Characterizing graph C*-correspondences
aa r X i v : . [ m a t h . OA ] O c t CHARACTERIZING GRAPH C ∗ -CORRESPONDENCES S. KALISZEWSKI, NURA PATANI, AND JOHN QUIGG
Abstract.
Every separable nondegenerate C ∗ -correspondence overa commutative C ∗ -algebra with discrete spectrum is isomorphic toa graph correspondence. Let E be a directed graph with vertex set V , edge set E , and rangeand source maps r, s : E → V . The graph correspondence is thenondegenerate C ∗ -correspondence X E over c ( V ) defined by X E = (cid:8) ξ : E → C (cid:12)(cid:12) the map v X s ( e )= v | ξ ( e ) | is in c ( V ) (cid:9) , with module actions and c ( V )-valued inner product given by( a · ξ · b )( e ) = a ( r ( e )) ξ ( e ) b ( s ( e )) and h ξ, η i ( v ) = X s ( e )= v ξ ( e ) η ( e )for a, b ∈ c ( V ), ξ, η ∈ X E , e ∈ E , and v ∈ V . (We cite [3] as a generalreference on graph algebras, and [3, Chapter 8] in particular for graphcorrespondences.)A graph correspondence X E contains at least as much informationabout E as the graph algebra C ∗ ( E ), since the Cuntz-Pimsner al-gebra O X E is isomorphic to C ∗ ( E ) ([3, Example 8.13]; see also [1,Example 1, p. 4303]). Indeed, many properties of E are directly re-flected in properties of X E : for example, X E is full in the sense thatspan h X E , X E i = c ( V ) if and only if the graph E has no sinks; thehomomorphism c ( V ) → L ( X E ) associated to the left module opera-tion maps into the compacts K ( X E ) if and only if no vertex receivesinfinitely many edges; c ( V ) maps faithfully into L ( X E ) if and only if E has no sources.In this paper we further investigate this connection between graphsand C ∗ -correspondences. In Section 1, our main result (Theorem 1)expands and elaborates on a remark of Schweizer ([5, § C ∗ -correspondence over C ∗ -algebras c ( X ) and c ( Y ) is unitarily equivalent to a correspondence Date : November 13, 2018.2000
Mathematics Subject Classification.
Primary 46L08; Secondary 05C20.
Key words and phrases. directed graph, C ∗ -correspondence. arising from a “diagram” from X to Y . (Schweizer’s diagrams can beput in our context by taking V = X ∪ Y .) Specifically, given a (sep-arable, nondegenerate) C ∗ -correspondence X over c ( V ), we constructa graph E such that X E ∼ = X as C ∗ -correspondences. We also show(Proposition 1.4) that for any graph F with vertex set V , applying ourconstruction to X F gives a graph E such that E ∼ = F via a vertex-fixinggraph isomorphism.In Section 2 we show that the assignment E X E can be extendedto a functor Γ between certain categories of graphs and correspon-dences. Γ is very nearly a category equivalence: it is essentially surjec-tive, faithful, and “essentially full” (see Theorem 2.3), but not full. Itis also injective on objects, and reflects isomorphisms (Theorem 2.2).Although our construction of a graph from a C ∗ -correspondence isfairly elementary, we believe our results give valuable insight into graphcorrespondences. In work currently in progress, we are applying thetechniques illustrated here to the more general context of topologi-cal graphs. We are also interested in the analogous problem for k -graphs, although since there are k -graph C ∗ -algebras which are notisomorphic to any directed-graph C ∗ -algebra, our main result in thispaper implies that for k -graphs it will not be adequate to consider C ∗ -correspondences over c of the vertex set. Notation and Terminology.
Throughout this paper, V will denotea countable set. All graphs considered will have vertex set V , and (bydefinition) countably many edges. We use χ S to denote the character-istic function of a set S , and we abbreviate this to χ x when S = { x } isa singleton. We will write p v for the minimal projection χ v ∈ c ( V ).A C ∗ -correspondence X over a C ∗ -algebra A is said to be nondegen-erate if the set of products S = { a · ξ | a ∈ A, ξ ∈ X } has dense spanin X .If X and Y are C ∗ -correspondences over A , a morphism from X to Y is a linear map ψ : X → Y satisfying(0.1) ψ ( a · ξ · b ) = a · ψ ( ξ ) · b and h ψ ( ξ ) , ψ ( η ) i = h ξ, η i for a, b ∈ A and ξ, η ∈ X . Note that such morphisms will be isometric,and in particular injective.Note that for a graph E , X E is densely spanned by the linearlyindependent set { χ e | e ∈ E } , whose elements we refer to as generators of the correspondence X E . The c ( V )-valued inner product on X E is HARACTERIZING GRAPH C ∗ -CORRESPONDENCES 3 characterized on generators by h χ e , χ f i = ( p s ( e ) if e = f e = f. The Isomorphisms
Theorem 1.1.
Let V be a countable set. Every separable nondegen-erate C ∗ -correspondence over c ( V ) is isomorphic to the graph corre-spondence of a directed graph with vertex set V .Proof. For ease of notation, set A = c ( V ). Let X be a separablenondegenerate C ∗ -correspondence over A , For each v ∈ V put X v = X · p v . Because finite sums of the p v ’s form a bounded approximate identityin A (and the right action of A on X is automatically nondegenerate),each ξ ∈ X can be approximated by sums of the form ξ · (cid:16)X v ∈ F p v (cid:17) = X v ∈ F ξ · p v ∈ span [ v ∈ F X v , with F ⊆ V finite; so the X v ’s span a dense subspace of X .Moreover, for u = v in V we have p u p v = 0 in A , and hence for ξ ∈ X u and η ∈ X v we have h ξ, η i = h ξ · p u , η · p v i = p u h ξ, η i p v = h ξ, η i p u p v = 0 . It follows by a standard argument that the subspaces X v are linearlyindependent.Now with the norm inherited from X , each X v is in fact a Hilbertspace, with inner product h· , ·i v (conjugate-linear in the first variable)given by h ξ, η i v = h ξ, η i ( v ) . For u ∈ V put X uv = p u · X · p v = p u · X v ⊆ X v . Then for u = w in V the closed subspaces X uv and X wv of X v areorthogonal because, for ξ ∈ X uv and η ∈ X wv , h ξ, η i v = h ξ, η i ( v ) = h p u · ξ, p w · η i ( v ) = h ξ, p u p w · η i ( v ) = 0 . Because the left action of A on X (hence on X v ) is nondegenerate,(arguing as above) the family { X uv | u ∈ V } of mutually orthogonalsubspaces spans a dense subspace of X v . S. KALISZEWSKI, NURA PATANI, AND JOHN QUIGG
For each u, v ∈ V choose an orthonormal basis E uv of the Hilbertspace X uv , and put E = [ u,v ∈ V E uv . Since X is separable, every E uv is countable and hence E is countable.Since the subspaces X uv are linearly independent, the union above is adisjoint union, so we can define r, s : E → V by r ( e ) = u and s ( e ) = v if e ∈ E uv . This gives a graph E = ( V, E , r, s ).We now show that X ∼ = X E as C ∗ -correspondences over A . Forthis we need a linear surjection ψ : X → X E satisfying (0.1). Notethat by construction, E is a linearly independent subset of X , andfor each e ∈ E we have χ e ∈ X E by definition of X E . Thus, setting X = span E , we can define ψ : X → X E by linearly extending therule ψ ( e ) = χ e from E to X .Now it is straightforward to check the action and inner product iden-tities (0.1) for a = p u , b = p v , and ξ, η ∈ E , and the identities thenextend to a, b ∈ c c ( V ) and ξ, η ∈ X by linearity. In particular, ψ isisometric on X , so since X is dense in X , ψ extends to an isometric c ( V )-bimodule map on X by a standard argument. Moreover, ψ issurjective because the image of E has dense span in X E . (cid:3) Example 1.2.
Let V be a countable set, and let σ be a map of V intoitself. This gives rise to a natural C ∗ -correspondence X over A = c ( V )defined by X = c ( V ), with actions and inner product given in termsof the usual operations in c ( V ) by a · ξ · b = ( a ◦ σ ) ξb and h ξ, η i = ξη for a, b ∈ A and ξ, η ∈ X . Note that a ◦ σ ∈ ℓ ∞ ( V ) = M ( c ( V )), sothe left action is well-defined. (This construction motivated the moregeneral discussion in [5, § X v = X · p v = { ξχ v | ξ ∈ c ( V ) } = span χ v , and then, since the left action of p u ∈ A on χ v ∈ X is ( χ u ◦ σ ) χ v = χ σ − ( u ) χ v = χ v if σ ( v ) = u , and is zero otherwise, we have X uv = ( span χ v if σ ( v ) = u HARACTERIZING GRAPH C ∗ -CORRESPONDENCES 5 So in this example, there is a natural choice of orthonormal basis foreach X uv (provided σ ( v ) = u ), namely E uv = { χ v } . (Each χ v is aunit vector in X because the C ∗ -correspondence norm on X is justthe supremum norm on c ( V ).) Since E uv = ∅ when σ ( v ) = u , theresulting graph E = ( V, E , r, s ) associated to X has E = { χ v | v ∈ V } , r ( χ v ) = σ ( v ) , s ( χ v ) = v. Note that E is isomorphic to the graph E σ with vertex set V whichis defined using σ in a natural way by defining an edge from v to u precisely when σ ( v ) = u : E σ = { ( σ ( v ) , v ) | v ∈ V } , r ( σ ( v ) , v ) = σ ( v ) , s ( σ ( v ) , v ) = v. Example 1.3.
Let V = { u, v } and define σ : V → V by σ ( u ) = v and σ ( v ) = u . Let X be the C ∗ -correspondence over c ( V ) associated to σ as in Example 1.2. Then X uu and X vv are zero-dimensional, and wecan choose unit vectors χ v ∈ X uv = span χ v and χ u ∈ X vu = span χ u .We have E = { χ u , χ v } with r ( χ v ) = s ( χ u ) = u and s ( χ v ) = r ( χ v ) = v ,so the graph associated to the correspondence X is u vχ u χ v The directed graph E σ associated to σ is u v ( v, u )( u, v )In the proof of Theorem 1.1 we constructed a graph E from a C ∗ -correspondence X and showed that X E recovers X , up to isomorphism,from E . Given a graph F , the same construction recovers F , up toisomorphism, from X F . Proposition 1.4.
Let F = ( V, F , r, s ) be a directed graph, and let X F be the associated graph correspondence. Then every graph E con-structed from X F as in the proof of Theorem is isomorphic to F via a graph isomorphism which fixes the vertex set. S. KALISZEWSKI, NURA PATANI, AND JOHN QUIGG
Proof.
For ease of notation, set X = X F . For u, v ∈ V , ξ ∈ X and f ∈ F , we have( p u · ξ · p v )( f ) = χ u ( r ( f )) ξ ( f ) χ v ( s ( f ))= ( ξ ( f ) if r ( f ) = u and s ( f ) = v , and it follows that X uv = p u · X · p v ∼ = ℓ (cid:0) { f ∈ F | r ( f ) = u and s ( f ) = v } (cid:1) . Thus, if E uv is an orthonormal basis for X uv , there exists a bijection φ uv : E uv → { f ∈ F | r ( f ) = u and s ( f ) = v } , and it is straightforward to check that if E = ( V, E , r, s ) is the graphconstructed from X as in the proof of Theorem 1.1 by setting E = S u,v ∈ V E uv , then there is a graph isomorphism φ : E → F such that φ = φ uv on E uv and φ fixes V . (cid:3) The Graph Correspondence Functor
Let G = G ( V ) be the category whose objects are directed graphswith vertex set V , and whose morphisms are the injective graph mor-phisms which are the identity on vertices. Also let C = C ( V ) denotethe category of nondegenerate C ∗ -correspondences over the C ∗ -algebra c ( V ) (we impose the nondegeneracy condition because it is automat-ically satisfied for graph correspondences). The morphisms in C arejust the morphisms of C ∗ -correspondences as in (0.1).For a morphism φ : E → F in G , consider the linear map ψ whichtakes span { χ e | e ∈ E } ⊆ X E to X F and is defined on generators by ψ ( χ e ) = χ φ ( e ) . It is easily verified that ψ satisfies (0.1) (the assumption that φ beinjective on edges ensures that ψ is isometric), and thus extends bycontinuity to a morphism ψ φ from X E to X F in C . Then it is routineto verify that the map Γ : G → C defined on objects by Γ( E ) = X E and on morphisms by Γ( φ ) = ψ φ is a functor. We call this the graph-correspondence functor .Note that Theorem 1.1 says precisely that Γ is essentially surjec-tive : every object of C is isomorphic to an object in the image of Γ.Moreover, it is clear from the definition that Γ is faithful , that is, Γis injective on each hom-set. If it were the case that Γ were also full (surjective on each hom-set) then Γ would be a category equivalencebetween G and C . However, this is too much to ask: for one thing, theconstruction of the proof of Theorem 1.1 involves an arbitrary choice HARACTERIZING GRAPH C ∗ -CORRESPONDENCES 7 of basis (this is why we have avoided using notation like “ X E X ”),although it is trying hard to serve as an adjoint for Γ.Also, it is not hard to see directly that Γ can fail to be full, as in thefollowing example. Example 2.1.
Let E be the graph with a single vertex and a single loopedge, so that we can identify c ( V ) with the complex numbers. Then X = Γ( E ) is a one-dimensional Hilbert space, and a morphism fromthe correspondence X to itself consists of multiplication by a complexnumber of modulus one. However, there is only one morphism fromthe graph E to itself, so there are endomorphisms of X which are notof the form Γ( φ ) for any endomorphism φ of E .In spite of the above negative result, Γ makes a surprisingly closeconnection between G and C . For instance, it follows from Theorem 2.2below that (i) Γ is injective on objects (and hence Γ( G ) is a subcategoryof C which is isomorphic to G , since Γ is faithful), and (ii) Γ reflectsisomorphisms . Theorem 2.2.
Let V be a countable set, and let E and F be objectsof G ( V ) . (i) If Γ( E ) = Γ( F ) in C ( V ) , then E = F in G ( V ) . (ii) If φ : E → F in G ( V ) is such that Γ( φ ) : Γ( E ) → Γ( F ) is anisomorphism in C ( V ) , then φ is an isomorphism in G ( V ) .Proof. (i) Suppose E = F in Obj G . If there exists e ∈ E \ F , then χ e is an element of Γ( E ) which is not in Γ( F ); similarly, if f ∈ F \ E ,then χ f ∈ Γ( F ) \ Γ( E ). Thus if E = F , we have Γ( E ) = Γ( F ).On the other hand, if E = F , then since E = F we may choose e ∈ E = F such that either r E ( e ) = r F ( e ) or s E ( e ) = s F ( e ). In eithercase, if we set u = r E ( e ) and v = s E ( e ), then p u · χ e · p v = χ e in Γ( E ),but p u · χ e · p v = 0 in Γ( F ). Thus Γ( E ) = Γ( F ).(ii) For any φ : E → F in G and any u, v ∈ V we have φ ( uE v ) ⊆ uF v, where we have set uE v = { e ∈ E | r ( e ) = u and s ( e ) = v } andsimilarly uF v = { f ∈ F | r ( f ) = u and s ( f ) = v } . If we further let E uv = { χ e | e ∈ uE v } ⊆ Γ( E ) and F uv = { χ f | f ∈ uF v } ⊆ Γ( F ) , it follows thatΓ( φ )( E uv ) = { Γ( φ )( χ e ) | e ∈ uE v } = { χ φ ( e ) | e ∈ uE v } ⊆ F uv . Now E uv and F uv are orthonormal bases for the Hilbert spaces p u · Γ( E ) · p v and p u · Γ( F ) · p v , respectively (see the proof of Proposition 1.4). If S. KALISZEWSKI, NURA PATANI, AND JOHN QUIGG Γ( φ ) is an isomorphism in C , then Γ( φ ) restricts to a Hilbert spaceisomorphism between these two spaces, and this forces Γ( φ )( E uv ) = F uv . It follows that φ ( uE v ) = uF v , and hence (because morphismsin G are injective and fix vertices by definition) φ is an isomorphismin G . (cid:3) In light of the next theorem, Γ could be called essentially full : everymorphism in C is “isomorphic” in the obvious sense to a morphismin the range of Γ. (This usage extends the sense in which the term“essentially full” appears in [4], where it was applied to a functor whichwas surjective on objects.) Theorem 2.3.
Let V be a countable set. For each morphism ψ : X → Y in C ( V ) there exist objects E and F and a morphism φ : E → F in G ( V ) , and isomorphisms υ E and υ F in C ( V ) , such that the followingdiagram commutes in C ( V ) : X υ E ∼ = / / ψ (cid:15) (cid:15) Γ( E ) Γ( φ ) (cid:15) (cid:15) Y υ F ∼ = / / Γ( F ) . Proof.
Let E be a graph associated to X as in the proof of Theorem 1.1,and let υ E : X → X E = Γ( E ) be the isomorphism constructed there.So for each u, v ∈ V , we have chosen an orthonormal basis E uv for X uv ,and E = S u,v ∈ V E uv . Now ψ ( E uv ) is an orthonormal set in Y uv since ψ is isometric and because ψ ( p u · X · p v ) = p u · ψ ( X ) · p v . So we may extend ψ ( E uv ) to an orthonormal basis F uv of Y uv and set F = [ u,v ∈ V F uv . As in the proof of Theorem 1.1, this is a disjoint union so we may define r F , s F : F → V by r F ( f ) = u, s F ( f ) = v, if f ∈ F uv to get a graph F = ( V, F , r F , s F ) ∈ Obj G , and then the rule f χ f extends to an isomorphism υ F : Y → X F = Γ( F ) in C .Now define φ : E → F by letting φ = ψ on E , and letting φ bethe identity on V . Note that φ is injective on E since ψ is, so φ is amorphism in G . HARACTERIZING GRAPH C ∗ -CORRESPONDENCES 9 To show that the diagram commutes, it suffices to show that Γ( φ ) ◦ υ E = υ F ◦ ψ on E , since the linear span of E is dense in X . But for e ∈ E the definitions of the maps involved giveΓ( φ ) ◦ υ E ( e ) = Γ( φ )( χ e ) = χ φ ( e ) = χ ψ ( e ) = υ F ( ψ ( e )) = υ F ◦ ψ ( e ) . (cid:3) Remark 2.4.
Combining Theorem 2.2(ii) with Theorem 2.3, it is nothard to see that Γ is injective on isomorphism classes; that is, if Γ( E ) ∼ =Γ( F ) in C ( V ), then E ∼ = F in G ( V ). References [1] T. Katsura,
A class of C ∗ -algebras generalizing both graph algebras and home-omorphism C ∗ -algebras. I. Fundamental results , Trans. Amer. Math. Soc. (2004), no. 11, 4287–4322 (electronic).[2] M. V. Pimsner, A class of C ∗ -algebras generalizing both Cuntz-Krieger algebrasand crossed products by Z , Free probability theory (D.-V. Voiculescu, ed.), FieldsInst. Commun., vol. 12, Amer. Math. Soc., 1997, pp. 189–212.[3] I. Raeburn, Graph algebras , CBMS Regional Conference Series in Mathematics,vol. 103, Published for the Conference Board of the Mathematical Sciences,Washington, DC, 2005.[4] J. Rosick´y,
Generalized Brown representability in homotopy categories , TheoryAppl. Categ. (2005), no. 19, 451–479 (electronic).[5] J. Schweizer, Crossed products by C ∗ -correspondences and Cuntz-Pimsner alge-bras , C ∗∗