C*-correspondences associated to generalizations of ... fileC -correspondences associated to...

C ∗-correspondences associated togeneralizations of directed graphs

Nura PataniJoint work with S. Kaliszewski and J. Quigg

School of Mathematical & Statistical SciencesArizona State University

West Coast Operator Algebra SeminarSeptember 2, 2010

Kaliszewski, Patani*, Quigg (ASU) WCOAS, September 2, 2010 1 / 22

Preliminaries Definitions

Directed graphs & graph correspondences

Definition (Directed graph)

A quadruple E = (V ,E , s, r) where V ,E are countable sets andr , s : E → V are maps.

Definition (Graph correspondence)

XE = {ξ ∈ CE |the map v 7→∑

s(e)=v

|ξ(e)|2 is in c0(V )}

with c0(V )-valued inner product

〈ξ, η〉c0(V )(v) =∑

s(e)=v

ξ(e)η(e)

and c0(V )-bimodule actions

(f · ξ · g)(e) = f (r(e))ξ(e)g(s(e)).

s(e)=v

|ξ(e)|2 is in c0(V )}

〈ξ, η〉c0(V )(v) =∑

s(e)=v

ξ(e)η(e)

(f · ξ · g)(e) = f (r(e))ξ(e)g(s(e)).

s(e)=v

|ξ(e)|2 is in c0(V )}

〈ξ, η〉c0(V )(v) =∑

s(e)=v

ξ(e)η(e)

(f · ξ · g)(e) = f (r(e))ξ(e)g(s(e)).

s(e)=v

|ξ(e)|2 is in c0(V )}

〈ξ, η〉c0(V )(v) =∑

s(e)=v

ξ(e)η(e)

(f · ξ · g)(e) = f (r(e))ξ(e)g(s(e)).

Preliminaries Motivation

Motivation

Theorem (Kaliszewski, P, Quigg)

Every separable nondegenerate C ∗-correspondence over a commutativeC ∗-algebra with discrete spectrum V is isomorphic to the graphcorrespondence of a directed graph with vertex set V .

The k-graph case Definitions

k-graphs

Definition (Higher-rank graph)

A countable category Λ together with a functor d : Λ→ Nk satisfying theunique factorization property: ∀λ ∈ Λ, m, n ∈ Nk with d(λ) = m + n,there are unique µ, ν ∈ Λ such that λ = µν, d(µ) = m, and d(ν) = n.

Remark

k-graphs are essentially product systems over Nk of directed graphs.

k-graphs

Definition (Higher-rank graph)

A countable category Λ together with a functor d : Λ→ Nk satisfying theunique factorization property: ∀λ ∈ Λ, m, n ∈ Nk with d(λ) = m + n,there are unique µ, ν ∈ Λ such that λ = µν, d(µ) = m, and d(ν) = n.

Remark

k-graphs are essentially product systems over Nk of directed graphs.

Product systems

Definition (Product system)

Let S be a semigroup with identity e. Let G be a tensor category (see[Fowler and Sims, 2002]). A product system over S taking values in G is apair (X , β):

X = {Xt}t∈S , a collection of objects in G,

β = {βs,t}s,t∈S , a collection of isomorphisms βs,t : Xs ⊗ Xt → Xst

satisfying certain conditions including an associativity condition.

Product systems

The k-graph case Product Systems

Product systems

Example

1 S = Nk

⊗ = ∗, fibred product of graphs,Objects in G are directed graphs with fixed vertex set V ,Isomorphisms are bijections that preserve range and source maps,

Product system over Nk of directed graphs

2 Let A be a C ∗-algebra.

⊗ = ⊗A balanced tensor product over A,Objects in G are A-correspondences,Isomorphisms are linear isomorphisms that preserve inner product,

Product system over S of A-correspondences

Product systems

Example

1 S = Nk

Product systems

Example

1 S = Nk

Product systems

Example

1 S = Nk

Product systems

Example

1 S = Nk

Product systems

Example

1 S = Nk

Product systems

Example

1 S = Nk

Product systems

Example

1 S = Nk

Product systems

Example

1 S = Nk

Product systems

Example

1 S = Nk

Product systems

Example

1 S = Nk

The “Skeleton” of a Product System

Theorem (Fowler, Sims)

Every product system (X , β) over Nk determines a pair (Y ,T ), theskeleton of (X , β):

Y = {Yi = Xei}i∈{1,2,··· ,k} is a collection of objects in X , and

T = {Ti ,j}1≤i<j≤k is a collection of isomorphisms

Ti ,j = β−1ej ,ei◦ βei ,ej : Yi ⊗ Yj → Yj ⊗ Yi

Moreover, every pair (Y ,T ) satisfying T−1p,q = Tq,p and

(Tq,r ⊗ 1p)(1q ⊗ Tp,r )(Tp,q ⊗ 1r ) = (1r ⊗ Tp,q)(Tp,r ⊗ 1q)(1p ⊗ Tq,r )

gives rise to a product system (X , β) over Nk and this gives a completeisomorphism invariant for product systems.

k-graph correspondence

Definition (Higher-rank graph correspondence)

Given a higher rank graph (Λ, d) with Λ0 = V , for each m ∈ Nk set

Em = (V , d−1(m), s, r)

and let Xm be the graph correspondence associated to Em.

The higher rank graph correspondence is the product system (X , β) overNk of c0(V )-correspondences whose skeleton is (Y ,T ) where

Yi = Xei for i = 1, . . . , k, and

Ti ,j(χe ⊗ χf ) = χf ⊗ χe where f and e are the unique edges in Eej

and Eei , respectively, such that f e = ef in Eei+ej .

Em = (V , d−1(m), s, r)

Yi = Xei for i = 1, . . . , k, and

Em = (V , d−1(m), s, r)

Yi = Xei for i = 1, . . . , k, and

Em = (V , d−1(m), s, r)

Yi = Xei for i = 1, . . . , k, and

Isomorphic product systems

Let (X , β), (Z , γ) be isomorphic product systems over Nk ofA-correspondences, and let (Y ,T ), (W ,R) be their respective skeletons.Then there are A-correspondence isomorphisms θi : Yi →Wi such that thediagram commutes:

Yi ⊗ Yj Yj ⊗ Yi

Wi ⊗Wj Wj ⊗Wi

θi ⊗ θj θj ⊗ θi

Isomorphic product systems

Let (X , β), (Z , γ) be isomorphic product systems over Nk ofA-correspondences, and let (Y ,T ), (W ,R) be their respective skeletons.Then there are A-correspondence isomorphisms θi : Yi →Wi such that thediagram commutes:

Yi ⊗ Yj Yj ⊗ Yi

Wi ⊗Wj Wj ⊗Wi

θi ⊗ θj θj ⊗ θi

The k-graph case One-Dimensional Product Systems

Characterization of 1-Dimensional Product Systems

Theorem

For each product system (X , β) over N2 of 1-dimensionalC-correspondences, there exists ωX ∈ T such that the assignment(X , β) 7→ ωX passes to a complete invariant for isomorphism classes ofsuch product systems.

Moreover, (X , β) is isomorphic to a 2-graph correspondence if and only ifωX = 1.

Sketch of Proof

Let (X , β), (Z , γ) be isomorphic product systems over N2 of 1-dimensionalC-correspondences with skeletons (Y ,T ), (W ,R), respectively. Since thecorrespondences in the skeletons are 1-dimensional, T1,2,R1,2 must bemultiples ωX , ωZ of the flip maps.

Isomorphisms of c0(V )-correspondences are inner product preserving,hence isometric. Thus ωX , ωZ ∈ T.

The isomorphism condition gives us that this is a complete isomorphisminvariant.

Sketch of Proof

Sketch of Proof (cont’d)

Suppose (X , β) is a product system over N2 of 1-dimensionalC-correspondences arising from a 2-graph.

The isomorphism T1,2 must correspond to a graph isomorphismR1,2 : E1 ∗ E2 → E2 ∗ E1 where Ei is the directed graph whosecorrespondence is Xi , ∗ is the fibred product of graphs.

The isomorphism R1,2 can only be the flip map, hence ωX = 1.

Corollary

Let V have cardinality one. There exist uncountably many non-isomorphicproduct systems over N2 of c0(V )-correspondences that are notisomorphic to 2-graph correspondences.

Proof.

We show that the range of ω is all of T.Fix ω ∈ T. Set Y1 = Y2 = C (with the usual C-correspondence structure)and define T1,2 : Y1 ⊗ Y2 → Y2 ⊗ Y1 by

T1,2 = ωΣ.

It is straightforward to show that (Y ,T ) is the skeleton of a productsystem (X , β) over N2 of C-correspondences such that ωX = ω.

Corollary

Proof.

T1,2 = ωΣ.

Corollary

Proof.

T1,2 = ωΣ.

Corollary

Proof.

T1,2 = ωΣ.

The topological graph case Definitions

Definitions

Definition (Topological graph)

A quadruple E = (V ,E , s, r) where V ,E are locally compact Hausdorffspaces, r : E → V is continuous, and s : E → V is a localhomeomorphism.

Definition (Topological graph correspondence)

Given a topological graph E = (V ,E , s, r), the topological graphcorrespondence is the space

XE = {ξ ∈ C (E )| the map v 7→∑

e∈s−1(v)

|ξ(e)|2 is in C0(V )}

with C0(V )-valued inner product and C0(V )-bimodule actions

〈ξ, η〉C0(V )(v) =∑

e∈s−1(v)

ξ(e)η(e), (f · ξ · g)(e) = f (r(e))ξ(e)g(s(e)).

Definitions

XE = {ξ ∈ C (E )| the map v 7→∑

e∈s−1(v)

|ξ(e)|2 is in C0(V )}

〈ξ, η〉C0(V )(v) =∑

e∈s−1(v)

Definitions

XE = {ξ ∈ C (E )| the map v 7→∑

e∈s−1(v)

|ξ(e)|2 is in C0(V )}

〈ξ, η〉C0(V )(v) =∑

e∈s−1(v)

Isomorphic Graph Correspondences

Let V be a locally compact Hausdorff space and suppose E and F aretopological graphs with vertex space V .

Definition (Vertex-fixing isomorphishm)

A vertex-fixing isomorphism is a homeomorphism Φ : E 1 → F 1 such thatthe following diagram commutes:

V E 1roo s //

φ��

``@@@@@@@@ s

>>~~~~~~~~

A vertex fixing isomorphism E ∼= F gives rise to an isomorphism XE∼= XF

of C0(V )-correspondences.

V E 1roo s //

φ��

``@@@@@@@@ s

>>~~~~~~~~

V E 1roo s //

φ��

``@@@@@@@@ s

>>~~~~~~~~

The topological graph case Picard groups

Picard Groups

Let V be a locally compact Hausdorff space.

Definition (Pic V )

The group of isomorphism classes of complex line bundles over V withgroup operation fibrewise tensor product.

Definition (Pic C0(V ))

The group of isomorphism classes of C0(V )-imprimitivity bimodules withgroup operation balanced tensor product over C0(V ).

Definition (Symmetric C0(V )-bimodule)

A C0(V )-imprimitivity bimodule is symmetric if

f · ξ = ξ · f for all f ∈ C0(V ), ξ ∈ X .

Picard Groups

Definition (Pic V )

Picard Groups

Definition (Pic V )

Picard Groups

Definition (Pic V )

Picard Groups (cont’d)

Proposition (Raeburn)

Pic C0(V ) is isomorphic to the algebraic Picard group, comprising ofisomorphism classes of invertible C0(V )-bimodules.

The symmetric imprimitivity bimodules determine a normal subgroup ofPic C0(V ) which, by Raeburn’s version of Swan’s theorem, is isomorphicto Pic V .

Theorem (Abadie, Exel)

Pic C0(V ) is a semidirect product of its symmetric part and theautomorphism group of C0(V ).

Picard Invariant

Definition (Picard Invariant)

The element of Pic C0(V ) determined by a C0(V )-imprimitivity bimoduleX is called the Picard invariant of X .

The Picard invariant is trivial if it is the identity element of Pic C0(V ).That is, if X is in the class of the trivial C0(V )-imprimitivity bimodule,C0(V ) itself (with the usual C0(V )-imprimitivity bimodule structure).

Picard Invariant

Definition (Picard Invariant)

The element of Pic C0(V ) determined by a C0(V )-imprimitivity bimoduleX is called the Picard invariant of X .

The Picard invariant is trivial if it is the identity element of Pic C0(V ).That is, if X is in the class of the trivial C0(V )-imprimitivity bimodule,C0(V ) itself (with the usual C0(V )-imprimitivity bimodule structure).

The topological graph case Topological Graph Correspondences

The Picard Invariant of a Topological GraphCorrespondence

Theorem

Let X be a symmetric C0(V )-imprimitivity bimodule. Then X isisomorphic to a topological graph correspondence if and only if its Picardinvariant is trivial. Thus, if C0(V ) has nontrivial Picard group, then thereare C0(V )-correspondences that are not isomorphic to topological graphcorrespondences.

Sketch of Proof

⇐: Suppose the Picard invariant of X is trivial. Then

X ∼= C0(V ) ∼= XE , where E = (V ,V , id, id)

⇒: Let E be a topological graph with vertex space V such that X ∼= XE .

It suffices to find a vertex-fixing isomorphism E ∼= (V ,V , id, id).

It suffices to show that s : E 1 → V is a homeomorphism.

It suffices to show that |s−1(v)| ≤ 1 for each v ∈ V .

That |s−1(v)| ≤ 1 follows from properties of the Rieffel Correspondence.

Sketch of Proof

Thank you!

References

R. Abadie, B.and Exel, Hilbert C∗-bimodules over commutative C∗-algebrasand an isomorphism condition for quantum Heisenberg manifolds, Rev.Math. Phys. 9 (1997), no. 4, 411–423.

N. J. Fowler and A. Sims, Product systems over right-angled Artinsemigroups, Trans. Amer. Math. Soc. 354 (2002), 1487–1509.

S. Kaliszewski, N. Patani, and J. Quigg, Characterizing graphcorrespondences, to appear Houston J. Math.

I. Raeburn, On the Picard group of a continuous trace C∗-algebra, Trans.Amer. Math. Soc. 263 (1981), 183–205.

I. Raeburn and D. P. Williams, Morita equivalence and continuous-traceC∗-algebras, Math. Surveys and Monographs, vol. 60, AmericanMathematical Society, Providence, RI, 1998.

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