Simplicity Criterion

# Simplicity criterion for C∗-algebras associated with topological group quivers

Shawn J. Cann Department of Mathematics and Statistics, University of Regina, Regina, Sk, S4S 0A2
###### Abstract.

Topological quivers generalize the notion of directed graphs in which the sets of vertices and edges are locally compact (second countable) Hausdorff spaces. Associated to a topological quiver is a -correspondence, and in turn, a Cuntz-Pimsner algebra Given a locally compact group and and endomorphisms on one may construct a topological quiver with vertex set and edge set In [12], the author examined the Cuntz-Pimsner algebra and found generators (and their relations) of In this paper, the author translates a known criterion for simplicity of topological quivers into a precise criterion for the simplicity of topological group relations.

## 1. Introduction and Notation

### 1.1. Introduction

In 2005, Muhly and Tomforde [14] defined a generalization of a directed graph called a topological quiver. This is a 5-tuple where and are locally compact (second countable) Hausdorff spaces, and are continuous maps from to with open, and is a system of Radon measures. One can then create a corresponding Cuntz-Pimsner -algebra One rather important property of a -algebra is that of simplicity and Muhly and Tomforde calculate that is simple if and only if all open subsets with the property implies must be either or the empty set and if the set of all base points of loops in with no exit has empty interior. In general this can be a tedious and often difficult procedure to check.

In [11, 12, 13], the author defines a particularly interesting topological quiver

 Q=(Γ,Ωα,β(Γ),r,s,λ)

where is a locally compact group, and are endormorphism of

 Ωα,β(Γ)={(x,y)∈Γ×Γ∣∣α(y)=β(x)}

and is an appropriate family of Radon measures. Then forms the relative Cuntz-Pimsner Algebra denoted

 Oα,β(Γ).

In [12], the spatial structure including generators, relations, and co-limit structure are examined. In [13], a six-term exact sequence is prepared for the -groups of and is used to calculate the -groups for where is the -torus and and are integral matrices with non-zero determinant. With all this structure in place, the -algebra is a classifiable (by its -groups) Kirchberg algebra is it is also simple.

The aim of this paper is to recast the Muhly and Tomforde conditions into an easier condition for the -algebras described above and furthermore, to determine the simplicity of solely by the properties of and That is, it will be shown that one merely needs to check the density of the connected component of the identity.

### 1.2. Notation

The sets of natural numbers, integers, rationals numbers, real numbers and complex numbers will be denoted by , , , , and respectively. Convention: does not contain zero. will denote the set denotes the set and Finally, denotes the abelian group and denotes the torus Whenever convenient, view by

For a topological space , the closure of is denoted Given a locally compact Hausdorff space , let

1. be the continuous complex functions on ;

2. be the continuous and bounded complex functions on ;

3. be the continuous complex functions on vanishing at infinity;

4. be the continuous complex functions on with compact support.

The supremum norm is denoted and defined by

 ||f||∞=supx∈X{|f(x)|}

for each continuous map For a continuous function denote the open support of by and the support of by

For -algebras and , is isomorphic to will be written for example, we use Moreover, denotes the -fold direct sum Given a group and a ring , a normal subgroup, , of is denoted and an ideal, , of is denoted Note if is a -algebra then the term ideal denotes a closed two-sided ideal. Furthermore, () and () denotes the set of endomorphisms of () and automorphisms of ), respectively.

Let then denotes the endomorphism of defined by

 α#(f)=f∘αfor each f∈C(X).

The set of by matrices with coefficients in a set will be denoted and for any the transpose of is denoted .

## 2. Preliminairies

### 2.1. Hilbert C∗-modules

Further details can be found in [10, 17].

###### Definition 2.1.

[10] If is a -algebra, then a (right) Hilbert -module is a Banach space together with a right action of on and an -valued inner product satisfying

1. and

for all , and (if the context is clear, we denote simply by ). For Hilbert -modules and , call a function adjointable if there is a function such that for all and . Let denote the set of adjointable (-linear) operators from to . If , then is a -algebra (see [10].) Let denote the closed two-sided ideal of compact operators given by

 K(E,F):=¯¯¯¯¯¯¯¯¯¯¯span{θE,Fξ,η∣∣ξ∈E,η∈F}

where

 θE,Fξ,η(ζ)=ξ⟨η,ζ⟩Afor each ζ∈E.

Similarly, and (or if understood) denotes . For Hilbert -module , the linear span of , denoted , once closed is a two-sided ideal of . Note that is dense in ([10]). The Hilbert module refers to the Hilbert module over itself, where for all .

###### Definition 2.2.

[8] A subset is called a basis provided the following reconstruction formula holds for all

 ξ=∑i∈Iui⋅⟨ui,ξ⟩(in E,||⋅||.)

If as well, call an orthonormal basis of .

###### Definition 2.3.

[3, 4] If and are -algebras, then an -correspondence is a right Hilbert -module together with a left action of on given by a -homomorphism , for and . We may occasionally write, to denote an -correspondence and instead of . Furthermore, if and are -correspondences, then a morphism consists of -homomorphisms and a linear map satisfying

for all and .

###### Notation 2.4.

When , we refer to as a -correspondence over . For a -correspondence over and a -correspondence over , a morphism will be denoted by .

###### Definition 2.5.

[14] If is the Hilbert module where is a -algebra with the inner product then call a morphism of Hilbert modules a representation of into

###### Remark 2.6.

Note that a representation of need only satisfying and of definition 2.3 as it was unnecessary to require (iii) (see [12, Remark 2.7]).

A morphism of Hilbert modules yields a -homomorphism by

 ΨT(θEξ,η)=θFT(ξ),T(η)

for and if , and are morphisms of Hilbert modules then . In the case where a -algebra, we may first identify as , and a representation of in a -algebra yields a -homomorphism given by

 ΨT(θξ,η)=T(ξ)T(η)∗.
###### Definition 2.7.

[14] For a -correspondence over , denote the ideal of by and let where is the ideal . If and are -correspondences over and respectively and , a morphism is called coisometric on if

 ΨT(ϕA(a))=ϕB(π(a))

for all , or just coisometric, if .

###### Notation 2.8.

We denote to be the -algebra generated by and where is a representation of in a -algebra . Furthermore, if is a -homomorphism of -algebras, then denotes the representation of .

###### Definition 2.9.

[14] A morphism coisometric on an ideal is said to be universal if whenever is a representation coisometric on , there exists a -homomorphism with . The universal -algebra is called the relative Cuntz-Pimsner algebra of determined by the ideal and denoted by . If , then is denoted by and called the universal Toeplitz -algebra for . We denote by .

### 2.2. Topological Quivers

###### Definition 2.10.

[14] A topological quiver (or topological directed graph) is a diagram

 \begindc\commdiag[5]\obj(10,0)E\obj(0,0)X\obj(20,0)Y\morEXs[−1,0]\morEYr\enddc

where and are second countable locally compact Hausdorff spaces, and are continuous maps with open, along with a family of Radon measures on satisfying

1. for all , and

2. for

###### Remark 2.11.

If then write in lieu of

###### Remark 2.12.

The author provides a broad history and a series of examples of topological quivers in [11, 12].

Given a topological quiver , one may associate a correspondence of the -algebra to the -algebra . Define left and right actions

 (a⋅ξ⋅b)(e)=a(s(e))ξ(e)b(r(e))

by and respectively on . Furthermore, define the -valued inner product

 ⟨ξ,η⟩(y)=∫r−1(y)¯¯¯¯¯¯¯¯¯¯ξ(α)η(α)dλy(α)

for , and let be the completion of with respect to the norm

 ||ξ||=||⟨ξ,ξ⟩1/2||∞=||λy(|ξ|2)||1/2∞.
###### Definition 2.13.

Given topological quiver over a space , define the -algebra, associated with to be the Cuntz-Pimnser -algebra of the correspondence over .

### 2.3. Topological Group Quivers

###### Definition 2.14.

[12, 11] Let be a (second countable) locally compact group and let be continuous. Define the closed subgroup, of

 Ωα,β(Γ)={(x,y)∈Γ×Γ∣∣α(y)=β(x)}

and let where and are the group homomorphisms defined by

 r(x,y)=xands(x,y)=y

for each and for is the measure on

 r−1(x)={x}×α−1(β(x))

defined by

 λx(B)=μ(y−1s(B∩r−1(x))∩kerα)(for any% y∈α−1(β(x)))

for each measurable where is a left Haar measure (normalized if possible) on (a closed normal subgroup of hence, a locally compact group). Note if then and so This measure is well-defined,

 supp λx={x}×ykerα={x}×α−1(β(x))=r−1(x)

and is a continuous compactly supported function (cf. [12, Definition 3.1].

Call a topological group relation. Define to be the -correspondence and form the Cuntz-Pimsner algebra

 Oα,β(Γ):=C∗(Qα,β(Γ))=O(JEα,β(Γ),Eα,β(Γ))

and the Toeplitz-Pimsner algebra

 Tα,β(Γ):=T(Qα,β(Γ)).
###### Remark 2.15.

It will be implicitly assumed that is second countable. Furthermore, since is locally compact Hausdorff, is closed and locally compact. Moreover, whenever is a local homeomorphism, is discrete and hence, is counting measure (normalized when .)

###### Example 2.16 ([12]).

For the compact abelian group note ([19]); that is, an element is of the form for some where

 σF(e2πit)=e2πiFt

for each To simplify notation, use and in place of and whenever convenient. For instance,

 QF,G(Td):=QσF,σG(Td)

and the -correspondence

where . We will consider the cases when these maps are surjective; that is, and are non-zero.

Let where . Then and so, the -valued inner product becomes

 ⟨ξ,η⟩(x)=1|detF|∑σF(y)=σG(x)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ξ(x,y)η(x,y)

for and This is a finite sum since the number of solutions, to given any is The left action defined by

 ϕ(a)ξ(x,y)=a(y)ξ(x,y)

for and is injective (cf. [12, Remark 3.18]).

###### Remark 2.17.

It was shown in [12, Corollary 3.20] that one may assume the matrix is positive diagonal.

Let where for each and let denote the -th row of , Further, let and let

 I(F)={ν=(νj)dj=1∈Zd∣∣0≤νj≤aj−1}.

The -valued inner product becomes

 ⟨ξ,η⟩(x)=1N∑σF(y)=σG(x)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ξ(x,y)η(x,y)

for all and

Given , define by

 uν(x,y)=yν=d∏j=1yνj

for It was shown in [12] that is a basis for and also the following:

###### Theorem 2.18.

[12, Theorem 3.23] Let where and let be the -th row vector of . Further, let denote the set . Then is the universal -algebra generated by isometries and (full spectrum) commuting unitaries that satisfy the relations

1. for all

2. for all and

where denotes Furthermore, is the universal -algebra generated by isometries and commuting unitaries that satisfy relations (1)-(3)

## 3. Simplicity Criterion

We now recast known conditions for the simplicity of Cuntz-Pimsner algebras for the correspondences in our specific context.

###### Definition 3.1.

[14] Let be a topological quiver. A path in is a finite sequence denoted with for all with length . Denote the set of paths of length by .

If and then call a loop of length . The vertex is refered to as the base point of the loop . An exit of a loop is any where for some ()

###### Definition 3.2.

[14] Say has condition () if the set of base points of loops in with no exits has empty interior.

###### Definition 3.3.

[14] Let be a topological quiver. An open subset is hereditary if for some guarantees that . Say a hereditary subset is saturated if and imply .

###### Definition 3.4.

[14] A topological quiver, , is minimal if the only saturated hereditary open subsets of are and

###### Theorem 3.5.

[14, Theorem 10.2] If is a topological quiver then is simple if and only if is minimal and satisfies condition ().

We now focus our attention to determining the necessary and sufficient conditions to ensure minimality and condition () for the topological group quiver . The following lemma may be found in [19].

###### Lemma 3.6.

[19] Every continuous surjective endomorphism of a compact group is measure-preserving.
Proof. Let denote the normalized Haar measure of and define a probability measure for all Borel subsets of . The measure is regular. We have

 μ(α(x)⋅B)=m(α−1(α(x)⋅B))=m(x⋅α−1(B))=m(α−1(B))=μ(B)

for all Borel subsets of . Since maps onto , we see that is left translation invariant and thus, by uniqueness of the Haar measure, ; that is, for all Borel subset of .

###### Theorem 3.7.

Let . If either

1. is not an injection, or

2. and are surjections, is not an injection and

then satisfies condition ().
Proof. If is a loop of length in , then has the following form:

 e=e1...en

where

 ek=(xk,xk−1) for k=1,...,n,
 r(en)=xn=x0=s(e1)

and

 xk=r(ek)=s(ek+1)∈α−1(β(xk+1)) for k=1,...,n−1.

But if is not an injection then there is a non-trivial . Furthermore, for any , let Then and so, with the property that and Hence, has an exit and as was arbitrary, all loops have exits. Thus, the set of base points of loops without exits is empty and hence, has empty interior.

If, alternatively, is an automorphism (satisfies (2) and not (1)), then there exists an inverse of , call it and so, define

 Bn={x∈Γ∣∣(β−1α)n(x)=x}(base % points of loops of length n)

for each It is now evident that the set of base points of loops, is

 L=⋃n∈NBn.

Furthermore, note is closed since is continuous, and hence is measurable. Let for the ease of this argument. Note is a continuous surjective endomorphism, hence by the above lemma, is measure-preserving. Furthermore,

 α−10(Bn)=kerα⋅Bn.

Indeed, if then and hence, So for some and thus, implies and hence, Conversely, if for some then and thus,

Also, for with , and , implies

 x=αn0(x)=αn0(z1x)=αn0(z2y)=αn0(y)=y

and thus, that is, it is shown that is disjoint from whenever for

Hence, for Haar measure ,

 m(Bn)=m(α−10(Bn))=m(kerα⋅Bn)=m(⨆z∈kerαz⋅Bn)=∑z∈kerαm(Bn)=|kerα|m(Bn).

Since , it must be that that is, must have empty interior. Moreover, since is a Baire space, has empty interior. Therefore, satisfies condition .

###### Remark 3.8.

(1) of Theorem 3.7 does not require compact.

Now, with the standing assumption that and are surjective endomorphisms on a compact group with , we intend to characterize when is minimal. But first, a lemma is needed.

###### Lemma 3.9.

Let and be surjective endomorphisms on a compact group with finite kernels. An open set is a saturated hereditary subset of if and only if
Proof. Let be a saturated hereditary open subset of , then certainly,

 r(s−1(y))⊆U if and only if y∈U.

Now realize and so, . Hence, recharacterize by if and only if Furthermore, if then, by the surjectivity of , there exists such that But note that since otherwise, if then Hence,

 Uc⊆β−1(α(Uc)).

Next, if then and thus, . Conversely, first note so (remember .) Now if with , then

 u′∈α−1(α(U))=kerα⋅U=U.

Hence, but since is surjective, and so, .

This now yields

 Uc⊆kerα⋅Uc=α−1(α(Uc))=α−1((α(U))c)⊆Uc

and thus, . Use the normalized Haar measure

 μ(Uc)=μ(α−1(α(Uc)))=μ(β−1(α(Uc)))

and since both and are closed with , it must be that . Hence,

 U=(β−1(α(Uc)))c⊆β−1(α(U))⊆U;

that is,