Non-surjective pullbacks of graph C*-algebras from non-injective pushouts of graphs

TL;DR

This paper identifies a broad class of graph C*-algebra homomorphisms whose pullbacks are AF algebras, providing a method to analyze quantum spaces via graph modifications, with applications to noncommutative topology.

Contribution

It introduces a new class of pullback constructions for graph C*-algebras that yield AF algebras and extends the results to more general graph C*-algebras involving sinks.

Findings

Pullback C*-algebras are AF in a substantial class of cases.

The results apply to quantum spaces like quantum spheres and teardrops.

An extension of the theorem to sink-extended graph C*-algebras is provided.

Abstract

We find a substantial class of pairs of $*$-homomorphisms between graph C*-algebras of the form $C^*(E)\hookrightarrow C^*(G)\twoheadleftarrow C^*(F)$ whose pullback C*-algebra is an AF graph C*-algebra. Our result can be interpreted as a recipe for determining the quantum space obtained by shrinking a quantum subspace. There is a variety of examples from noncommutative topology, such as quantum complex projective spaces (including the standard Podle\'s quantum sphere) or quantum teardrops, that instantiate the result. Furthermore, to go beyond AF graph C*-algebras, we consider extensions of graphs over sinks and prove an analogous theorem for the thus obtained graph C*-algebras.

Equations

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$$S_e^{\ast }{S}_f$$

$$\sum _{e\in s^{-1}(v)}{S}_e{S}_e^{\ast }$$

$$S_e{S}_e^{\ast }$$

$$S_{\alpha }^{\ast }{S}_{\beta }\ne 0⟺\left(\alpha ⪯\beta \text{or}\beta ⪯\alpha \right).$$

$$S_{\alpha }^{\ast }{S}_{\beta }=S_{\gamma }^{\ast }{S}_{\beta }^{\ast }{S}_{\beta }=S_{\gamma }^{\ast }\ne 0.$$

$$\alpha :=x_1\dots x_m,\beta :=y_1\dots y_r,x_1,\dots ,x_m,y_1,\dots ,y_r\in E^1.$$

$$0\ne S_{\alpha }^{\ast }{S}_{\beta }=S_{x_m}^{\ast }\dots S_{x_1}^{\ast }{S}_{y_1}\dots S_{y_r}$$

$$S_{\alpha }{S}_{\alpha }^{\ast }\le P_v\in C^{\ast }(E).$$

$$S_{\alpha }{S}_{\alpha }^{\ast }=S_{\beta }(S_e{S}_e^{\ast })S_{\beta }^{\ast }\le S_{\beta }{P}_w{S}_{\beta }^{\ast }=S_{\beta }{S}_{\beta }^{\ast },$$

$$v$$

$$w_1$$

$$1$$

$$v_1$$

$$I_H=\overline{\mathrm{span}}\{S_{\alpha }{S}_{\beta }^{\ast }\text{leavevmode}|\text{leavevmode}\alpha ,\beta \in \mathrm{Path}(E),r(\alpha )=r(\beta )\in H\}.$$

$$C^{\ast }(E)/I_H\cong C^{\ast }(E/H),\text{where}E/H:=(E^0\setminus H,r^{-1}(E^0\setminus H),s_H,r_H)$$

$$f^0\circ s_D=s_E\circ f^1,f^0\circ r_D=r_E\circ f^1.$$

$$C^{\ast }(D)\cong C^{\ast }(E/(E^0\setminus D^0)).$$

$$f(\alpha )⪯f(\beta )\Rightarrow \alpha ⪯\beta ;$$

$$s_F^{-1}(v)⟶s_E^{-1}(f(v)).$$

$$\forall v\in F^0:f_{\ast }(P_v):=P_{f(v)}\text{and}\forall x\in F^1:f_{\ast }(S_x):=S_{f(x)}.$$

$$P_{f(v)}{P}_{f(w)}={\delta }_{f(v),f(w)}{P}_{f(v)}={\delta }_{v,w}{P}_{f(v)}.$$

$$S_{f(e_1)}^{\ast }{S}_{f(e_2)}\ne 0\Rightarrow e_1=e_2,$$

$$C^{\ast }(E_1)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\ker {\pi }_2\subseteq f_{\ast }(\ker {\pi }_1).$$

$$\ker {\pi }_1=\overline{\mathrm{span}}\left\{S_{\alpha }{S}_{\beta }^{\ast }\text{leavevmode}|\text{leavevmode}\alpha ,\beta \in \mathrm{Path}(E_1),\text{leavevmode}r(\alpha )=r(\beta )\in E_1^0\setminus F_1^0\right\},$$

$$\ker {\pi }_2=\overline{\mathrm{span}}\left\{S_{\gamma }{S}_{\delta }^{\ast }\text{leavevmode}|\text{leavevmode}\gamma ,\delta \text{leavevmode}\in \mathrm{Path}(E_2),\text{leavevmode}r(\gamma )=r(\delta )\in E_2^0\setminus F_2^0\right\}.$$

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$$C^{\ast }(E)\underset{C^{\ast }(X)}{\bullet }{C}^{\ast }(H):=(C^{\ast }(E)\underset{C^{\ast }(X)}{\ast }{C}^{\ast }(H))/⟨P_v{P}_w|v\in E^0\setminus {\iota }_E(X),w\in H^0\setminus {\iota }_H(X)⟩.$$

Full text

\textstyle{C^{*}(E)\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}(G)}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces C^{*}(F)}

whose pullback C*-algebra is an AF graph C*-algebra. Our result can be interpreted as a recipe for determining the quantum space obtained by shrinking a quantum subspace. There is a variety of examples from noncommutative topology, such as quantum complex projective spaces (including the standard Podleś quantum sphere) or quantum teardrops, that instantiate the result. Furthermore, to go beyond AF graph C*-algebras, we consider extensions of graphs over sinks and prove an analogous theorem for the thus obtained graph C*-algebras.

1. Introduction

The classical two-sphere $S^{2}$ can be obtained by shrinking the boundary of the disc $B^{2}$ to a point. In other words, there is a pushout diagram in the category of topological spaces

[TABLE]

Due to the contravariant duality of algebras and spaces, the diagram (1.1) amounts to an isomorphism $C(S^{2})\cong C(B^{2})\oplus_{C(S^{1})}\mathbb{C}$ of C*-algebras of complex-valued continuous functions on the two-sphere and the pushout $B^{2}\sqcup_{S^{1}}\{\ast\}$ respectively.

At the same time, the Toeplitz algebra $\mathcal{T}$ [6] can be viewed as a noncommutative deformation of $C(B^{2})$ (see [13, Theorem IV.7]). Therefore, the C*-algebra $C(S^{2}_{q0})$ of the standard Podleś sphere [16, (3a)] provides a noncommutative deformation of the diagram (1.1), namely we have the following pullback diagram in the category of C*-algebras

[TABLE]

The aim of this paper is to generalize the above pullback construction using the concept of a C*-algebra $C^{*}(E)$ of a directed graph $E$ (e.g., see [3]). Graph C*-algebras provide powerful tools in noncommutative topology, and many C*-algebras representing noncommutative deformations of topological spaces are isomorphic with C*-algebras of graphs [5, 11, 12]. These isomorphisms are usually quite complicated and they do not depend on the deformation parameter. Nevertheless, when such an isomorphism is established, it is easier to obtain solutions to many problems, especially concerning K-theory.

Our starting point is that all the C*-algebras in the diagram (1.2) can be viewed as C*-algebras of graphs. We present this pictorially as follows:

[TABLE]

(See the examples in Section 2 for details.)

The graph-algebraic decomposition (1.3) manifests a certain general phenomenon that can be explained in terms of non-injective pushouts of graphs. The goal of this paper is to explore this phenomenon to arrive at a general setting. To this end, we search for a new concept of morphisms of graphs, so as to ensure that, in the thus defined category of graphs, the assignment of graph algebras to graphs becomes a contravariant functor translating pushouts of graphs into pullbacks of graphs algebras. While this task seems to be completed in [10] (cf. [14, Corollary 3.4]) for injective pushouts of row-finite graphs (each vertex emits only finitely many edges), herein we handle a non-injective case without row-finiteness assumption.

To accommodate this naturally occuring non-injectivity, we replace the standard idea of mapping vertices to vertices and edges to edges by the more flexible idea of mapping finite paths to finite paths. We arrive at a general result for a class of unital AF graph C*-algebras including the standard Podleś sphere, complex quantum projective spaces [18, Definition on p. 109], and quantum teardrops [4]. Finally, we go beyond AF graph C*-algebras by extending their acyclic graphs over sinks.

2. Graph-algebraic preliminaries

A directed graph $E$ is a quadruple $(E^{0},E^{1},s,r)$, where $E^{0}$ is the set of vertices, $E^{1}$ is the set of edges (arrows), and $s,r:E^{1}\to E^{0}$ are the source map and the range (target) map respectively. Throughout the paper, we consider only directed graphs with countable sets of vertices and edges, and we will often simply refer to them as graphs.

Definition 2.1** *(Graph C-algebra).

The graph C-algebra $C^{*}(E)$ of a directed graph $E$ is the universal C*-algebra generated by mutually orthogonal projections \big{\{}P_{v}\;|\;v\in E^{0}\big{\}} and partial isometries \big{\{}S_{e}\;|\;e\in E^{1}\big{\}} satisfying the following conditions:*

[TABLE]

A vertex $v$ in $E$ is called a sink if and only if $s^{-1}(v)=\varnothing$. A vertex is called regular iff it is not a sink and it emits finitely many edges. A graph is called row finite iff all its vertices are either regular or sinks. By a finite path in $E$ we mean a sequence $(e_{1},\ldots,e_{n})$ of edges satisfying $r(e_{i})=s(e_{i+1})$ for all $i\in\{1,\ldots,n-1\}$. The length of a path is the number of edges in the sequence. We consider vertices as paths of length zero, and denote the set of finite paths by ${\rm Path}(E)$. The notation $S_{\alpha}$, along with the source and the range map, naturally extend to any $\alpha\in{\rm Path}(E)$. As we consider only finite paths throughout this paper, we will simply refer to them as paths.

A path $\alpha$ is called a loop if and only if $s(\alpha)=r(\alpha)$ and $\alpha$ is not a vertex. We say that a loop is short iff it is an edge.

Definition 2.2**.**

We call a path pointed iff its final edge is not a loop.

We say that a path $\alpha$ is a prolongation of a path $\beta$ if and only if $\alpha=\beta\gamma$ for a path $\gamma\in{\rm Path}(E)$ such that $r(\beta)=s(\gamma)$. We write $\beta\preceq\alpha$ when $\alpha$ is a prolongation of $\beta$. Observe that $\preceq$ gives a partial order on ${\rm Path}(E)$.

Lemma 2.3**.**

Let $\alpha$ and $\beta$ be finite paths in an arbitrary graph $E$. Then

[TABLE]

Proof.

Assume that $\beta\preceq\alpha$, i.e. that $\alpha=\beta\gamma$ with $r(\beta)=s(\gamma)$. Then, as $S^{*}_{\gamma}$ is an element of a linear basis of $C^{*}(E)$ for any $\gamma\in{\rm Path}(E)$ [1, Corollary 1.5.12], we obtain

[TABLE]

Much in the same way, we see that $S^{*}_{\alpha}S_{\beta}\neq 0$ when $\alpha\preceq\beta$.

Conversely, assume that $S^{*}_{\alpha}S_{\beta}\neq 0$ for some finite paths

[TABLE]

If $m\geq r$, then

[TABLE]

and (GA1) imply that $x_{i}=y_{i}$ for $i=1,\ldots,r$. This means that $\beta\preceq\alpha$. Otherwise, when $r\geq m$, we get that $\alpha\preceq\beta$. ∎

Next, to make the condition (GA3) easier to check, we prove the following lemma:

Lemma 2.4**.**

Let $E$ be an arbitrary graph and $\alpha$ a path in $E$ with its origin at $v\in E^{0}$. Then

[TABLE]

Proof.

Write $\alpha=\beta e$, where $e$ is an edge with its origin at $w\in E^{0}$ and $\beta$ is an initial subpath of $\alpha$ ending at $w$. Then

[TABLE]

where the middle inequality is due to (GA3). Now, the claim follows by the induction on the length of $\alpha$. ∎

To get ready for examples in the last section, we present graph-algebraic presentations of some well-known C*-algebras.

Example 2.5**.**

The algebra $\mathbb{C}$ of complex numbers is isomorphic with the graph C-algebra of the graph with one vertex and no edges.*

[TABLE]

Example 2.6**.**

The C-algebra $C(S^{1})$ of all continuous complex-valued functions on the circle is the universal unital C*-algebra generated by a single unitary $u$. It is isomorphic with the graph C*-algebra of the graph given below through the isomorphism given by $u\mapsto S_{e}$.*

[TABLE]

Example 2.7**.**

The Toeplitz algebra $\mathcal{T}$ [6] is the universal unital C-algebra generated by a single isometry $s$. It is isomorphic with the graph C*-algebra of the graph given below through the isomorphism given by $s\mapsto S_{t_{1}}+S_{t_{2}}$.*

[TABLE]

Example 2.8**.**

The Cuntz algebra $\mathcal{O}_{m}$ [7] is the universal unital C-algebra generated by isometries $s_{1}$, $\ldots$, $s_{m}$ subject to the relation $\sum_{i=1}^{m}s_{i}s^{*}_{i}=1$. It is isomorphic with the graph C*-algebra of the graph $R_{m}$ given below through the isomorphism given by $s_{i}\mapsto S_{e_{i}}$.*

[TABLE]

Example 2.9**.**

Let $q\in[0,1)$. The C*-algebra $C(S^{2}_{q0})$ [16, (3a)] of the standard Podleś quantum sphere coincides with the C*-algebra of the Vaksman–Soibelman quantum complex projective line $C(\mathbb{C}{\rm P}^{1}_{q})$ [18, p. 109], which has a graph-algebraic presentation as the graph C*-algebra of the graph given below (see [11, Section 2.3]):*

[TABLE]

Here the arrow decorated by $(\infty)$ denotes countably infinitely many arrows.**

We end this section by recalling some standard results that we will use throughout the paper. Let $E$ be a directed graph. A subset $H\subseteq E^{0}$ is called hereditary iff, for any $v\in H$ such that there is a path starting at $v$ and ending at $w\in E^{0}$, we have $w\in H$. If $H$ is hereditary, then the ideal $I_{H}$ generated by the projections associated with the elements of $H$ is of the form (cf. the equation (1) in [3]):

[TABLE]

Here $\overline{\rm span}$ denotes the closed linear span.

Assume additionally that there are no vertices that emit infinitely many arrows into $H$ and finitely many (but not zero) arrows outside of $H$. Assume also that $H$ is saturated, i.e. that there does not exist a regular vertex $v\notin H$ such that $r(s^{-1}(v))\subseteq H$. Then, the quotient algebra $C^{*}(E)/I_{H}$ is again a graph C*-algebra (cf. the discussion below the equation (1) in [3]):

[TABLE]

and $s_{H}$ and $r_{H}$ are the restrictions-corestrictions of $s$ and $r$ respectively.

3. Non-surjective pullbacks of graph C*-algebras

In this section we prove a non-surjective pullback theorem generalizing the diagram (1.2). First, we need some preliminaries on graphs and their morphisms.

Let $D=(D^{0},D^{1},s_{D},r_{D})$ and $E=(E^{0},E^{1},s_{E},r_{E})$ be directed graphs. A morphism of graphs $f:D\to E$ is a pair of mappings $f^{0}:D^{0}\to E^{0}$ and $f^{1}:D^{1}\to E^{1}$ satisfying

[TABLE]

If there is an injective morphism of graphs $D\to E$, we say that $D$ is a subgraph of $E$ and write $D\subseteq E$.

Definition 3.1**.**

An injective graph morphism $\iota:D\to E$ is called an admissible inclusion iff the following conditions are satisfied:

1. (A1)

$E^{0}\setminus\iota^{0}(D^{0})$* is hereditary and saturated,* 2. (A2)

$\iota^{1}(D^{1})=r_{E}^{-1}(\iota^{0}(D^{0}))$, 3. (A3)

no vertex in $E^{0}$ emits infinitely many edges into $E^{0}\setminus D^{0}$ while emitting finitely many (but not zero) edges into $D^{0}$.

Next, let us state the following elementary fact (cf. (2.7) and the discussion preceding it).

Proposition 3.2**.**

Let $D\subseteq E$ be an admissible inclusion. Then, we have an isomorphism of graph C-algebras*

[TABLE]

To phrase our main result, it is convenient to view graphs as small categories whose objects are vertices and morphisms are finite paths. Then functors between such categories are what we want as morphisms between graphs. Using the thus understood functors as morphisms, we generalize the Cuntz–Krieger graph category [8, p. 172] (cf. [1, Definition 1.6.2]) by allowing egdes to be mapped to finite paths intead of only edges.

Lemma 3.3**.**

Let $f\colon F\to E$ be a functor between graphs such that:

1. (1)

$f$* is compatible with the prolongation relation as follows*

[TABLE] 2. (2)

for any vertex $v$ that emits at least one and at most finitely many edges, $f$ restricts-corestricts to a bijection

[TABLE]

Then $f$ induces a $*$-homomorphism $f_{*}\colon C^{*}(F)\to C^{*}(E)$ given by

[TABLE]

Proof.

Since graph C*-algebras are universal, it suffices to show that all defining relations are preserved. For starters, since the condition (1) implies the injectivity of $f$, we infer that the set of mutually orthogonal projections is sent to the set of mutually orthogonal projections:

[TABLE]

Next, to show that (GA1) is preserved, it suffices to prove the implication

[TABLE]

which follows from combining Lemma 2.3 with the condition (1). Finally, showing that (GA2) and (GA3) are preserved is also straightforward: the former follows directly from the condition (2) and the latter from Lemma 2.4. ∎

We are now ready to prove our first main result:

Theorem 3.4**.**

Let $F_{i}\subseteq E_{i}$, $i=1,2$, be admissible inclusions of graphs such that

1. (1)

$E_{1}$* has no loops, $E_{2}$ has no short loops at vertices in $E_{2}^{0}\setminus F_{2}^{0}$, and $E^{0}_{1}=E^{0}_{2}$, $F^{0}_{1}=F^{0}_{2}$;* 2. (2)

there is a functor $f\colon E_{1}\to E_{2}$ such that: it satisfies the condition (1) in Lemma 3.3, it is $\mathrm{id}$ on objects, and its image is the set of all pointed paths.

Then the induced $*$-homomorphisms exist and render the diagram

[TABLE]

a pullback diagram of C-algebras. (If $E_{1}^{0}$ is finite, then this is a pullback diagram of unital C*-algebras.) Here $\pi_{1}$ and $\pi_{2}$ are the canonical surjections (3.2), $f_{*}$ is a $*$-homomorphism of Lemma 3.3, and $f|_{*}$ is its restriction-corestriction.*

Proof.

We begin by proving that $f_{*}$ and $f|_{*}$ are well-defined injective $*$-homomorphisms. To see that $f_{*}$ is well defined, by Lemma 3.3 and the assumption (2), it suffices to check the condition (2) of Lemma 3.3. To this end, take any regular vertex $v\in E^{0}_{1}$ and any edge $e\in s^{-1}(v)$. Then, as the image of $f$ is the set of pointed paths, $f(e)$ is a pointed path from $v$ to $r(e)$.

Suppose that $f(e)$ factorizes through a third vertex $w$. Then we can write $f(e)=\alpha\beta$, where $\alpha$ is a pointed path from $v$ to $w$ and $\beta$ is a pointed path from $w$ to $r(e)$. Indeed, deleting any intial subpath from a pointed path always yields a pointed path, and making all loops based at $w$ part of $\beta$ makes $\alpha$ a pointed path. Furthermore, as $f$ is surjective on the set of pointed paths, we can write $f(e)=\alpha\beta=f(\alpha^{\prime})f(\beta^{\prime})=f(\alpha^{\prime}\beta^{\prime})$. Combining it with the injectivity of $f$, which follows from the condition (1) in Lemma 3.3, we get a contradiction $e=\alpha^{\prime}\beta^{\prime}$ (the edge $e$ is not a path factorizing through the vertex $w$). Hence $f(e)$ is a pointed path from $v$ to $r(e)$ that does not factorize through any third vertex.

If there is a loop in $E_{2}$ based at $v$, then there are infinitely many non-factorizing pointed paths from $v$ to $r(e)$, and (because $f$ is a functor) none of them can be the image of a path that factorizes through a third vertex. Consequently, as $f$ is bijective when corestricted to the set of pointed paths, and there are no loops in $E_{1}$, there must be infinitely many edges in $E_{1}$ from $v$ to $r(e)$, which contradicts the assumption that $v$ is a regular vertex in $E_{1}$. Hence, there is no loop in $E_{2}$ based at $v$, so $f(e)$ is an edge.

Next, if $f(\alpha)\in E^{1}_{2}$, then $\alpha\in E^{1}_{1}$ because $f$ is an injective functor that is $\mathrm{id}$ on the set of veritices. Indeed, suppose that $\alpha=e_{1}\dots e_{n}$, where $e_{i}$’s are edges. Then $f(\alpha)=f(e_{1}\dots e_{n})=f(e_{1})\dots f(e_{n})$ is of length at least $n$, as $f(e_{i})$ cannot be a vertex. Hence $n=1$, i.e. $\alpha$ is an edge, so any edge emitted from $v$ in $E_{2}$ comes from an edge emitted from $v$ in $E_{1}$. Combining this with the injectivity of $f$ and the above established fact that $f(e)$ is an edge, we conclude that the condition (2) in Lemma 3.3 is satisfied.

Thus we obtain a well-defined $*$-homomorphism $f_{*}$ that is injective by [17, Corollary 1.3] because $E_{1}$ has no loops. Furthermore, by the admissibility condition Definition 3.1(A2), it is clear that $f$ restricted to the subgraph $F_{1}$ corestricts to $F_{2}$ yielding a restriction-corestriction $f|_{*}$ of $f_{*}$. The $*$-homomorphism $f|_{*}$ is injective because $f_{*}$ is injective.

It is straightforward to check that the maps $\pi_{1}$, $\pi_{2}$, $f_{*}$ and $f|_{*}$ make the diagram (3.5) commutative. Therefore, as $\pi_{1}$ and $\pi_{2}$ are surjective and $f_{*}$ and $f|_{*}$ are injective, due to [15, 3.1 Proposition], to show that (3.5) is a pullback diagram, it suffices to prove that

[TABLE]

To obtain the above inclusion, we use the characterization of ideals associated to hereditary subsets (2.6):

[TABLE]

By the assumption (1), all paths in $E_{2}$ terminating in $E_{2}^{0}\setminus F_{2}^{0}$ are pointed, so they are in the image of $f$. Therefore, as $E_{2}^{0}\setminus F_{2}^{0}=E_{1}^{0}\setminus F_{1}^{0}$ by the assumption (1), we conclude that the inclusion (3.6) holds at the algebraic level. Finally, as any $*$-homomorphism between C*-algebras is a continuous map whose image is closed, we infer the desired inclusion at the C*-level. ∎

4. Extending graphs over sinks

To generalize the diagram (1.2) even further (e.g. to allow loops in $E_{1}$ in the pullback theorem of the previous section), we first need to determine suitable conditions under which the graph-algebra construction preserves pushouts of graphs over sinks.

The general setup assumptions (GS) are as follows:

• •

$E$ and $H$ are graphs;

• •

$X$ is a set regarded as a graph with no edges;

• •

$\iota_{E}:X\to E^{0}$ and $\iota_{H}:X\to H^{0}$ are injective maps defining the pushout

[TABLE]

• •

$E\sqcup_{X}H:=(E^{0}{\sqcup}_{X}H^{0},E^{1}\sqcup H^{1},\pi\circ(s_{E}\sqcup s_{H}),\pi\circ(r_{E}\sqcup r_{H}))$, where $\pi$ is the canonical quotient map.

Next, let $\iota_{E*}:C^{*}(X)\to C^{*}(E)$ and $\iota_{H*}:C^{*}(X)\to C^{*}(H)$ be the induced $*$-homomorphisms (see Lemma 3.3). Define

[TABLE]

Here we divide the amalgamated free product by the ideal generated by the product of non-identified projections.

Lemma 4.1**.**

Assume that at least one of the maps $\iota_{H}$ and $\iota_{E}$ takes its values in the sinks of the respective graph. Then the natural assignment of elements defines an isomorphism of C-algebras:*

[TABLE]

Proof.

Since $\iota_{E}:X\to E^{0}$ or $\iota_{H}:X\to H^{0}$ takes values in the sinks of $E$ or $H$, respectively, all edge relations in $C^{*}(E\sqcup_{X}H)$ involving vertices in the image of $X$ are of one of two types: either they refer to edges only in $E^{1}$, or to edges only in $H^{1}$. Hence, there are $*$-homomorphisms

[TABLE]

given the natural assignment of elements. Furthermore, as $(E\sqcup_{X}H)^{0}=E^{0}\sqcup_{X}H^{0}$, they induce a surjective $*$-homomorphism

[TABLE]

Finally, as the kernel of $\pi_{\sqcup}$ coincides with the kernel of the defining surjection

[TABLE]

the claim follows. ∎

Now, consider three graphs $E_{1}$, $E_{2}$ and $H$ with injective maps $\iota_{E_{1}}\!:\!X\to E_{1}^{0}$, $\iota_{E_{2}}\!:\!X\to E_{2}^{0}$, and $\iota_{H}\!:\!X\to H^{0}$. Assume also that $\iota_{E_{1}}(X)$ and $\iota_{E_{2}}(X)$ consist of sinks of the two respective graphs $E_{1}$ and $E_{2}$. Now, consider a C*-algebra homomorphism

[TABLE]

annihilating the vertex projections of $\iota_{E_{1}}(X)\subseteq E_{1}^{0}$. Then $\delta$ and the zero map $C^{*}(H)\to A$ induce a $*$-homomorphism on the amalgamated product that annihilates the kernel of $\pi_{\bullet}$. Hence, by Lemma 4.1, $\delta$ extends to

[TABLE]

Lemma 4.2**.**

Let $j^{1}_{H}\colon C^{*}(H)\to C^{*}(E_{1}\sqcup_{X}H)$ be the map defined in (4.3). Then

[TABLE]

Proof.

The inclusion $\ker\delta^{\prime}\supseteq\langle j_{E_{1}}(\ker\delta)\rangle+\langle j^{1}_{H}(C^{*}(H))\rangle$ is clear by the construction of $\delta^{\prime}$. For the other inclusion, note that, as $\delta^{\prime}$ annihilates $C^{*}(H)$, it factors as

[TABLE]

where the last map is induced by $\delta:C^{*}(E_{1})\to A$. The inclusion follows from this factorization. Finally, as

[TABLE]

we infer that

[TABLE]

which ends the proof. ∎

Assume now that, under the general setup assumptions (GS) and the assumptions preceding (4.6), we have a pullback diagram

[TABLE]

of $*$-homomorphism of C*-algebras. Here $\phi$ is an injective $*$-homomorphism sending all vertex projections to vertex projections and all partial isometries associated with edges to partial isometries associated with paths, intertwining $\iota_{E_{1}}$ with $\iota_{E_{2}}$, and sending the projections labeled by $E_{1}^{0}\setminus\iota_{E_{1}}(X)$ to projections labeled by $E_{2}^{0}\setminus\iota_{E_{2}}(X)$. We also assume that all partial isometries in $C^{*}(E_{2})$ associated with paths ending in $\iota_{E_{2}}(X)$ are in the image of $\phi$. Furthermore, we assume that $\delta$ is a $*$-homomorphism annihilating the vertex projections labeled by $\iota_{E_{1}}(X)$, and $A$ and $B$ are arbitrary C*-algebras fitting into the pullback diagram (4.11) for some $*$-homomorphisms $\theta$ and $\rho$.

Note that the assumptions made on $\phi$ allow us to define its extension

[TABLE]

Indeed, we can use the isomorphism (4.2) and observe that the conditions on $\phi$ allow us to extend it by $\mathrm{id}\colon C^{*}(H)\to C^{*}(H)$ to

[TABLE]

This brings us to the second main result of the paper:

Theorem 4.3**.**

Under the general setup assumptions (GS) and the additional assumptions preceding (4.6), the pullback diagram (4.11) of $*$-homomorphisms of C-algebras induces the following pullback diagram of $*$-homomorphisms of C*-algebras:*

[TABLE]

Here $\delta^{\prime}$ and $\theta^{\prime}$ are defined by (4.7), and $\psi$ is defined by (4.13).

Proof.

The commutativity of the diagram (4.14) is immediate by construction. To prove that it is a pullback diagram, first we establish the injectivity of $\psi$. It follows from: the injectivity of $\phi$, the assumption that $\phi$ does not annihilate vertex projections, the fact that loops without exit in $E_{1}\sqcup_{X}H$ remain loops without exit both in $E_{1}$ and $H$, and the general Cuntz–Krieger uniqueness theorem [17, Theorem 1.2].

Next, using the injectivity of $\psi$ and appealing to [15, 3.1 Proposition], we note that to conclude the proof of the theorem, it suffices to check the following two conditions:

[TABLE]

The first condition is immediate from our assumption that (4.11) is a pullback. Indeed, the analogous equation holds for the unprimed maps $\theta$ and $\delta$, and the images of these maps coincide with those of $\theta^{\prime}$ and $\delta^{\prime}$ respectively because the primed maps are obtained from the unprimed maps by extending them by the zero map on $C^{*}(H)$.

To show the second condition, we apply Lemma 4.2 and (4.10) to obtain:

[TABLE]

Here both $j^{1}_{H}$ and $j^{2}_{H}$ are defined as in (4.3). Now, since (4.11) is a pullback diagram, we have $\ker\theta\subseteq\phi(\ker\delta)$. Furthermore, it follows from the construction of $\psi$ and $\delta^{\prime}$ that

[TABLE]

Hence $j_{E_{2}}(\ker\theta)\subseteq\psi(\ker\delta^{\prime})$, so, by (4.18), we are left having to argue that

[TABLE]

To this end, note first that

[TABLE]

because $j^{1}_{H}(C^{*}(H))\subseteq\ker\delta^{\prime}$ and $\psi\circ j^{1}_{H}=j^{2}_{H}$. Furthermore, observe that

[TABLE]

and

[TABLE]

Next, since all partial isometries in $C^{*}(E_{2})$ associated with paths ending in $\iota_{E_{2}}(X)$ are in the image of $\phi$ by assumption, we conclude that

[TABLE]

Indeed, it boils down to showing that

[TABLE]

Since any graph C*-algebra is the closed linear span of elements of the form $S_{\alpha}S^{*}_{\beta}$ with $r(\alpha)=r(\beta)$ (see [1, Corollary 1.5.12]), we are looking at $\alpha,\beta\in\mathrm{Path}(E_{2}\sqcup_{X}H)$ such that $P_{v}S_{\alpha}S^{*}_{\beta}$ or $S_{\alpha}S^{*}_{\beta}P_{v}$ can be non-zero. This means that $s(\alpha)=v$ or $s(\beta)=v$. However, as $v$ is a sink in $E_{2}$, we infer that $\alpha\in\mathrm{Path}(H)$ or $\beta\in\mathrm{Path}(H)$. Hence $\alpha\in\mathrm{Path}(H)$ and $r(\beta)\in H^{0}$, or $\beta\in\mathrm{Path}(H)$ and $r(\alpha)\in H^{0}$. Furthermore, any $E_{2}$-subpath $\gamma$ of any path ending in $H^{0}$ has to end in $\iota_{E_{2}}(X)$. Consequently, $S_{\gamma}\in\phi(C^{*}(E_{1}))$, so $j_{E_{2}}(S_{\gamma})\in\psi(C^{*}(E_{1}\sqcup_{X}H))$. As $j^{2}_{H}(S_{\delta})\in\psi(C^{*}(E_{1}\sqcup_{X}H))$ for any $\delta\in\mathrm{Path}(H)$, we conclude that all elements $S_{\alpha}S^{*}_{\beta}$ that can multiply nontrivially with $P_{v}$ are in the image of $\psi$, which proves (4.25).

Now, taking advantage of the assumption that $\delta(P_{v})=0$ for any $v\in\iota_{E_{1}}(X)$, we infer that

[TABLE]

Hence, by (4.17),

[TABLE]

Consequently, combining (4.23), (4.24) and (4.27), we arrive at

[TABLE]

which, together with (4.21) and (4.22) proves (4.20). ∎

5. Examples and applications

This section is devoted to the study of special cases of Theorem 3.4 and Theorem 4.3 leading to interesting examples in noncommutative topology.

5.1. The standard Podleś quantum sphere

Observe that the assumptions of Theorem 3.4 are true for the standard Podleś quantum sphere. Here $C^{*}(E_{1})=C(S^{2}_{q0})$, $C^{*}(F_{1})=\mathbb{C}$, $C^{*}(E_{2})=\mathcal{T}$ and $C^{*}(F_{2})=C(S^{1})$ (see the diagram (1.3)).

Our next example generalizes a simple gluing construction in topology. Recall that the real projective plane $\mathbb{R}{\rm P}^{2}$ may be represented as a closed hemisphere with the antipodal points on the equator identified. If we further identify all those antipodal points, we obtain the sphere $S^{2}$. Here we present a $q$-deformed analog of this procedure.

The C*-algebra $C(\mathbb{R}{\rm P}^{2}_{q})$ of the quantum real projective plane $\mathbb{R}{\rm P}^{2}_{q}$ [9, Section 4] admits a graph-algebraic presentation (see [11, Section 3.2]) as the C*-algebra of the graph given below:

[TABLE]

Due to Theorem 3.4 and $C(S^{1})\cong C(\mathbb{R}{\rm P}^{1})$, we have the following pullback diagram:

[TABLE]

Observe that the diagram (5.2) reflects the aforementioned procedure of shrinking the copy of $\mathbb{R}{\rm P}^{1}$ inside $\mathbb{R}{\rm P}^{2}$ to a point.

5.2. The quantum teardrop $\mathbb{W}{\rm P}^{1}_{q}(1,2)$

The classical teardrop $\mathbb{W}{\rm P}^{1}(1,2)$ may be represented as the wedge of two spheres, namely we have the following pushout diagram:

[TABLE]

To obtain a noncommutative counterpart of the diagram (5.3), we need to introduce a different kind of a noncommutative sphere. The C*-algebra $C(S^{2}_{q\infty})$ of the equatorial Podleś quantum sphere $S^{2}_{q\infty}$ [16, (3b)] admits a graph-algebraic presentation (see [11, Section 3.1]) as the C*-algebra of the graph given below.

[TABLE]

Theorem 3.4 applies and we obtain the pullback diagram

[TABLE]

which can be regarded as a noncommutative deformation of the diagram (5.3).

5.3. The quantum complex projective spaces $\mathbb{C}{\rm P}^{n}_{q}$

The CW-complex decomposition of complex projectives spaces may be described in terms of pushout diagrams

[TABLE]

Let us recall the graph-algebraic presentation of $q$-deformations of the spaces in the diagram (5.6).

• •

The C*-algebra $C(\mathbb{C}{\rm P}^{n}_{q})$ of the quantum complex projective space $\mathbb{C}{\rm P}^{n}_{q}$ [18] is the graph C*-algebra of a graph that, for $n=3$, is given below (see [11, Section 4.3]):

[TABLE]

• •

The C*-algebra $C(B^{2n}_{q})$ of the Hong–Szymański quantum even-dimensional ball $B^{2n}_{q}$ [12] is the graph C*-algebra of a graph that, for $n=3$, is given below (see [12, Section 3.1]):

[TABLE]

• •

The C*-algebra $C(S^{2n-1}_{q})$ of the Vaskman–Soibelman quantum odd-dimensional sphere $S^{2n-1}_{q}$ [18, Definition on p. 106] is the graph C*-algebra of a graph that, for $n=4$, is given below (see [11, Section 4.1]):

[TABLE]

Applying Theorem 3.4, we obtain the pullback diagram

[TABLE]

Note that the diagram (5.10) was obtained in [2, Proposition 4.1] using equivariant pullback structures.

5.4. The quantum teardrops $\mathbb{W}{\rm P}^{1}_{q}(1,n)$

Let $n\in\mathbb{N}\setminus\{0\}$. Consider the following graph $W_{n}$:

[TABLE]

Observe that $C^{*}(W_{1})\cong C(S^{2}_{q0})$. Moreover, one can show (see [5, Section 3]) that, in general, the graph C*-algebra $C^{*}(W_{n})$ is isomorphic with the C*-algebra $C(\mathbb{W}{\rm P}^{1}_{q}(1,n))$ [4, Section 3]. We will also need the following $n$-sink extension $R^{n}_{m}$ of the graph (2.5):

[TABLE]

Here the notation $(i_{j})$ means that there are $i_{j}\in\mathbb{N}\setminus\{0\}$ many edges from $r_{0}$ to $r_{j}$. Now, due to Theorem 3.4, we obtain the pullback diagram

[TABLE]

Let us now consider the graph $G^{n}$ defined as a pushout of $W_{n}$ (see (5.11)) and an another graph $H$ over the sinks of $W_{n}$. The only restriction on the graph $H$ is that there exists an inclusion $\{r_{1},\ldots,r_{n}\}\subseteq H^{0}$. The graph $G^{n}$ is represented pictorially as follows:

[TABLE]

Next, we consider an analogous construction for the graph $R^{n}_{m}$ (see (5.12)) using the same graph $H$, and we denote the resulting graph by $E^{n}_{m}$. The graph $E^{n}_{m}$ is represented pictorially as follows:

[TABLE]

Theorem 4.3 applies and, for any $n,m\in\mathbb{N}\setminus\{0\}$, we obtain the following pullback diagram:

[TABLE]

Acknowledgement

The work on this project was partially supported by NCN grant 2015/19/B/ST1/03098 (Piotr M. Hajac, Mariusz Tobolski) and by NSF grant DMS-1801011 (Alexandru Chirvasitu). It is a pleasure to thank Sarah Reznikoff for a helpful discussion. P.M.H. is also grateful to SUNY Buffalo for its hospitality and financial support.