Non-surjective pullbacks of graph C*-algebras from non-injective pushouts of graphs
NNON-SURJECTIVE PULLBACKS OF GRAPH C*-ALGEBRASFROM NON-INJECTIVE PUSHOUTS OF GRAPHS
ALEXANDRU CHIRVASITU, PIOTR M. HAJAC, AND MARIUSZ TOBOLSKI
Abstract.
We find a substantial class of pairs of ∗ -homomorphisms between graphC*-algebras of the form C ∗ ( E ) (cid:31) (cid:127) (cid:47) (cid:47) C ∗ ( G ) C ∗ ( F ) (cid:111) (cid:111) (cid:111) (cid:111) whose pullback C*-algebra is an AFgraph C*-algebra. Our result can be interpreted as a recipe for determining the quantumspace obtained by shrinking a quantum subspace. There is a variety of examples fromnoncommutative topology, such as quantum complex projective spaces (including thestandard Podle´s quantum sphere) or quantum teardrops, that instantiate the result.Furthermore, to go beyond AF graph C*-algebras, we consider extensions of graphs oversinks and prove an analogous theorem for the thus obtained graph C*-algebras. Introduction
The classical two-sphere S can be obtained by shrinking the boundary of the disc B toa point. In other words, there is a pushout diagram in the category of topological spaces S {∗} (cid:62) (cid:62) B . (cid:97) (cid:97) S (cid:96) (cid:96) (cid:61) (cid:61) (1.1)Due to the contravariant duality of algebras and spaces, the diagram (1.1) amounts toan isomorphism C ( S ) ∼ = C ( B ) ⊕ C ( S ) C of C*-algebras of complex-valued continuousfunctions on the two-sphere and the pushout B (cid:116) S {∗} respectively.At the same time, the Toeplitz algebra T [6] can be viewed as a noncommutativedeformation of C ( B ) (see [13, Theorem IV.7]). Therefore, the C*-algebra C ( S q ) ofthe standard Podle´s sphere [16, (3a)] provides a noncommutative deformation of thediagram (1.1), namely we have the following pullback diagram in the category of C*-alge-bras C ( S q ) (cid:124) (cid:124) (cid:35) (cid:35) C (cid:34) (cid:34) T . (cid:123) (cid:123) C ( S ) (1.2) a r X i v : . [ m a t h . QA ] D ec A. CHIRVASITU, P. M. HAJAC, AND M. TOBOLSKI
The aim of this paper is to generalize the above pullback construction using the con-cept of a C*-algebra C ∗ ( E ) of a directed graph E (e.g., see [3]). Graph C*-algebrasprovide powerful tools in noncommutative topology, and many C*-algebras representingnoncommutative deformations of topological spaces are isomorphic with C*-algebras ofgraphs [5, 11, 12]. These isomorphisms are usually quite complicated and they do notdepend on the deformation parameter. Nevertheless, when such an isomorphism is estab-lished, 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 asC*-algebras of graphs. We present this pictorially as follows: C ∗ (cid:32) ( ∞ ) (cid:33) (cid:37) (cid:37) (cid:123) (cid:123) C ∗ ( ) (cid:36) (cid:36) C ∗ . (cid:121) (cid:121) C ∗ (cid:18) (cid:19) (1.3)(See the examples in Section 2 for details.)The graph-algebraic decomposition (1.3) manifests a certain general phenomenon thatcan be explained in terms of non-injective pushouts of graphs. The goal of this paper isto explore this phenomenon to arrive at a general setting. To this end, we search for anew concept of morphisms of graphs, so as to ensure that, in the thus defined categoryof graphs, the assignment of graph algebras to graphs becomes a contravariant functortranslating pushouts of graphs into pullbacks of graphs algebras. While this task seemsto 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 withoutrow-finiteness assumption.To accommodate this naturally occuring non-injectivity, we replace the standard ideaof mapping vertices to vertices and edges to edges by the more flexible idea of mappingfinite paths to finite paths. We arrive at a general result for a class of unital AF graphC*-algebras including the standard Podle´s sphere, complex quantum projective spaces [18,Definition on p. 109], and quantum teardrops [4]. Finally, we go beyond AF graph C*-alge-bras by extending their acyclic graphs over sinks.2. Graph-algebraic preliminaries A directed graph E is a quadruple ( E , E , s, r ), where E is the set of vertices, E is theset of edges (arrows), and s, r : E → E are the source map and the range (target) map ON-SURJECTIVE PULLBACKS OF GRAPH C*-ALGEBRAS 3 respectively. Throughout the paper, we consider only directed graphs with countable setsof 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 (cid:8) P v | v ∈ E (cid:9) and partial isometries (cid:8) S e | e ∈ E (cid:9) satisfying the following conditions: S ∗ e S f = δ e,f P r ( e ) for all e, f ∈ E , (GA1) (cid:88) e ∈ s − ( v ) S e S ∗ e = P v for all v ∈ E such that < | s − ( v ) | < ∞ , (GA2) S e S ∗ e ≤ P s ( e ) for all e ∈ E . (GA3)A vertex v in E is called a sink if and only if s − ( v ) = ∅ . 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 itsvertices are either regular or sinks. By a finite path in E we mean a sequence ( e , . . . , e n )of edges satisfying r ( e i ) = s ( e i +1 ) for all i ∈ { , . . . , n − } . The length of a path is thenumber of edges in the sequence. We consider vertices as paths of length zero, and denotethe set of finite paths by Path( E ). The notation S α , along with the source and the rangemap, naturally extend to any α ∈ Path( E ). As we consider only finite paths throughoutthis paper, we will simply refer to them as paths.A path α is called a loop if and only if s ( α ) = r ( α ) and α is not a vertex. We say thata 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 α is a prolongation of a path β if and only if α = βγ for a path γ ∈ Path( E ) such that r ( β ) = s ( γ ). We write β (cid:22) α when α is a prolongation of β .Observe that (cid:22) gives a partial order on Path( E ). Lemma 2.3.
Let α and β be finite paths in an arbitrary graph E . Then S ∗ α S β (cid:54) = 0 ⇐⇒ ( α (cid:22) β or β (cid:22) α ) . Proof.
Assume that β (cid:22) α , i.e. that α = βγ with r ( β ) = s ( γ ). Then, as S ∗ γ is an elementof a linear basis of C ∗ ( E ) for any γ ∈ Path( E ) [1, Corollary 1.5.12], we obtain S ∗ α S β = S ∗ γ S ∗ β S β = S ∗ γ (cid:54) = 0 . (2.1)Much in the same way, we see that S ∗ α S β (cid:54) = 0 when α (cid:22) β .Conversely, assume that S ∗ α S β (cid:54) = 0 for some finite paths α := x . . . x m , β := y . . . y r , x , . . . , x m , y , . . . , y r ∈ E . (2.2)If m ≥ r , then 0 (cid:54) = S ∗ α S β = S ∗ x m . . . S ∗ x S y . . . S y r (2.3)and (GA1) imply that x i = y i for i = 1 , . . . , r . This means that β (cid:22) α . Otherwise, when r ≥ m , we get that α (cid:22) β . (cid:4) A. CHIRVASITU, P. M. HAJAC, AND M. TOBOLSKI
Next, to make the condition (GA3) easier to check, we prove the following lemma:
Lemma 2.4.
Let E be an arbitrary graph and α a path in E with its origin at v ∈ E .Then S α S ∗ α ≤ P v ∈ C ∗ ( E ) . Proof.
Write α = βe , where e is an edge with its origin at w ∈ E and β is an initialsubpath of α ending at w . Then S α S ∗ α = S β ( S e S ∗ e ) S ∗ β ≤ S β P w S ∗ β = S β S ∗ β , (2.4)where the middle inequality is due to (GA3). Now, the claim follows by the induction onthe length of α . (cid:4) To get ready for examples in the last section, we present graph-algebraic presentationsof some well-known C*-algebras.
Example 2.5.
The algebra C of complex numbers is isomorphic with the graph C*-algebra of the graph with one vertex and no edges. v Example 2.6.
The C*-algebra C ( S ) of all continuous complex-valued functions on thecircle is the universal unital C*-algebra generated by a single unitary u . It is isomor-phic with the graph C*-algebra of the graph given below through the isomorphism givenby u (cid:55)→ S e . ve Example 2.7.
The Toeplitz algebra T [6] is the universal unital C*-algebra generated bya single isometry s . It is isomorphic with the graph C*-algebra of the graph given belowthrough the isomorphism given by s (cid:55)→ S t + S t . w w t t Example 2.8.
The Cuntz algebra O m [7] is the universal unital C*-algebra generatedby isometries s , . . . , s m subject to the relation (cid:80) mi =1 s i s ∗ i = 1. It is isomorphic withthe graph C*-algebra of the graph R m given below through the isomorphism given by s i (cid:55)→ S e i . 1... e e m (2.5) ON-SURJECTIVE PULLBACKS OF GRAPH C*-ALGEBRAS 5
Example 2.9.
Let q ∈ [0 , C ( S q ) [16, (3a)] of the standard Podle´squantum sphere coincides with the C*-algebra of the Vaksman–Soibelman quantum com-plex projective line C ( C P q ) [18, p. 109], which has a graph-algebraic presentation as thegraph C*-algebra of the graph given below (see [11, Section 2.3]): v v ( ∞ )Here the arrow decorated by ( ∞ ) denotes countably infinitely many arrows.We end this section by recalling some standard results that we will use throughout thepaper. Let E be a directed graph. A subset H ⊆ E is called hereditary iff, for any v ∈ H such that there is a path starting at v and ending at w ∈ E , we have w ∈ H . If H ishereditary, then the ideal I H generated by the projections associated with the elementsof H is of the form (cf. the equation (1) in [3]): I H = span { S α S ∗ β | α, β ∈ Path( E ) , r ( α ) = r ( β ) ∈ H } . (2.6)Here 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 / ∈ H such that r ( s − ( v )) ⊆ H . Then,the quotient algebra C ∗ ( E ) /I H is again a graph C*-algebra (cf. the discussion below theequation (1) in [3]): C ∗ ( E ) /I H ∼ = C ∗ ( E/H ) , where E/H := ( E \ H, r − ( E \ H ) , s H , r H ) (2.7)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 , D , s D , r D ) and E = ( E , E , s E , r E ) be directed graphs. A morphism ofgraphs f : D → E is a pair of mappings f : D → E and f : D → E satisfying f ◦ s D = s E ◦ f , f ◦ r D = r E ◦ f . (3.1)If there is an injective morphism of graphs D → E , we say that D is a subgraph of E andwrite D ⊆ E . Definition 3.1.
An injective graph morphism ι : D → E is called an admissible inclusion iff the following conditions are satisfied: (A1) E \ ι ( D ) is hereditary and saturated, (A2) ι ( D ) = r − E ( ι ( D )) , (A3) no vertex in E emits infinitely many edges into E \ D while emitting finitelymany (but not zero) edges into D . A. CHIRVASITU, P. M. HAJAC, AND M. TOBOLSKI
Next, let us state the following elementary fact (cf. (2.7) and the discussion preceding it).
Proposition 3.2.
Let D ⊆ E be an admissible inclusion. Then, we have an isomorphismof graph C*-algebras C ∗ ( D ) ∼ = C ∗ ( E/ ( E \ D )) . (3.2)To phrase our main result, it is convenient to view graphs as small categories whoseobjects are vertices and morphisms are finite paths. Then functors between such cat-egories are what we want as morphisms between graphs. Using the thus understoodfunctors 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 : F → E be a functor between graphs such that: (1) f is compatible with the prolongation relation as follows f ( α ) (cid:22) f ( β ) ⇒ α (cid:22) β ;(2) for any vertex v that emits at least one and at most finitely many edges, f restricts-corestricts to a bijection s − F ( v ) −→ s − E ( f ( v )) . Then f induces a ∗ -homomorphism f ∗ : C ∗ ( F ) → C ∗ ( E ) given by ∀ v ∈ F : f ∗ ( P v ) := P f ( v ) and ∀ x ∈ F : f ∗ ( S x ) := S f ( x ) . Proof.
Since graph C*-algebras are universal, it suffices to show that all defining relationsare preserved. For starters, since the condition (1) implies the injectivity of f , we inferthat the set of mutually orthogonal projections is sent to the set of mutually orthogonalprojections: P f ( v ) P f ( w ) = δ f ( v ) ,f ( w ) P f ( v ) = δ v,w P f ( v ) . (3.3)Next, to show that (GA1) is preserved, it suffices to prove the implication S ∗ f ( e ) S f ( e ) (cid:54) = 0 ⇒ e = e , (3.4)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 fromthe condition (2) and the latter from Lemma 2.4. (cid:4) We are now ready to prove our first main result:
Theorem 3.4.
Let F i ⊆ E i , i = 1 , , be admissible inclusions of graphs such that (1) E has no loops, E has no short loops at vertices in E \ F , and E = E , F = F ; (2) there is a functor f : E → E such that: it satisfies the condition (1) in Lemma 3.3,it is id on objects, and its image is the set of all pointed paths. ON-SURJECTIVE PULLBACKS OF GRAPH C*-ALGEBRAS 7
Then the induced ∗ -homomorphisms exist and render the diagram C ∗ ( E ) π (cid:121) (cid:121) f ∗ (cid:37) (cid:37) C ∗ ( F ) f | ∗ (cid:37) (cid:37) C ∗ ( E ) π (cid:121) (cid:121) C ∗ ( F ) (3.5) a pullback diagram of C*-algebras. (If E is finite, then this is a pullback diagram of unitalC*-algebras.) Here π and π are the canonical surjections (3.2) , f ∗ is a ∗ -homomorphismof 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 checkthe condition (2) of Lemma 3.3. To this end, take any regular vertex v ∈ E and anyedge e ∈ s − ( v ). Then, as the image of f is the set of pointed paths, f ( e ) is a pointedpath from v to r ( e ).Suppose that f ( e ) factorizes through a third vertex w . Then we can write f ( e ) = αβ ,where α is a pointed path from v to w and β is a pointed path from w to r ( e ). Indeed,deleting any intial subpath from a pointed path always yields a pointed path, and makingall loops based at w part of β makes α a pointed path. Furthermore, as f is surjectiveon the set of pointed paths, we can write f ( e ) = αβ = f ( α (cid:48) ) f ( β (cid:48) ) = f ( α (cid:48) β (cid:48) ). Combiningit with the injectivity of f , which follows from the condition (1) in Lemma 3.3, we get acontradiction e = α (cid:48) β (cid:48) (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 based at v , then there are infinitely many non-factorizing pointedpaths from v to r ( e ), and (because f is a functor) none of them can be the image of a paththat factorizes through a third vertex. Consequently, as f is bijective when corestrictedto the set of pointed paths, and there are no loops in E , there must be infinitely manyedges in E from v to r ( e ), which contradicts the assumption that v is a regular vertexin E . Hence, there is no loop in E based at v , so f ( e ) is an edge.Next, if f ( α ) ∈ E , then α ∈ E because f is an injective functor that is id onthe set of veritices. Indeed, suppose that α = e . . . e n , where e i ’s are edges. Then f ( α ) = f ( e . . . e n ) = f ( e ) . . . f ( e n ) is of length at least n , as f ( e i ) cannot be a vertex.Hence n = 1, i.e. α is an edge, so any edge emitted from v in E comes from an edgeemitted from v in E . Combining this with the injectivity of f and the above establishedfact 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 has no loops. Furthermore, by the admissibility condition Definition 3.1(A2),it is clear that f restricted to the subgraph F corestricts to F yielding a restriction-corestriction f | ∗ of f ∗ . The ∗ -homomorphism f | ∗ is injective because f ∗ is injective. A. CHIRVASITU, P. M. HAJAC, AND M. TOBOLSKI
It is straightforward to check that the maps π , π , f ∗ and f | ∗ make the diagram (3.5)commutative. Therefore, as π and π 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 thatker π ⊆ f ∗ (ker π ) . (3.6)To obtain the above inclusion, we use the characterization of ideals associated to heredi-tary subsets (2.6):ker π = span (cid:8) S α S ∗ β | α, β ∈ Path( E ) , r ( α ) = r ( β ) ∈ E \ F (cid:9) , (3.7)ker π = span (cid:8) S γ S ∗ δ | γ, δ ∈ Path( E ) , r ( γ ) = r ( δ ) ∈ E \ F (cid:9) . (3.8)By the assumption (1), all paths in E terminating in E \ F are pointed, so they are inthe image of f . Therefore, as E \ F = E \ F by the assumption (1), we conclude thatthe inclusion (3.6) holds at the algebraic level. Finally, as any ∗ -homomorphism betweenC*-algebras is a continuous map whose image is closed, we infer the desired inclusion atthe C*-level. (cid:4) Extending graphs over sinks
To generalize the diagram (1.2) even further (e.g. to allow loops in E in the pullbacktheorem of the previous section), we first need to determine suitable conditions underwhich 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; • ι E : X → E and ι H : X → H are injective maps defining the pushout E (cid:116) X H E (cid:59) (cid:59) H ; (cid:99) (cid:99) X ι E (cid:100) (cid:100) ι H (cid:58) (cid:58) (4.1) • E (cid:116) X H := ( E (cid:116) X H , E (cid:116) H , π ◦ ( s E (cid:116) s H ) , π ◦ ( r E (cid:116) r H )), where π is the canonicalquotient map.Next, let ι E ∗ : C ∗ ( X ) → C ∗ ( E ) and ι H ∗ : C ∗ ( X ) → C ∗ ( H ) be the induced ∗ -homo-morphisms (see Lemma 3.3). Define C ∗ ( E ) • C ∗ ( X ) C ∗ ( H ) := ( C ∗ ( E ) ∗ C ∗ ( X ) C ∗ ( H )) / (cid:104) P v P w | v ∈ E \ ι E ( X ) , w ∈ H \ ι H ( X ) (cid:105) . Here we divide the amalgamated free product by the ideal generated by the product ofnon-identified projections.
ON-SURJECTIVE PULLBACKS OF GRAPH C*-ALGEBRAS 9
Lemma 4.1.
Assume that at least one of the maps ι H and ι E takes its values in the sinksof the respective graph. Then the natural assignment of elements defines an isomorphismof C*-algebras: C ∗ ( E ) • C ∗ ( X ) C ∗ ( H ) −→ C ∗ ( E (cid:116) X H ) . (4.2) Proof.
Since ι E : X → E or ι H : X → H takes values in the sinks of E or H , respectively,all edge relations in C ∗ ( E (cid:116) X H ) involving vertices in the image of X are of one of twotypes: either they refer to edges only in E , or to edges only in H . Hence, there are ∗ -homomorphisms j E : C ∗ ( E ) −→ C ∗ ( E (cid:116) X H ) and j H : C ∗ ( H ) −→ C ∗ ( E (cid:116) X H ) (4.3)given the natural assignment of elements. Furthermore, as ( E (cid:116) X H ) = E (cid:116) X H , theyinduce a surjective ∗ -homomorphism π (cid:116) : C ∗ ( E ) ∗ C ∗ ( X ) C ∗ ( H ) −→ C ∗ ( E (cid:116) X H ) . (4.4)Finally, as the kernel of π (cid:116) coincides with the kernel of the defining surjection π • : C ∗ ( E ) ∗ C ∗ ( X ) C ∗ ( H ) −→ C ∗ ( E ) • C ∗ ( X ) C ∗ ( H ) , (4.5)the claim follows. (cid:4) Now, consider three graphs E , E and H with injective maps ι E : X → E , ι E : X → E ,and ι H : X → H . Assume also that ι E ( X ) and ι E ( X ) consist of sinks of the two respec-tive graphs E and E . Now, consider a C*-algebra homomorphism δ : C ∗ ( E ) −→ A (4.6)annihilating the vertex projections of ι E ( X ) ⊆ E . Then δ and the zero map C ∗ ( H ) → A induce a ∗ -homomorphism on the amalgamated product that annihilates the kernel of π • .Hence, by Lemma 4.1, δ extends to δ (cid:48) : C ∗ ( E (cid:116) X H ) −→ A. (4.7) Lemma 4.2.
Let j H : C ∗ ( H ) → C ∗ ( E (cid:116) X H ) be the map defined in (4.3) . Then ker δ (cid:48) = j E (ker δ ) + (cid:104) j H ( C ∗ ( H )) (cid:105) . Proof.
The inclusion ker δ (cid:48) ⊇ (cid:104) j E (ker δ ) (cid:105) + (cid:104) j H ( C ∗ ( H )) (cid:105) is clear by the construction of δ (cid:48) .For the other inclusion, note that, as δ (cid:48) annihilates C ∗ ( H ), it factors as C ∗ ( E (cid:116) X H ) → C ∗ ( E (cid:116) X H ) / (cid:104) j H ( C ∗ ( H )) (cid:105) ∼ = C ∗ ( E ) / (cid:104) ι E ∗ ( C ∗ ( X )) (cid:105) → A, (4.8)where the last map is induced by δ : C ∗ ( E ) → A . The inclusion follows from thisfactorization. Finally, as (cid:104) j E (ker δ ) (cid:105) = j E (ker δ ) + (cid:104) j E (ker δ ) (cid:105) ∩ (cid:104) j H ( C ∗ ( H )) (cid:105) , (4.9)we infer that (cid:104) j E (ker δ ) (cid:105) + (cid:104) j H ( C ∗ ( H )) (cid:105) = j E (ker δ ) + (cid:104) j H ( C ∗ ( H )) (cid:105) , (4.10)which ends the proof. (cid:4) Assume now that, under the general setup assumptions (GS) and the assumptionspreceding (4.6), we have a pullback diagram C ∗ ( E ) δ (cid:124) (cid:124) φ (cid:37) (cid:37) A ρ (cid:35) (cid:35) C ∗ ( E ) θ (cid:121) (cid:121) B (4.11)of ∗ -homomorphism of C*-algebras. Here φ is an injective ∗ -homomorphism sending allvertex projections to vertex projections and all partial isometries associated with edgesto partial isometries associated with paths, intertwining ι E with ι E , and sending theprojections labeled by E \ ι E ( X ) to projections labeled by E \ ι E ( X ). We also assumethat all partial isometries in C ∗ ( E ) associated with paths ending in ι E ( X ) are in theimage of φ . Furthermore, we assume that δ is a ∗ -homomorphism annihilating the vertexprojections labeled by ι E ( X ), and A and B are arbitrary C*-algebras fitting into thepullback diagram (4.11) for some ∗ -homomorphisms θ and ρ .Note that the assumptions made on φ allow us to define its extension ψ : C ∗ ( E (cid:116) X H ) −→ C ∗ ( E (cid:116) X H ) . (4.12)Indeed, we can use the isomorphism (4.2) and observe that the conditions on φ allow usto extend it by id : C ∗ ( H ) → C ∗ ( H ) to C ∗ ( E ) • C ∗ ( X ) C ∗ ( H ) −→ C ∗ ( E ) • C ∗ ( X ) C ∗ ( H ) . (4.13)This brings us to the second main result of the paper: Theorem 4.3.
Under the general setup assumptions (GS) and the additional assumptionspreceding (4.6) , the pullback diagram (4.11) of ∗ -homomorphisms of C*-algebras inducesthe following pullback diagram of ∗ -homomorphisms of C*-algebras: C ∗ ( E (cid:116) X H ) δ (cid:48) (cid:123) (cid:123) ψ (cid:38) (cid:38) A ρ (cid:36) (cid:36) C ∗ ( E (cid:116) X H ) . θ (cid:48) (cid:119) (cid:119) B (4.14) Here δ (cid:48) and θ (cid:48) are defined by (4.7) , and ψ is defined by (4.13) .Proof. The commutativity of the diagram (4.14) is immediate by construction. To provethat it is a pullback diagram, first we establish the injectivity of ψ . It follows from: theinjectivity of φ , the assumption that φ does not annihilate vertex projections, the factthat loops without exit in E (cid:116) X H remain loops without exit both in E and H , and thegeneral Cuntz–Krieger uniqueness theorem [17, Theorem 1.2]. ON-SURJECTIVE PULLBACKS OF GRAPH C*-ALGEBRAS 11
Next, using the injectivity of ψ and appealing to [15, 3.1 Proposition], we note that toconclude the proof of the theorem, it suffices to check the following two conditions: ρ − (cid:16) θ (cid:48) (cid:16) C ∗ ( E (cid:116) X H ) (cid:17)(cid:17) = δ (cid:48) (cid:16) C ∗ ( E (cid:116) X H ) (cid:17) , (4.15)ker θ (cid:48) ⊆ ψ (ker δ (cid:48) ) . (4.16)The first condition is immediate from our assumption that (4.11) is a pullback. Indeed,the analogous equation holds for the unprimed maps θ and δ , and the images of thesemaps coincide with those of θ (cid:48) and δ (cid:48) respectively because the primed maps are obtainedfrom 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:ker δ (cid:48) = (cid:104) j E (ker δ ) (cid:105) + (cid:104) j H ( C ∗ ( H )) (cid:105) , (4.17)ker θ (cid:48) = j E (ker θ ) + (cid:104) j H ( C ∗ ( H )) (cid:105) . (4.18)Here both j H and j H are defined as in (4.3). Now, since (4.11) is a pullback diagram, wehave ker θ ⊆ φ (ker δ ). Furthermore, it follows from the construction of ψ and δ (cid:48) that j E ( φ (ker δ )) = ψ ( j E (ker δ )) ⊆ ψ (ker δ (cid:48) ) . (4.19)Hence j E (ker θ ) ⊆ ψ (ker δ (cid:48) ), so, by (4.18), we are left having to argue that (cid:104) j H ( C ∗ ( H )) (cid:105) ⊆ ψ (ker δ (cid:48) ) . (4.20)To this end, note first that j H ( C ∗ ( H )) ⊆ ψ (ker δ (cid:48) ) (4.21)because j H ( C ∗ ( H )) ⊆ ker δ (cid:48) and ψ ◦ j H = j H . Furthermore, observe that (cid:104) j H ( C ∗ ( H )) (cid:105) = j H ( C ∗ ( H )) + (cid:104) j H ( C ∗ ( H )) (cid:105) ∩ (cid:104) j E ( C ∗ ( E )) (cid:105) (4.22)and (cid:104) j H ( C ∗ ( H )) (cid:105) ∩ (cid:104) j E ( C ∗ ( E )) (cid:105) ⊆ (cid:104){ j E ( P v ) | v ∈ ι E ( X ) }(cid:105) . (4.23)Next, since all partial isometries in C ∗ ( E ) associated with paths ending in ι E ( X ) arein the image of φ by assumption, we conclude that (cid:104){ j E ( P v ) | v ∈ ι E ( X ) }(cid:105) = ψ ( (cid:104){ j E ( P v ) | v ∈ ι E ( X ) }(cid:105) ) . (4.24)Indeed, it boils down to showing that (cid:104) ψ ( { j E ( P v ) | v ∈ ι E ( X ) } ) (cid:105) ⊆ ψ ( (cid:104){ j E ( P v ) | v ∈ ι E ( X ) }(cid:105) ) . (4.25)Since any graph C*-algebra is the closed linear span of elements of the form S α S ∗ β with r ( α ) = r ( β ) (see [1, Corollary 1.5.12]), we are looking at α, β ∈ Path( E (cid:116) X H ) such that P v S α S ∗ β or S α S ∗ β P v can be non-zero. This means that s ( α ) = v or s ( β ) = v . However,as v is a sink in E , we infer that α ∈ Path( H ) or β ∈ Path( H ). Hence α ∈ Path( H )and r ( β ) ∈ H , or β ∈ Path( H ) and r ( α ) ∈ H . Furthermore, any E -subpath γ of any path ending in H has to end in ι E ( X ). Consequently, S γ ∈ φ ( C ∗ ( E )), so j E ( S γ ) ∈ ψ ( C ∗ ( E (cid:116) X H )). As j H ( S δ ) ∈ ψ ( C ∗ ( E (cid:116) X H )) for any δ ∈ Path( H ), we conclude that all elements S α S ∗ β that can multiply nontrivially with P v are in the imageof ψ , which proves (4.25).Now, taking advantage of the assumption that δ ( P v ) = 0 for any v ∈ ι E ( X ), we inferthat (cid:104){ j E ( P v ) | v ∈ ι E ( X ) }(cid:105) ⊆ (cid:104) j E (ker δ ) (cid:105) . (4.26)Hence, by (4.17), ψ ( (cid:104){ j E ( P v ) | v ∈ ι E ( X ) }(cid:105) ) ⊆ ψ ( (cid:104) j E (ker δ ) (cid:105) ) ⊆ ψ (ker δ (cid:48) ) . (4.27)Consequently, combining (4.23), (4.24) and (4.27), we arrive at (cid:104) j H ( C ∗ ( H )) (cid:105) ∩ (cid:104) j E ( C ∗ ( E )) (cid:105) ⊆ ψ (ker δ (cid:48) ) , (4.28)which, together with (4.21) and (4.22) proves (4.20). (cid:4) Examples and applications
This section is devoted to the study of special cases of Theorem 3.4 and Theorem 4.3leading to interesting examples in noncommutative topology.5.1.
The standard Podle´s quantum sphere.
Observe that the assumptions of The-orem 3.4 are true for the standard Podle´s quantum sphere. Here C ∗ ( E ) = C ( S q ), C ∗ ( F ) = C , C ∗ ( E ) = T and C ∗ ( F ) = C ( S ) (see the diagram (1.3)).Our next example generalizes a simple gluing construction in topology. Recall that thereal projective plane R P may be represented as a closed hemisphere with the antipodalpoints on the equator identified. If we further identify all those antipodal points, weobtain the sphere S . Here we present a q -deformed analog of this procedure.The C*-algebra C ( R P q ) of the quantum real projective plane R P q [9, Section 4] admitsa graph-algebraic presentation (see [11, Section 3.2]) as the C*-algebra of the graph givenbelow: (5.1)Due to Theorem 3.4 and C ( S ) ∼ = C ( R P ), we have the following pullback diagram: C ( S q ) (cid:124) (cid:124) (cid:37) (cid:37) C (cid:34) (cid:34) C ( R P q ) . (cid:121) (cid:121) C ( R P ) (5.2) ON-SURJECTIVE PULLBACKS OF GRAPH C*-ALGEBRAS 13
Observe that the diagram (5.2) reflects the aforementioned procedure of shrinking thecopy of R P inside R P to a point.5.2. The quantum teardrop W P q (1 , The classical teardrop W P (1 ,
2) may berepresented as the wedge of two spheres, namely we have the following pushout diagram: W P (1 , {∗} (cid:58) (cid:58) S . (cid:100) (cid:100) S (cid:101) (cid:101) (cid:57) (cid:57) (5.3)To obtain a noncommutative counterpart of the diagram (5.3), we need to introducea different kind of a noncommutative sphere. The C*-algebra C ( S q ∞ ) of the equatorialPodle´s quantum sphere S q ∞ [16, (3b)] admits a graph-algebraic presentation (see [11,Section 3.1]) as the C*-algebra of the graph given below. (5.4)Theorem 3.4 applies and we obtain the pullback diagram C ( W P q (1 , (cid:121) (cid:121) (cid:39) (cid:39) C (cid:37) (cid:37) C ( S q ∞ ) , (cid:119) (cid:119) C ( S ) (5.5)which can be regarded as a noncommutative deformation of the diagram (5.3).5.3. The quantum complex projective spaces C P nq . The CW-complex decomposi-tion of complex projectives spaces may be described in terms of pushout diagrams C P n C P n − (cid:57) (cid:57) B n . (cid:100) (cid:100) S n − (cid:100) (cid:100) (cid:58) (cid:58) (5.6) Let us recall the graph-algebraic presentation of q -deformations of the spaces in the dia-gram (5.6). • The C*-algebra C ( C P nq ) of the quantum complex projective space C P nq [18] is thegraph C*-algebra of a graph that, for n = 3, is given below (see [11, Section 4.3]):( ∞ ) ( ∞ )( ∞ ) ( ∞ )( ∞ )( ∞ ) (5.7) • The C*-algebra C ( B nq ) of the Hong–Szyma´nski quantum even-dimensional ball B nq [12] is the graph C*-algebra of a graph that, for n = 3, is given below (see [12,Section 3.1]): (5.8) • The C*-algebra C ( S n − q ) of the Vaskman–Soibelman quantum odd-dimensionalsphere S n − 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]): (5.9)Applying Theorem 3.4, we obtain the pullback diagram C ( C P nq ) (cid:38) (cid:38) (cid:120) (cid:120) C ( C P n − q ) (cid:38) (cid:38) C ( B nq ) . (cid:121) (cid:121) C ( S n − q ) (5.10)Note that the diagram (5.10) was obtained in [2, Proposition 4.1] using equivariant pull-back structures. ON-SURJECTIVE PULLBACKS OF GRAPH C*-ALGEBRAS 15
The quantum teardrops W P q (1 , n ). Let n ∈ N \ { } . Consider the followinggraph W n : r ∞ ) r ∞ ) r . . . ( ∞ ) r n -1( ∞ ) r n (5.11)Observe that C ∗ ( W ) ∼ = C ( S q ). Moreover, one can show (see [5, Section 3]) that, ingeneral, the graph C*-algebra C ∗ ( W n ) is isomorphic with the C*-algebra C ( W P q (1 , n )) [4,Section 3]. We will also need the following n -sink extension R nm of the graph (2.5): r ... r r . . .r n − r n ( i ) ( i ) ( i n -1 )( i n ) (5.12)Here the notation ( i j ) means that there are i j ∈ N \ { } many edges from r to r j . Now,due to Theorem 3.4, we obtain the pullback diagram C ( W P q (1 , n )) (cid:39) (cid:39) (cid:121) (cid:121) C (cid:37) (cid:37) C ∗ ( R nm ) . (cid:119) (cid:119) O m (5.13)Let us now consider the graph G n defined as a pushout of W n (see (5.11)) and ananother graph H over the sinks of W n . The only restriction on the graph H is that thereexists an inclusion { r , . . . , r n } ⊆ H . The graph G n is represented pictorially as follows: ( ∞ ) ( ∞ ) . . . ( ∞ ) ( ∞ ) H (5.14)Next, we consider an analogous construction for the graph R nm (see (5.12)) using thesame graph H , and we denote the resulting graph by E nm . The graph E nm is representedpictorially as follows: ( i ) ( i ) . . . ( i n − )( i n ) H ... (5.15) Theorem 4.3 applies and, for any n, m ∈ N \{ } , we obtain the following pullback diagram: C ∗ ( G n ) (cid:37) (cid:37) (cid:124) (cid:124) C (cid:35) (cid:35) C ∗ ( E nm ) . (cid:120) (cid:120) O m (5.16) 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 Chirv-asitu). It is a pleasure to thank Sarah Reznikoff for a helpful discussion. P.M.H. is alsograteful to SUNY Buffalo for its hospitality and financial support.
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E-mail address : [email protected] (P. M. Hajac) Instytut Matematyczny, Polska Akademia Nauk, ul. ´Sniadeckich 8, Warszawa,00-656 Poland; and Department of Mathematics, University of Colorado Boulder, 2300Colorado Avenue, Boulder, CO 80309-0395, USA
E-mail address : [email protected] (M. Tobolski) Instytut Matematyczny, Polska Akademia Nauk, ul. ´Sniadeckich 8, Warszawa,00-656 Poland
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