod_{\mathcal{U}}A_{j}$ is simple. Since Cuntz semigroups of $C^{*}$-algebras satisfy (O6), and since the Cuntz semigroup of a $C^{*}$-algebra is simple if and only if the $C^{*}$-algebra is simple, we can apply Subsection 8.2 to deduce that there exist $n\geq 1$ and $F\in\mathcal{U}$ such that the scaled $\mathrm{Cu}$-semigroups $\operatorname{Cu}(A_{j})$ have $(C_{n})$ for all $j\in F$. Let $F_{0}$ be set of $j\in F$ such that $A_{j}$ is simple and purely infinite, and for $k=1,\ldots,n$ let $F_{k}$ be the set of $j\in F$ such that $A_{j}\cong M_{k}(\mathbb{C})$. By Lemma 8.5, we have $F=F_{0}\cup\ldots\cup F_{n}$, and since $\mathcal{U}$ is an ultrafilter, we have $F_{k}\in\mathcal{U}$ for precisely one $k$, which proves the claim. ∎

It is worth mentioning that a similar characterization is obtained by Kirchberg [Kir06] for central sequence $C^{*}$-algebras, proving that $(A_{\mathcal{U}}\cap A^{\prime})/\text{Ann}_{A}(A_{\mathcal{U}})$ is simple if and only if $A\otimes\mathcal{K}\cong\mathcal{K}$ or $A$ is simple purely infinite and nuclear.

8.3. Comparability

We show that several comparability properties pass to (ultra)products of $\mathrm{Cu}$-semigroups. In particular, in Subsection 8.3 we recover an unpublished result of A. Tikuisis, that was brought to our attention by L. Robert (private communication). We refer to [APT18, Section 5.6, p.72ff], for definitions of the unperforation properties considered in the next results.

Proposition \theprpCt.

The following properties pass to products, direct sums, and ultraproducts of $\mathrm{Cu}$-semigroups: unperforation, almost (near) unperforation.

Proof.

It is straightforward to verify that each of the considered properties passes to ideals and quotients. Thus, it is enough to consider products. We only prove the statement for unperforation, as the other situations are similar.

Let $(S_{j})_{j\in J}$ be a family of unperforated $\mathrm{Cu}$-semigroups, and let $x,y\in\prod_{j}S_{j}$ and $n\geq 1$ satisfy $nx\leq ny$. Choose $\textbf{x}_{t}=(x_{t,j})_{j}$ and $\textbf{y}_{t}=(y_{t,j})_{j}$ such that $x=[(\textbf{x}_{t})_{t\leq 0}]$ and $y=[(\textbf{y}_{t})_{t\leq 0}]$. Given $t<0$, there is $s<0$ such that $n\textbf{x}_{t}\ll_{\mathrm{pw}}n\textbf{y}_{s}$, that is, $nx_{t,j}\ll ny_{s,j}$ for all $j\in J$. Therefore $x_{t,j}\ll y_{s,j}$ for all $j$, which implies $\textbf{x}_{t}\ll_{\mathrm{pw}}\textbf{y}_{s}$. Hence, $x\leq y$, as desired. ∎

Corollary \thecorCt.

Given a family of $C^{*}$-algebras whose Cuntz semigroups are unperforated (almost unperforated, nearely unperforated), the Cuntz semigroups of their product, direct sum, and ultraproduct over some ultrafilter are also unperforated (almost unperforated, nearly unperforated).

Proof.

This is a direct consequence of Subsection 8.3, using Theorems 5, 6 or 7.1, and using that each of the properties passes to ideals, and hence from (ultra)products to scaled (ultra)products. ∎

Proposition \theprpCt.

Let $\mathcal{U}$ be an ultrafilter on a set $J$, and let $(S_{j})_{j\in J}$ be a family of totally ordered $\mathrm{Cu}$-semigroups. Then $\prod_{\mathcal{U}}S_{j}$ is totally ordered.

Proof.

Let $x,y\in\prod_{j}S_{j}$. We need to show that $\pi_{\mathcal{U}}(x)\leq\pi_{\mathcal{U}}(y)$ or $\pi_{\mathcal{U}}(y)\leq\pi_{\mathcal{U}}(x)$. Choose $\textbf{x}_{t}=(x_{t,j})_{j}$ and $\textbf{y}_{t}=(y_{t,j})_{j}$ such that $x=[(\textbf{x}_{t})_{t\leq 0}]$ and $y=[(\textbf{y}_{t})_{t\leq 0}]$. For every $\varepsilon>0$ and $t<0$ let us define the following set:

[TABLE]

If $F_{\varepsilon,t}\in\mathcal{U}$ for some $\varepsilon>0$ and $t<0$, then $\pi_{\mathcal{U}}([(\textbf{x}_{t-\varepsilon})_{t\leq 0}])\leq\pi_{\mathcal{U}}([(\textbf{y}_{t})_{t\leq 0})$. Hence, if for all $\varepsilon>0$ there is $t<0$ with $F_{\varepsilon,t}\in\mathcal{U}$, then $\pi_{\mathcal{U}}(x)\leq\pi_{\mathcal{U}}(y)$. Otherwise, there is $\varepsilon>0$ with $F_{\varepsilon,t}\notin\mathcal{U}$ for all $t<0$. Since the $S_{j}$ are totally ordered, we get

[TABLE]

for all $t<0$, which then implies $\pi_{\mathcal{U}}(y)\leq\pi_{\mathcal{U}}(x)$. ∎

Corollary \thecorCt.

Let $\mathcal{U}$ be an ultrafilter over some set, and let $A$ be a strongly self-absorbing, stably finite $C^{*}$-algebra satisfying the UCT. Then the Cuntz semigroup of the ultrapower $A_{\mathcal{U}}$ is totally ordered.

Proof.

By [APT18, Paragraph 7.6.1, p.136], the Cuntz semigroup $\operatorname{Cu}(A)$ is totally ordered. By Subsection 8.3, $\prod_{\mathcal{U}}\operatorname{Cu}(A)$ is totally ordered. Now, the statement follows using that $\operatorname{Cu}(\prod_{\mathcal{U}}A)$ is an ideal in $\prod_{\mathcal{U}}\operatorname{Cu}(A)$; see Subsection 7.1. ∎

Remark \thermkCt.

Recall that a functional on a $\mathrm{Cu}$-semigroup $S$ is a $\mathrm{PoM}$-morphism $S\to[0,\infty]$ that preserves suprema of increasing sequences. Given a family $(S_{j})_{j\in J}$ of $\mathrm{Cu}$-semigroups and an ultrafilter $\mathcal{U}$ on $J$, it is natural to study functionals on the ultraproduct $\prod_{\mathcal{U}}S_{j}$. Given a functional $\varphi_{j}\colon S_{j}\to[0,\infty]$, for each $j\in J$, there is a naturally induced functional on $\prod_{\mathcal{U}}S_{j}$. The construction of this functional and the question whether the set of such functionals is dense in the set of all functionals on $\prod_{\mathcal{U}}S_{j}$ will be analysed in detail in forthcoming work, [APRT20].