An Infinite C*-algebra with a Dense, Stably Finite *-Subalgebra

TL;DR

This paper constructs a specific algebraic structure that is stably finite at the algebraic level but becomes infinite upon C*-completion, revealing nuanced differences between algebraic and topological properties.

Contribution

It introduces a unital pre-C*-algebra with a dense, stably finite *-subalgebra that becomes infinite after C*-completion, highlighting a novel example of divergence between algebraic and topological properties.

Findings

The pre-C*-algebra is stably finite.

Its C*-completion contains a non-unitary isometry.

The resulting C*-algebra is infinite.

Abstract

We construct a unital pre-C*-algebra $A_0$ which is stably finite, in the sense that every left invertible square matrix over $A_0$ is right invertible, while the C*-completion of $A_0$ contains a non-unitary isometry, and so it is infinite.

Equations

$$(s_1{t}_1\cdots s_n{t}_n{)}^{\ast }=t_n^{\ast }{s}_n^{\ast }\cdots t_1^{\ast }{s}_1^{\ast }$$

$$L_m(BC\ast S_{\infty })=\{u\in BC\ast S_{\infty }:\mathrm{len}u\le m\},L_m(S_{\infty })=\{u\in S_{\infty }:\mathrm{len}u\le m\}.$$

$$⟨\phi ({\delta }_y),{\delta }_w^{\prime }⟩\ne 0\iff y=w.$$

$$\phi ({\delta }_w)=\phi ({\delta }_v)\phi ({\delta }_{t_j})=\left(\sum _{u\in E}{\alpha }_u{\delta }_u\right)({\delta }_p+{\gamma }_j{\delta }_{t_j})=\sum _{u\in E}{\alpha }_u{\delta }_{up}+{\alpha }_u{\gamma }_j{\delta }_{u{t}_j},$$

$$\mathrm{len}(up)\le \mathrm{len}(u)+1\le m\text{and}\mathrm{len}(u{t}_j)=\mathrm{len}(u)+1\le m$$

$$\phi ({\delta }_y)=\phi ({\delta }_u)\phi ({\delta }_z)=\sum _{s\in F}{\beta }_s{\delta }_{sp}+{\beta }_s{\gamma }_j{\delta }_{s{t}_j}.$$

$$⟨\phi ({\delta }_y),{\delta }_w^{\prime }⟩={\beta }_v{\gamma }_j\text{mathbbm}{1}_{t_j,x}=⟨\phi ({\delta }_u),{\delta }_v^{\prime }⟩{\gamma }_j\text{mathbbm}{1}_{t_j,x}.$$

$$0=⟨\phi \left(\sum _{u\in F}{\alpha }_u{\delta }_u\right),{\delta }_w^{\prime }⟩=\sum _{u\in F}{\alpha }_u⟨\phi ({\delta }_u),{\delta }_w^{\prime }⟩={\alpha }_w⟨\phi ({\delta }_w),{\delta }_w^{\prime }⟩,$$

Full text

To appear in Proceedings of the American Mathematical Society.

1. Introduction

Let $A$ be a unital algebra. We say that $A$ is finite (also called directly finite or Dedekind finite) if every left invertible element of $A$ is right invertible, and we say that $A$ is infinite otherwise. This notion originates in the seminal studies of projections in von Neumann algebras carried out by Murray and von Neumann in the 1930s. At the $22^{\text{nd}}$ International Conference on Banach Algebras and Applications, held at the Fields Institute in Toronto in 2015, Yemon Choi raised the following questions:

1. (1)

Let $A$ be a unital, finite normed algebra. Must its completion be finite? 2. (2)

Let $A$ be a unital, finite pre-C*-algebra. Must its completion be finite?

Choi also stated Question (1) in [7, Section 6].

A unital algebra $A$ is said to be stably finite if the matrix algebra $M_{n}(A)$ is finite for each $n\in\mathbb{N}$. This stronger form of finiteness is particularly useful in the context of $K$-theory, and so it has become a household item in the Elliott classification programme for C*-algebras. The notions of finiteness and stable finiteness differ even for C*-algebras, as was shown independently by Clarke [8] and Blackadar [4] (or see [5, Exercise 6.10.1]). A much deeper result is due to Rørdam [9, Corollary 7.2], who constructed a unital, simple C*-algebra which is finite (and separable and nuclear), but not stably finite.

We shall answer Question (2), and hence Question (1), in the negative by proving the following result:

Theorem 1.1**.**

There exists a unital, infinite C-algebra which contains a dense, unital, stably finite -subalgebra.

Let $A$ be a unital *-algebra. Then there is a natural variant of finiteness in this setting, namely we say that $A$ is -finite if whenever we have $u\in A$ satisfying $u^{*}u=1$, then $uu^{*}=1$. However, it is known (see, e.g., [10, Lemma 5.1.2]) that a C-algebra is finite if and only if it is *-finite, so in this article we shall not need to refer to *-finiteness again.

2. Preliminaries

Our approach is based on semigroup algebras. Let $S$ be a monoid, that is, a semigroup with an identity, which we shall usually denote by $e$. By an involution on $S$ we mean a map from $S$ to $S$, always denoted by $s\mapsto s^{*}$, satisfying $(st)^{*}=t^{*}s^{*}$ and $s^{**}=s\;(s,t\in S)$. By a *-monoid we shall mean a pair $(S,*)$, where $S$ is a monoid, and $*$ is an involution on $S$. Given a *-monoid $S$, the semigroup algebra $\mathbb{C}S$ becomes a unital *-algebra simply by defining $\delta_{s}^{*}=\delta_{s^{*}}\;(s\in S)$, and extending conjugate-linearly.

Next we shall recall some basic facts about free products of *-monoids, unital -algebras, and their C-representations.

Let $S$ and $T$ be monoids, and let $A$ and $B$ be unital algebras. Then we denote the free product (i.e. the coproduct) of $S$ and $T$ in the category of monoids by $S*T$, and similarly we denote the free product of the unital algebras $A$ and $B$ by $A*B$. It follows from the universal property satisfied by free products that, for monoids $S$ and $T$, we have $\mathbb{C}(S*T)\cong(\mathbb{C}S)*(\mathbb{C}T)$.

Given *-monoids $S$ and $T$, we can define an involution on $S*T$ by

[TABLE]

for $n\in\mathbb{N},s_{1}\in S,s_{2},\ldots,s_{n}\in S\setminus\{e\},t_{1},\ldots,t_{n-1}\in T\setminus\{e\},$ and $t_{n}\in T.$ The resulting *-monoid, which we continue to denote by $S*T$, is the free product in the category of *-monoids. We can analogously define an involution on the free product of two unital *-algebras, and again the result is the free product in the category of unital *-algebras. We then find that $\mathbb{C}(S*T)\cong(\mathbb{C}S)*(\mathbb{C}T)$ as unital *-algebras.

Let $A$ be a -algebra. If there exists an injective -homomorphism from $A$ into some C-algebra, then we say that $A$ admits a faithful C-representation. In this case, $A$ admits a norm such that the completion of $A$ in this norm is a C*-algebra, and we say that $A$ admits a C-completion*. Our construction will be based on C*-completions of *-algebras of the form $\mathbb{C}S$, for $S$ a *-monoid.

We shall denote by $S_{\infty}$ the free *-monoid on countably many generators; that is, as a monoid $S_{\infty}$ is free on some countably-infinite generating set ${\{t_{n},s_{n}:n\in\mathbb{N}\}}$, and the involution is determined by $t_{n}^{*}=s_{n}\>(n\in\mathbb{N})$. For the rest of the text we shall simply write $t_{n}^{*}$ in place of $s_{n}$. We define $BC$ to be the bicyclic monoid $\langle p,q:pq=e\rangle$. This becomes a *-monoid when an involution is defined by $p^{*}=q$, and the corresponding $*$-algebra $\mathbb{C}{}BC$ is infinite because $\delta_{p}\delta_{q}=\delta_{e}$, but $\delta_{q}\delta_{p}=\delta_{qp}\neq\delta_{e}.$

Lemma 2.1**.**

The following unital -algebras admit faithful C-representations:

1. (i)

$\mathbb{C}(BC),$** 2. (ii)

$\mathbb{C}(S_{\infty})$.

Proof.

(i) Since $BC$ is an inverse semigroup, this follows from [2, Theorem 2.3].

(ii) By [3, Theorem 3.4] $\mathbb{C}S_{2}$ admits a faithful C*-representation, where $S_{2}$ denotes the free monoid on two generators $S_{2}=\langle a,b\rangle$, endowed with the involution determined by $a^{*}=b$. There is a *-monomorphism $S_{\infty}\hookrightarrow S_{2}$ defined by $t_{n}\mapsto a(a^{*})^{n}a\ (n\in\mathbb{N})$ and this induces a *-monomorphism $\mathbb{C}S_{\infty}\hookrightarrow\mathbb{C}S_{2}$. The result follows. ∎

By a state on a unital *-algebra $A$ we mean a linear functional $\mu\colon A\rightarrow\mathbb{C}$ satisfying $\langle a^{*}a,\mu\rangle\geq 0\ (a\in A)$ and $\langle 1,\mu\rangle=1$. We say that a state $\mu$ is faithful if $\langle a^{*}a,\mu\rangle>0\ (a\in A\setminus\{0\})$. A unital -algebra with a faithful state admits a faithful C-representation via the GNS representation associated with the state.

The following theorem appears to be folklore in the theory of free products of $C^{*}$-algebras; it can be traced back at least to the seminal work of Avitzour [1, Proposition 2.3] (see also [6, Section 4] for a more general result).

Theorem 2.2**.**

Let $A$ and $B$ be unital $*$-algebras which admit faithful states. Then their free product $A*B$ also admits a faithful state, and hence it has a faithful C-representation.*

We make use of this result in our next lemma.

Lemma 2.3**.**

The unital -algebra $\mathbb{C}(BC*S_{\infty})$ admits a faithful C-representation.

Proof.

We first remark that a separable C*-algebra $A$ always admits a faithful state. To see this, note that the unit ball of $A^{*}$ with the weak*-topology is a compact metric space, and hence also separable. It follows that the set of states $S(A)$ is weak*-separable. Taking $\{\rho_{n}:n\in\mathbb{N}\}$ to be a dense subset of $S(A)$, we then define $\rho=\sum_{n=1}^{\infty}2^{-n}\rho_{n}$, which is easily seen to be a faithful state on $A$.

By Lemma 2.1, both $\mathbb{C}(BC)$ and $\mathbb{C}(S_{\infty})$ admit C*-completions. Since both of these algebras have countable dimension, their C*-completions are separable, and, as such, each admits a faithful state, which we may then restrict to obtain faithful states on $\mathbb{C}BC$ and $\mathbb{C}S_{\infty}$. By Theorem 2.2, $(\mathbb{C}BC)*(\mathbb{C}S_{\infty})\cong{\mathbb{C}(BC*S_{\infty})}$ admits a faithful C*-representation. ∎

3. Proof of Theorem 1.1

The main idea of the proof is to embed $\mathbb{C}S_{\infty}$, which is finite, as a dense -subalgebra of some C-completion of $\mathbb{C}(BC*S_{\infty})$, which will necessarily be infinite. In fact we have the following:

Lemma 3.1**.**

*The -algebra $\mathbb{C}S_{\infty}$ is stably finite.

Proof.

As we remarked in the proof of Lemma 2.1, $\mathbb{C}S_{\infty}$ embeds into $\mathbb{C}S_{2}$. It is also clear that, as an algebra, $\mathbb{C}S_{2}$ embeds into $\mathbb{C}F_{2}$, where $F_{2}$ denotes the free group on two generators. Hence $\mathbb{C}S_{\infty}$ embeds into $\operatorname{vN}(F_{2})$, the group von Neumann algebra of $F_{2}$, which is stably finite since it is a C*-algebra with a faithful tracial state. It follows that $\mathbb{C}S_{\infty}$ is stably finite as well. ∎

We shall next define a notion of length for elements of $BC*S_{\infty}$. Indeed, each $u\in{(BC*S_{\infty})\setminus\{e\}}$ has a unique expression of the form $w_{1}w_{2}\cdots w_{n}$, for some $n\in\mathbb{N}$ and some $w_{1},\ldots,w_{n}\in\left(BC\setminus\{e\}\right)\cup\{t_{j},t_{j}^{*}:j\in\mathbb{N}\},$ satisfying $w_{i+1}\in\{t_{j},t_{j}^{*}:j\in\mathbb{N}\}$ whenever $w_{i}\in BC\setminus\{e\}\ (i=1,\ldots,n-1)$. We then define ${\rm len\,}u=n$ for this value of $n$, and set ${\rm len\,}e=0$. This also gives a definition of length for elements of $S_{\infty}$ by considering $S_{\infty}$ as a submonoid of $BC*S_{\infty}$ in the natural way. For $m\in\mathbb{N}_{0}$ we set

[TABLE]

We now describe our embedding of $\mathbb{C}S_{\infty}$ into $\mathbb{C}(BC*S_{\infty}$). By Lemma 2.3, $\mathbb{C}(BC*S_{\infty})$ has a C*-completion $(A,\|\cdot\|)$. Let $\gamma_{n}=(n\|\delta_{t_{n}}\|)^{-1}\ (n\in\mathbb{N})$ and define elements $a_{n}$ in ${\mathbb{C}(BC*S_{\infty})}$ by $a_{n}=\delta_{p}+\gamma_{n}\delta_{t_{n}}\ (n\in\mathbb{N})$, so that $a_{n}\rightarrow\delta_{p}$ as $n\rightarrow\infty$. Using the universal property of $S_{\infty}$ we may define a unital *-homomorphism $\varphi\colon\mathbb{C}S_{\infty}\rightarrow\mathbb{C}(BC*S_{\infty})$ by setting $\varphi(\delta_{t_{n}})=a_{n}\ (n\in\mathbb{N})$ and extending to $\mathbb{C}S_{\infty}$. In what follows, given a monoid $S$ and $s\in S$, $\delta_{s}^{\prime}$ will denote the linear functional on $\mathbb{C}S$ defined by $\langle\delta_{t},\delta_{s}^{\prime}\rangle=\mathbbm{1}_{s,t}\quad(t\in S),$ where $\mathbbm{1}_{s,t}=1$ if $s=t$ and $\mathbbm{1}_{s,t}=0$ otherwise.

Lemma 3.2**.**

Let $w\in S_{\infty}$ with ${\rm len\,}w=m$. Then

1. (i)

$\varphi(\delta_{w})\in{\rm span\,}\{\delta_{u}:u\in L_{m}(BC*S_{\infty})\}$; 2. (ii)

for each $y\in L_{m}(S_{\infty})$ we have

[TABLE]

Proof.

We proceed by induction on $m$. When $m=0$, $w$ is forced to be $e$ and hence, as $\varphi$ is unital, $\varphi(\delta_{e})=\delta_{e}$, so that (i) is satisfied. In (ii), $y$ is also equal to $e$, so that (ii) is trivially satisfied as well.

Assume $m\geq 1$ and that (i) and (ii) hold for all elements of $L_{m-1}(S_{\infty})$. We can write $w$ as $w=vx$ for some $v\in S_{\infty}$ with ${\rm len\,}v=m-1$ and some $x\in\{t_{j},t_{j}^{*}:j\in\mathbb{N}\}$.

First consider (i). By the induction hypothesis, we can write $\varphi(\delta_{v})=\sum_{u\in E}\alpha_{u}\delta_{u}$, for some finite set $E\subset L_{m-1}(BC*S_{\infty})$ and some scalars $\alpha_{u}\in\mathbb{C}\ (u\in E)$. Suppose that $x=t_{j}$ for some $j\in\mathbb{N}$. Then

[TABLE]

which belongs to ${\rm span\,}\{\delta_{u}:u\in L_{m}(BC*S_{\infty})\}$ because

[TABLE]

for each $u\in L_{m-1}(BC*S_{\infty})$. The case $x=t_{j}^{*}$ is established analogously.

Next consider (ii). Let $y\in L_{m}(S_{\infty})$. If ${\rm len\,}y\leq m-1$ then, by (i), we know that $\varphi(\delta_{y})\in{\rm span\,}\{\delta_{u}:u\in L_{m-1}(BC*S_{\infty})\}\subset\ker\delta_{w}^{\prime}$. Hence in this case $y\neq w$ and $\langle\varphi(\delta_{y}),\delta_{w}^{\prime}\rangle=0$.

Now suppose instead that ${\rm len\,}y=m$, and write $y=uz$ for some ${u\in L_{m-1}(S_{\infty})}$ and $z\in\{t_{j},t_{j}^{*}:j\in\mathbb{N}\}$. By (i) we may write $\varphi(\delta_{u})=\sum_{s\in F}\beta_{s}\delta_{s}$ for some finite subset $F\subset L_{m-1}(BC*S_{\infty})$ and some scalars $\beta_{s}\in\mathbb{C}\ (s\in F)$, and we may assume that $v\in F$ (possibly with $\beta_{v}=0$). We prove the result in the case that $z=t_{j}$ for some $j\in\mathbb{N}$, with the argument for the case $z=t_{j}^{*}$ being almost identical. We have $\varphi(\delta_{z})=\delta_{p}+\gamma_{j}\delta_{t_{j}}$ and it follows that

[TABLE]

Observe that $sp\neq w$ for each $s\in F$. This is because we either have ${\rm len\,}(sp)<m={\rm len\,}(w)$, or else $sp$ ends in $p$ when considered as a word over the alphabet $\{p,p^{*}\}\cup\{t_{j},t_{j}^{*}:j\in\mathbb{N}\}$, whereas $w\in S_{\infty}$. Moreover, given $s\in F$, $st_{j}=w=vx$ if and only if $s=v$ and $t_{j}=x$. Hence

[TABLE]

As $\gamma_{j}>0$, this implies that $\langle\varphi(\delta_{y}),\delta_{w}^{\prime}\rangle\neq 0$ if and only if $\langle\varphi(\delta_{u}),\delta_{v}^{\prime}\rangle\neq 0$ and $t_{j}=x$, which, by the induction hypothesis, occurs if and only if $u=v$ and $t_{j}=x$. This final statement is equivalent to $y=w$. ∎

Corollary 3.3**.**

The map $\varphi$ is injective.

Proof.

Assume towards a contradiction that $\sum_{u\in F}\alpha_{u}\delta_{u}\in\ker\varphi$ for some non-empty finite set $F\subset S_{\infty}$ and $\alpha_{u}\in\mathbb{C}\setminus\{0\}\ (u\in F)$. Take $w\in F$ of maximal length. Then

[TABLE]

where the final equality follows from Lemma 3.2(ii). That lemma also tells us that $\langle\varphi(\delta_{w}),\delta_{w}^{\prime}\rangle\neq 0$, forcing $\alpha_{w}=0$, a contradiction. ∎

We can now prove our main theorem.

Proof of Theorem 1.1.

Recall that $(A,\|\cdot\|)$ denotes a C*-completion of $\mathbb{C}(BC*S_{\infty})$, which exists by Lemma 2.3, and $A$ is infinite since $\delta_{p},\delta_{q}\in A$. Let $A_{0}\subset A$ be the image of $\varphi$. Corollary 3.3 implies that $A_{0}\cong\mathbb{C}S_{\infty}$, which is stably finite by Lemma 3.1. Moreover, $\varphi(\delta_{t_{n}})=a_{n}\rightarrow\delta_{p}$ as $n\rightarrow\infty$, so that $\delta_{p}\in\overline{A_{0}},$ and we see also that $\delta_{t_{n}}=\frac{1}{\gamma_{n}}(a_{n}-\delta_{p})\in\overline{A_{0}}\ (n\in\mathbb{N}).$ The elements $\delta_{p}$ and $\delta_{t_{n}}\ (n\in\mathbb{N})$ generate $A$ as a C*-algebra, and since $\overline{A_{0}}$ is a C*-subalgebra containing them, we must have $A=\overline{A_{0}}$, which completes the proof. ∎

Acknowledgements

We are grateful to Yemon Choi for several helpful discussions whilst working on this problem. We are also grateful to the referee for his/her useful comments, in particular those concerning the references for Theorem 2.2.