aa r X i v : . [ m a t h . OA ] M a y TRACES ON TOPOLOGICAL GRAPH ALGEBRAS
CHRISTOPHER SCHAFHAUSER
Abstract.
Given a topological graph E , we give a complete description of tracial stateson the C ∗ -algebra C ∗ ( E ) which are invariant under the gauge action; there is an affinehomeomorphism between the space of gauge invariant tracial states on C ∗ ( E ) and Radonprobability measures on the vertex space E which are, in a suitable sense, invariantunder the action of the edge space E . It is shown that if E has no cycles, then everytracial state on C ∗ ( E ) is gauge invariant. When E is totally disconnected, the gaugeinvariant tracial states on C ∗ ( E ) are in bijection with the states on K (C ∗ ( E )). Introduction
The class of topological graph algebras was introduced by Katsura in [8] and furtherdeveloped in [9, 10, 11]. These C ∗ -algebras provide a simultaneous generalization of bothhomeomorphism algebras and the Cuntz-Kreiger graph algebras. This class of algebrascontains a large class of nuclear C ∗ -algebras including all AF-algebras, Kirchberg algebrassatisfying the Universal Coefficient Theorem (UCT), and separable, simple, unital, realrank zero A T -algebras. A large collection of examples is given in [9]. In fact, there iscurrently no example of a nuclear C ∗ -algebra satisfying the UCT which is known notto be defined by a topological graph, although Rørdam’s examples of simple, nuclear,C ∗ -algebras with finite and infinite projections constructed in [15] are likely candidates.In this paper, we consider the tracial states on topological graph algebras. Given ahomeomorphism σ of a compact metric space X , it is a standard fact that every Radonprobability measure µ on X which is invariant under σ induces a tracial state on thecrossed product C ( X ) ⋊ σ Z . For the Cuntz-Krieger graph algebras generated by a discretegraph, this problem was studied by Tomforde in [16]. Tomforde introduced the notion a graph trace which is a probability measure on the vertex set E of the graph E which is,in a suitable sense, invariant under the action of the edges E . We provide a simultaneousgeneralization of these two theorems. Below is a summary of our results. Theorem.
Let E be a topological graph. (1) There is an affine homeomorphism between the space of invariant measures on E and the space of gauge invariant tracial states on C ∗ ( E ) . (2) If E is totally disconnected, there is a affine homeomorphism between the spaceof invariant measures on E and the space of states on K (C ∗ ( E )) . (3) If E is has no cycles, every tracial state on C ∗ ( E ) is gauge invariant. Date : June 30, 2018.2010
Mathematics Subject Classification.
Primary: 46L05.
Key words and phrases.
Graph C ∗ -algebras, Topological graphs, Traces, K-theory. Given a tracial state τ on C ∗ ( E ), the composition C ( E ) C ∗ ( E ) C ι τ is a state on C ( E ) and hence is given by integration against a Radon probability measure µ on E . The invariance of µ can be verified directly using the Cuntz-Krieger relations inC ∗ ( E ).Most of the work involved is in proving the converse. The key observation is that givena tracial state τ on C ∗ ( E ), the GNS space L (C ∗ ( E ) , τ ) decomposes as a direct integralover the boundary path space ∂E developed by Yeend in [17] and [18]. The fibres of thedirect integral and the measure on the base space ∂E can be described explicitly in termsof the graph E and the invariant measure µ on E . Hence it is possible to build the GNSrepresentation of the desired tracial state using only the invariant measure on the graphand then the trace is constructed as a vector state on this space.More precisely, given a topological graph E , we define a Hilbert module L ( E ) over thealgebra C ( ∂E ) and a representation π : C ∗ ( E ) → B ( L ( E )). Given an invariant measure µ on the graph E , there is an induced measure ˜ µ on ∂E and extending scalars yields aHilbert space L ( E, µ ) := L ( E ) ⊗ C ( ∂E ) L ( ∂E, µ )and a representation π µ of C ∗ ( E ) on the space L ( E, µ ). Moreover, the representation π µ admits a cyclic vector ˆ u ∈ L ( E, µ ) such that τ := h ˆ u, π ( · )ˆ u i is a tracial state on C ∗ ( E ).This trace is indeed invariant under the gauge action and recovers the given measure µ when restricted to C ( E ).Describing all tracial states on C ∗ ( E ) would be a difficult task in general. However, itappears that every tracial state on C ∗ ( E ) will be gauge invariant in many important cases.For homeomorphism algebras, this is the case whenever the action is free and for discretegraphs, this is the case when the graph satisfies condition (K). In [10], Katsura introduceda freeness condition for topological graphs which yields a simultaneous generalization ofboth these notions. It is probably true that every tracial state on a topological graphalgebra C ∗ ( E ) is gauge invariant whenever E is free and, in particular, whenever C ∗ ( E )is simple. We were not able to prove this in full generality, but the result holds in thespecial case when E has no cycles.In the case when E is a totally disconnected topological graph, the gauge invarianttracial states on C ∗ ( E ) are precisely those which can be detected by the ordered groupK (C ∗ ( E )). As an abelian group, K (C ∗ ( E )) can be described explicitly via the Pimsner-Voiculescu sequence for topological graph algebras, but very little is known about theorder structure in general. When E is a minimal, totally disconnected graph and E iscompact, our results on traces can be used, at least in principle, to describe the positivecone in K (C ∗ ( E )) up to perforation. It is likely that K (C ∗ ( E )) is weakly unperforatedfor minimal, totally disconnected graphs E which would then give a complete descriptionof the positive cone on K (C ∗ ( E )).The paper is structured as follows. Section 2 recalls a few definitions and fundamentalresults on topological graphs in order to standardize notation. Section 3 is devoted to theboundary path space ∂E of E . Invariant measures on topological graphs are introducedin Section 4 and it is shown that invariant measures on E extend to invariant measures RACES ON TOPOLOGICAL GRAPH ALGEBRAS 3 on ∂E . In section 5, the representations of C ∗ ( E ) on the Hilbert module L ( E ) and theHilbert spaces L ( E, µ ) mentioned above are constructed. Section 6 contains most of ourmain results and the final section is devoted to totally disconnected graphs and K-theoreticcalculations.We end this introduction with some remarks on notation. On Hilbert modules andHilbert spaces, inner products are always linear in the second variable and conjugate linearin the first. For graph algebras, there are two competing conventions in the literature.We follow the conventions in Raeburn’s book [13], which agrees with the notation used byKatsura in [8] where topological graphs were introduced; in particular, an edge representsa partial isometry from the projection at the source of the edge into the projection at therange of the edge.We do not assume our spaces are second countable. Because of this, there are somemeasure theoretic technicalities which occur. The following extension of the MonotoneConvergence Theorem (see Proposition 7.12 in [5]) allows us to work around these issuesin the non-separable case. It will be used implicitly throughout the paper.
Monotone Convergence Theorem.
Suppose X is a locally compact Hausdorff spaceand µ is a Radon measure on X . If ( f n ) ⊆ C c ( X ) is an increasing net of positive functionsconverging pointwise to a (necessarily Borel) function f : X → [0 , ∞ ] , then Z X f dµ = lim n Z X f n dµ. Topological Graph Algebras
We recall briefly the definition of topological graphs and the C ∗ -algebras they generatein order to standardize notation. The reader is referred to Katsura’s paper [8] for details. Definition 2.1. A topological graph E = ( E , E , r, s ) consists of locally compact Haus-dorff spaces E and E , a continuous function r : E → E , and a local homeomorphism s : E → E . Elements of E and E are called vertices and edges respectively.There are two subsets of the vertex space E which have a special role in the theory. Definition 2.2.
There is a maximal open set E ⊆ E such that r restricts to a propersurjection r − ( E ) → E . Define E = E \ E . The elements of E and E arecalled regular and singular vertices, respectively. Definition 2.3. A path of length n in E consists of a word α · · · α n such that α i ∈ E and s ( α i ) = r ( α i +1 ) for each i = 1 , . . . , n −
1. Let E n denote the path of lengths n . Definethe finite path space of E by E ∗ = ` ∞ n =0 E n . We endow E n with relative product topologyand E ∗ with the topology generated by the E n . Extend the range and source maps to r, s : E ∗ → E by r ( α · · · α n ) = r ( α ) and s ( α · · · α n ) = s ( α n ) . Then r : E ∗ → E is continuous and s : E ∗ → E is a local homeomorphism.Define the infinite path space by E ∞ = { α α · · · : s ( α i ) = r ( α i +1 ) for every i ∈ N } Extend the range map to r : E ∞ → E by r ( α α · · · ) = r ( α ). CHRISTOPHER SCHAFHAUSER
Given a path α ∈ E ∗ ∪ E ∞ , we write | α | for the length of α . That is, | α | = n if α ∈ E n for n ∈ N and | α | = ∞ if α ∈ E ∞ .The most obvious choice of topology on E ∞ is the product topology. The defect isthat the product topology on E ∞ need not be locally compact. Yeend introduced a largerspace ∂E containing E ∞ and showed there is a natural locally compact topology on ∂E .See Section 3 below.For our purposes, it will be useful to partition the finite path space E ∗ into two subsets:paths starting at regular vertices and paths starting at singular vertices. We introducethe following notation. Definition 2.4.
Define E n sng = { α ∈ E n : s ( α ) ∈ E } and E n reg = { α ∈ E n : s ( α ) ∈ E } . The sets E ∗ sng and E ∗ reg are defined similarly.We now define a C ∗ -algebra C ∗ ( E ) associated to a topological graph E . First we recalla simple fact which will be need throughout the paper. The proof follows as in Lemmas1.4 and 1.5 of [8]. Lemma 2.5.
Suppose X and Y are locally compact Hausdorff spaces and ϕ : X → Y isa local homoeomorphism. If f ∈ C c ( X ) , the function Y → C , y X x ∈ ϕ − ( y ) f ( x ) is defined and is compactly supported and continuous. In fact, the sum occurring on theright is always finite. Given a topological graph E , the space C c ( E ) is a bimodule over C ( E ) with the leftand right actions given by a · ξ = ( a ◦ r ) ξ and ξ · a = ξ ( a ◦ s ) for a ∈ C ( E ) and ξ ∈ C c ( E ).Moreover, there is a C ( E )-valued inner product on C c ( E ) given by h ξ, η i ( v ) = X e ∈ s − ( v ) ξ ( e ) η ( e )for each ξ, η ∈ C c ( E ) and v ∈ E . Definition 2.6.
Let H ( E ) denote the C ∗ -correspondence obtained as the completion of C c ( E ) with respect to the inner product given above.The topological graph algebra C ∗ ( E ) is defined as the Cuntz-Pimsner algebra of theC ∗ -correspondence ( H ( E ) , C ( E )). We recall briefly how this is defined. Definition 2.7. A Toeplitz representation ( π , π ) of E on a C ∗ -algebra A consists of a ∗ -homomorphism π : C ( E ) → A and a linear map π : H ( E ) → A such that π ( ξ ) ∗ π ( η ) = π ( h ξ, η i ) and π ( a ) π ( ξ ) = π ( aξ )for all a ∈ C ( E ) and ξ, η ∈ H ( E ). RACES ON TOPOLOGICAL GRAPH ALGEBRAS 5
Given a Toeplitz representation ( π , π ) of E on A , for each k ≥
2, there is an inducedlinear map π k : H ( E ) ⊗ k → A given by π k ( ξ ⊗ . . . ⊗ ξ k ) = π ( ξ ) · · · π ( ξ k ) . For any k ∈ N , C c ( E k ) is a dense submodule of the tensor power H ( E ) ⊗ k ; given functions ξ , . . . , ξ k ∈ C c ( E ), the elementary tensor ξ ⊗ · · · ⊗ ξ k acts on E k by( ξ ⊗ · · · ⊗ ξ k )( α · · · α k ) = ξ ( α ) · · · ξ k ( α k )for all α · · · α k ∈ E k .For any locally compact Hausdorff space X , C b ( X ) denotes the C ∗ -algebra of continuousbounded functions on X . Proposition 2.8.
There is a ∗ -homomorphism λ : C b ( E ) → B ( H ( E )) given by pointwisemultiplication. Moreover, λ ( ξ ) ∈ K ( H ( E )) if and only if ξ ∈ C ( E ) . For any Hilbert module H and ξ, η ∈ H , let ξ ⊗ η ∗ denote the rank one operator on H defined by ( ξ ⊗ η ∗ )( ζ ) = ξ h η, ζ i for ζ ∈ H . Proposition 2.9.
Given a Toeplitz representation ( π , π ) of a topological graph E ona C ∗ -algebra A , there is an induced ∗ -homomorphism ϕ : K ( H ( E )) → A determined by ϕ ( ξ ⊗ η ∗ ) = π ( ξ ) π ( η ) ∗ . Definition 2.10.
A Toeplitz representation ( π , π ) of E is called a covariant representa-tion of E if ϕ ( λ ( a ◦ r )) = π ( a ) for all a ∈ C ( E ). The topological graph algebra C ∗ ( E )is defined as the universal C ∗ -algebra generated by a covariant representation of E .For a topological graph E , there is a natural action γ : T y C ∗ ( E ) which we nowdescribe. If π : C ( E ) → C ∗ ( E ) and π : H ( E ) → C ∗ ( E ) are the canonical maps, theaction γ is determined by γ z ( π ( a )) = π ( a ) and γ z ( π ( ξ )) = zπ ( ξ )for each a ∈ C ( E ), ξ ∈ H ( E ), and z ∈ T . The action γ is called the gauge action .We end this section with two technical lemmas which will be needed later. The firstfollows by combining Lemmas 1.15 and 1.16 in [8]. Lemma 2.11. If ζ ∈ C c ( E ) , there are n ∈ N and ξ i , η i ∈ C c ( E ) for i = 1 , . . . , n , suchthat (1) ζ = P ni =1 ξ i η i , (2) λ ( ζ ) = P ni =1 ξ i ⊗ η ∗ i , and (3) if e, e ′ ∈ E are distinct edges with s ( e ) = s ( e ′ ) , then ξ i ( e ) η i ( e ′ ) = 0 . Lemma 2.12.
Let ( π , π ) be a Toeplitz representation of a topological graph E and let ϕ denote the induced representation of K ( H ( E )) . If ζ ∈ C c ( E ) and a ∈ C c ( E ) with ≤ ζ ≤ a ◦ r , then ≤ ϕ ( λ ( ζ )) ≤ π ( a ) .Proof. Let H denote the Fock space associated to H ( E ). The Toeplitz representation( π , π ) of E on B ( H ) defined by π ( a )( φ ) = aφ and π ( ξ )( φ ) = ξ ⊗ φ CHRISTOPHER SCHAFHAUSER is the universal Toeplitz representation of E . Hence it is enough to verify the lemma forthis representation.Fix ξ i , η i ∈ C c ( E ) for 1 ≤ i ≤ n as in Lemma 2.11. Given φ ∈ H ( E ), φ ′ ∈ H , note that h ϕ ( λ ( ζ ))( φ ⊗ φ ′ ) , φ ⊗ φ ′ i = X i h ( ξ i ⊗ η ∗ i )( φ ) ⊗ φ ′ , φ ⊗ φ ′ i = h λ ( ζ )( φ ) ⊗ φ ′ , φ ⊗ φ ′ i≤ h λ ( a ◦ r )( φ ) ⊗ φ ′ , φ ⊗ φ ′ i = h π ( a )( φ ⊗ φ ′ ) , φ ⊗ φ ′ i . Moreover, when φ ∈ H ⊗ = C ( E ), we have h ϕ ( λ ( ζ ))( φ ) , φ i = 0 ≤ h π ( a )( φ ) , φ i . It follows that h ϕ ( λ ( ζ )) φ, φ i ≤ h π ( a ) φ, φ i for all φ ∈ H and hence ϕ ( λ ( ζ )) ≤ π ( a ). (cid:3) The Boundary Path Space
In [17] and [18], Yeend introduced a locally compact Hausdorff space ∂E associated toa (higher rank) topological graph E called the boundary path space . In the case of a rankone topological graph E with E = E , ∂E is exactly the infinite path space E ∞ withthe product topology. In general, the product topology on E ∞ will not be locally compactand the space ∂E is the appropriate replacement.Kumjian and Li gave a simpler description of ∂E for rank one topological graphs in[12] (see also [19] for the case when E is discrete). We follow their approach. For ourpurposes, it will be useful to have approximate versions ∂E n of the boundary path spacewhose projective limit is ∂E (see Proposition 3.3).Recall from Definition 2.4, E ∗ sng and E ∗ reg denote the set of paths whose source is asingular or regular vertex, respectively. Define ∂E n = E ∪ · · · ∪ E n − ∪ E n and ∂E = E ∗ sng ∪ E ∞ . For S ⊆ E ∗ and n ∈ N , define Z n ( S ) = { α ∈ ∂E n : r ( α ) ∈ S or α · · · α k ∈ S for some 1 ≤ k ≤ n } ⊆ ∂E n and define Z ( S ) ⊆ ∂E similarly.The spaces ∂E and ∂E n carry natural locally compact Hausdorff topologies (Proposition3.2). This was shown by Yeend in [17] for higher rank topological graphs using a differentdescription of ∂E . That our definition agrees with Yeend’s is Lemma 4.5 in [12]. Theresult for ∂E n has an identical proof. We give a careful description of these topologies anda proof that they are locally compact and Hausdorff since the proof is not easy to extractfrom [17] and the details are omitted in [12].First we need a topological lemma. The proof is an easy exercise in constructing subnets. Lemma 3.1.
Suppose X is a topological space, ( x i ) i ∈ I is a net in X , and x ∈ X . If forevery neighborhood U of x , the set { i ∈ I : x i ∈ U } is cofinal in I , then ( x i ) has a subnetconverging to x . Proposition 3.2.
For n ∈ N , the sets of the form Z n ( U ) \ Z n ( K ) (resp. Z ( U ) \ Z ( K ) ) foropen sets U ⊆ E ∗ and compact sets K ⊆ E ∗ form a basis for a locally compact Hausdorfftopology on ∂E n (resp. ∂E ).Moreover, for any compact set K ⊆ E ∗ , Z n ( K ) is compact in ∂E n and Z ( K ) is compactin ∂E . RACES ON TOPOLOGICAL GRAPH ALGEBRAS 7
Proof.
For notational convenience, we write ∂E ∞ := ∂E and Z ∞ ( S ) := Z ( S ) for S ⊆ E ∗ .Fix n ∈ N ∪ {∞} .First we show the sets described above form a basis for ∂E n . Suppose U, V ⊆ E ∗ areopen and K, L ⊆ E ∗ are compact and assume α ∈ ( Z n ( U ) \ Z n ( K )) ∪ ( Z n ( V ) \ Z n ( L )).Then there are k, ℓ ∈ N such that k, ℓ ≤ | α | , α · · · α k ∈ U and α · · · α ℓ ∈ V (if k = 0or ℓ = 0, these expressions are interpreted as r ( α )). Without loss of generality, assume k ≤ ℓ . Note that the set W := { β ∈ V : | β | ≥ k and β · · · β k ∈ U } is open in E ∗ . Also, α ∈ Z n ( W ) and Z n ( W ) ⊆ Z n ( U ) ∩ Z n ( V ). Moreover, K ∪ L iscompact and Z n ( K ) ∪ Z n ( L ) = Z n ( K ∪ L ). As α ∈ Z n ( W ) \ Z n ( K ∪ L ) ⊆ ( Z n ( U ) \ Z n ( K )) ∩ ( Z n ( V ) \ Z n ( L )) , the sets described in the statement of the proposition form a basis for ∂E n .To see the topology is Hausdorff, fix distinct paths α, β ∈ ∂E n . Let k = | α | and ℓ = | β | ,and assume, without loss of generality, k ≤ ℓ . If k = ℓ = ∞ , there is an m ∈ N suchthat α · · · α m = β · · · β m . Fix disjoint open neighborhoods U, V ⊆ E m of α · · · α m and β · · · β m , respectively. Then α ∈ Z n ( U ), β ∈ Z n ( V ), and Z n ( U ) ∩ Z n ( V ) = ∅ . If k < ∞ and α · · · α k = β · · · β k , we can proceed as above. The case that remains is when k < ℓ and α · · · α k = β · · · β k . Fix a relatively compact open set U ⊆ E k +1 containing β · · · β k +1 . Then Z n ( U ) and ∂E n \ Z n ( ¯ U ) are disjoint open sets containing β and α ,respectively. Hence ∂E n is Hausdorff.Next we show Z n ( K ) is compact for every compact set K ⊆ E ∗ . We may assume K ⊆ E k for some k ∈ N with k ≤ n . Let ( α i ) be a net in Z n ( K ). We first show thereis an α ∈ ∂E n such that for every that for every ℓ ∈ N with ℓ ≤ | α | , there is a subnetof ( α i · · · α iℓ ) converging to α · · · α ℓ . Note in particular that the choice of subnet maydepend on ℓ , but α does not.Consider the net ( α i · · · α ik ) ⊆ K . Since K is compact, there is a subnet convergingto a path α · · · α k ∈ K . If k = n or s ( α k ) ∈ E , then α = α · · · α k ∈ ∂E n andthe claim holds. Otherwise, there is a compact neighborhood L ⊆ E of s ( α k ). If( α j · · · α jk ) is a subnet of ( α i · · · α ik ) converging to α · · · α k , then for sufficiently large j , s ( α jk ) ∈ L ⊆ E . In particular, α j · · · α jk / ∈ ∂E n and hence | α j | > k . Now, for large j , α jk +1 ∈ r − ( L ). Since r − ( L ) is compact, there is a subnet of α jk +1 converging to some α k +1 . Then s ( α k +1 ) = r ( α k ) by the continuity of r and s and hence α · · · α k +1 ∈ E k +1 .Also, ( α i · · · α ik ) has a subnet converging to α · · · α k by construction. Continuing byinduction yields the desired path α ∈ ∂E n .Having constructed α , we now show the net ( α i ) has a subnet converging to α . Fix anopen set U ⊆ E ∗ and a compact set L ⊆ E ∗ with α ∈ Z n ( U ) \ Z n ( L ). We may assume U ∩ E p and L ∩ E p are empty for p > n . Then for some ℓ ∈ N , α · · · α ℓ ∈ U . As L iscompact, there is an integer m with ℓ ≤ m ≤ n such that E p ∩ L = ∅ for p > m . By theconstruction of α , there is a subnet ( α j ) of ( α i ) such that α j · · · α jm converges to α . . . α m in E m . Since ℓ ≤ m and U is open, for sufficiently large j , α j . . . α jℓ ∈ U . Moreover, foreach 0 ≤ p ≤ m and sufficiently large j , α j . . . α jp / ∈ L since L is closed. In particular, CHRISTOPHER SCHAFHAUSER α j ∈ Z n ( U ) \ Z n ( L ) for large j and hence { i ∈ I : α i ∈ Z n ( U ) \ Z n ( L ) } is cofinal in I . By Lemma 3.1, the net ( α i ) has a subnet converging to α . Hence Z n ( K )is compact.In only remains to show ∂E n is locally compact. Given any α ∈ ∂E n , choose a relativelycompact open set U ⊆ E containing r ( α ). Then Z n ( U ) is an open set containing α . Bythe work above, Z n ( ¯ U ) is compact and α ∈ Z n ( U ) ⊆ Z n ( ¯ U ). This completes the proof. (cid:3) For each n ≥
1, define maps ρ n : ∂E n → ∂E n − by ρ n ( α · · · α k ) = ( α · · · α n − k = nα · · · α k k < n and define ρ n, ∞ : ∂E → ∂E n similarly. Proposition 3.3.
The maps ρ n and ρ n, ∞ are proper, continuous, and surjective. More-over, the maps ρ ∞ ,n induce a homeomorphism ∂E −→ lim ←− ( ∂E n , ρ n ) . Proof.
It is easy to see the maps ρ n are surjective. Indeed, if α ∈ ∂E n − with s ( α ) ∈ E ,then α is in the range of ρ n . Otherwise, | α | = n − s ( α ) ∈ E . By the definitionof E , there is an e ∈ E with r ( e ) = s ( α ). Hence αe ∈ E n ⊆ ∂E n and ρ n ( αe ) = α .Applying this argument by induction shows ρ n, ∞ is surjective.To see continuity, note that if S ⊆ E ∗ and S ∩ E n = ∅ , then ρ − n ( Z n − ( S )) = Z n ( S ). Inparticular, if U ⊆ E ∗ is open and K ⊆ E ∗ is compact with U ∩ E n = K ∩ E n = ∅ , then ρ − n ( Z n − ( U ) \ Z n − ( K )) = Z n ( U ) \ Z n ( K ). So ρ n is continuous. If K ⊆ ∂E n − is compact,there are relatively compact open sets U , . . . , U m ⊆ E ∗ with K ⊆ S ni =1 Z n − ( U i ). Now, ρ − n ( K ) ⊆ n [ i =1 ρ − n ( Z n − ( U i )) ⊆ n [ i =1 Z n ( ¯ U i ) . As each Z ( ¯ U i ) is compact by Proposition 3.2 and ρ − n ( K ) is closed by the continuity of ρ n , ρ − n ( K ) is compact. Hence ρ n is proper. Nearly identical arguments show ρ n, ∞ iscontinuous and proper for each n .It remains to show ∂E is the projective limit of the ∂E n . Suppose X is a topologicalspace and f n : X → ∂E n are continuous maps such that ρ n +1 f n +1 = f n for each n ≥ f : X → ∂E such that ρ n, ∞ f = f n for each n ∈ N . It sufficesto verify f is continuous. Fix a relatively compact open set U ⊆ E ∗ and a compact set K ⊆ E ∗ . There is an n ∈ N such that U ∩ E m and K ∩ E m are empty for each m > n .Then Z ( U ) \ Z ( K ) = ρ − n, ∞ ( Z n ( U ) \ Z n ( K )). Now, f − ( Z ( U ) \ Z ( K )) = f − ( ρ − n, ∞ ( Z n ( U ) \ Z n ( K ))) = f − n ( Z n ( U ) \ Z n ( K ))is open by the continuity of f n . So f is continuous. (cid:3) RACES ON TOPOLOGICAL GRAPH ALGEBRAS 9
Definition 3.4.
For n ≥
1, define the backwards shift σ n : ∂E n \ E → ∂E n − by σ ( α · · · α k ) = ( s ( α ) k = 1 α · · · α k k ≥ . and define σ : ∂E \ E → ∂E similarly.Note that the maps σ n and ρ n satisfy the following compatibility conditions: σ n − ρ n = ρ n − σ n and σ n ρ n, ∞ = ρ n, ∞ σ. on ∂E n \ E and ∂E \ E , respectively. Proposition 3.5.
The maps σ n : ∂E n \ E → ∂E n − and σ : ∂E \ E → ∂E are localhomeomorphisms.Proof. As in the proof of Proposition 3.2, we let ∂E ∞ := ∂E , σ ∞ := σ , and Z ∞ ( S ) = Z ( S )for all S ⊆ E ∗ .First we claim the maps σ ′ = s : E → E and σ ′ n : E n → E n − , n ≥
2, given by σ ′ n ( α · · · α n ) = α · · · α n are continuous and open. Continuity is clear and the map σ ′ = s is open by the definitionof a topological graph. Fix n ≥ U ⊆ E n be open. Fix α ∈ U and consider a net( β i ) ⊆ E n − with β i → σ ′ n ( α ). It’s enough to show β i ∈ σ ′ n ( U ) for large i . Let V ⊆ E bean open neighborhood of α such that s | V is a homeomorphism. Then r ( β i ) → r ( σ ′ n ( α )) = s ( α ) ∈ s ( V ) . Since s is a local homeomorphism, s ( V ) is open and hence r ( β i ) ∈ s ( V ) for large i . Hence,for large i , there is an e i ∈ V with s ( e i ) = r ( β i ). As s | V is a homeomorphism and s ( e i ) → s ( α ), we have e i → α . Hence e i β i → α . So for sufficiently large i , e i β i ∈ U and β i = σ ′ n ( e i β i ) ∈ σ ′ n ( U ). This shows σ ′ n ( U ) is open. Hence σ ′ n is open and continuous.Let σ ′ : E ∗ \ E → E ∗ denote the map induced by the σ ′ n , n ≥
1. Then σ ′ is open andcontinuous. Given an open set U ⊆ E ∗ and a compact set K ⊆ E ∗ , and n ∈ N ∪ {∞} , wehave σ n (( Z n ( U ) \ Z n ( K )) \ E ) = Z n − ( σ ′ n ( U \ E )) \ Z n − ( σ ′ n ( K \ E )) . As σ ′ n is open and continuous, σ ′ n ( U \ E ) ⊆ E ∗ is open and σ ′ n ( K \ E ) ⊆ E ∗ is compact.Therefore, σ n is open.To show continuity, suppose α i ∈ ∂E \ E is a net converging to α ∈ ∂E \ E . Let U ⊆ E ∗ be an open set and K ⊆ E ∗ be a compact set with σ n ( α ) ∈ Z n − ( U ) \ Z n − ( K ).For some m ≥ | α | ≥ m and α · · · α m ∈ U (where this expression is interpreted as s ( α ) when m = 1). It follows that | α i | ≥ m for all large i and α i · · · α im → α · · · α m in E m − . Hence σ n ( α i ) ∈ Z n − ( U ) for sufficiently large i . A similar argument shows σ n ( α i ) / ∈ Z n − ( K ) for sufficiently large i and hence σ n ( α i ) → σ n ( α ) and σ is continuous.Having shown σ n is open and continuous, it only remains to show σ is locally injective.Fix α ∈ ∂E n \ E . Let V ⊆ E be an open neighborhood of α such that s | V is injective.Then Z n ( V ) is an open neighborhood of α ∈ ∂E n \ E . Given β, β ′ ∈ Z ( V ) with σ n ( β ) = σ n ( β ′ ), we have β , β ′ ∈ V and s ( β ) = r ( σ n ( β )) = r ( σ n ( β ′ )) = s ( β ′ ). So β = β ′ . Now, β = β σ n ( β ) = β ′ σ n ( β ′ ) = β ′ . (cid:3) Invariant Measures on Topological Graphs
Our goal is to describe the tracial states on a topological graph algebra C ∗ ( E ). Notethat any tracial state on C ∗ ( E ) yields a Radon probability measure on E by composingthe tracial state with the canonical map π : C ( E ) → C ∗ ( E ) and applying the ReiszRepresentation Theorem. Definition 4.1 below describes all measures on E which arisein this way (see Theorem 6.3). We first introduce some notation.Suppose X and Y are locally compact Hausdorff spaces and ϕ : X → Y is a localhomeomorphism. Given f ∈ C c ( X ), the function Y → C : y X x ∈ ϕ − ( y ) f ( x )is continuous and has compact support by Lemma 2.5. Given a Radon measure µ on Y ,let ϕ ∗ µ denote the unique Radon measure on X satisfying Z X f dϕ ∗ µ = Z Y X x ∈ ϕ − ( y ) f ( x ) dµ ( y )for all f ∈ C c ( X ).In particular, note that if E is a topological graph, then s : E → E is a localhomeomorphism. Thus a Radon measure µ on E induces a Radon measure s ∗ µ on E . Definition 4.1. An invariant measure µ on E is a Radon probability measure µ on E such that for all positive functions f ∈ C c ( E ),(1) Z E f ◦ r ds ∗ µ ≤ Z E f dµ, with equality when supp( f ) ⊆ E . Let T ( E ) denote the set of invariant measures on E . Remark 4.2. If µ is an invariant measure on E , then s ∗ µ has mass at most 1. To see this,note that (1) holds for all positive f ∈ C b ( E ) by the Monotone Convergence Theorem.Then taking f = 1 in (1) shows ( s ∗ µ )( E ) ≤ µ ( E ) = 1. The measure s ∗ µ will not havemass 1 in general. It is easy to construct examples of this even with finite graphs.We end this section by showing invariant measures on a topological graph E inducemeasures on the boundary path space ∂E satisfying a natural invariance condition. Firstwe need a measure theoretic lemma. Lemma 4.3.
Let µ be an invariant measure on E . For any function f ∈ C b ( E ) , (2) Z r − ( E ) f ◦ r ds ∗ µ = Z E f dµ, and for any positive function f ∈ C c ( E ) , (3) Z r − ( E ) f ◦ r ds ∗ µ ≤ Z E f dµ. RACES ON TOPOLOGICAL GRAPH ALGEBRAS 11
Proof.
We first prove (2). For any positive function f ∈ C c ( E ), this is an immediateconsequence of Definition 4.1. The result follows by the Monotone Convergence Theoremand the linearity of the integral using that E is an open subset of E .Now suppose f ∈ C c ( E ) is positive. By the Tietze Extension Theorem, we mayextend f to a positive function, still denoted f , in C c ( E ). As f | E is a bounded,continuous function on E , (3) follows by subtracting (2) from (1). (cid:3) Proposition 4.4.
Suppose E is a topological graph and µ is an invariant measure on E .There is a Radon probability measure ˜ µ on ∂E such that Z ∂E f ◦ r d ˜ µ = Z E f dµ for each f ∈ C c ( E )(4) and Z ∂E \ E f dσ ∗ ˜ µ = Z ∂E f d ˜ µ for each f ∈ C c ( ∂E \ E ) . (5) Proof.
For n ∈ N , we define measure ˜ µ n on ∂E n inductively as follows. Set ˜ µ = µ andgiven ˜ µ n , let ˜ µ n +1 denote the Radon measure on ∂E n +1 determined by(6) Z ∂E n +1 f d ˜ µ n +1 = Z ∂E n +1 \ E f dσ ∗ n +1 ˜ µ n + Z E f dµ − Z r − ( E ) f ◦ r ds ∗ µ for every f ∈ C c ( ∂E n +1 ). Note in particular that if f ∈ C c ( ∂E n +1 ), then f | E ∈ C c ( E )since E is closed in ∂E n +1 and hence ˜ µ n +1 is positive by (3).Next we show(7) Z ∂E n +1 f ◦ ρ n +1 d ˜ µ n +1 = Z ∂E n f d ˜ µ n for every f ∈ C c ( ∂E n ). Assume first n = 0 and f ∈ C c ( ∂E ). Then since ρ ( v ) = v for all v ∈ E , Z ∂E f ◦ ρ d ˜ µ = Z E f ◦ r ds ∗ µ + Z E f dµ − Z r − ( E ) f ◦ r ds ∗ µ = Z r − ( E ) f ◦ r ds ∗ µ + Z E f dµ (2) = Z E f dµ + Z E f dµ = Z E f dµ. So (7) holds when n = 0.Suppose n ≥ n −
1. Let f ∈ C c ( ∂E n \ E ) be given. Since σ n : ∂E n \ E → ∂E n − is a local homeomorphism, the function g : ∂E n − → C definedby g ( α ) = X σ n ( β )= α f ( β ) = X s ( e )= r ( α ) f ( eα ) is continuous and has compact support by Lemma 2.5. Now we compute Z ∂E n +1 f ◦ ρ n +1 dσ ∗ n +1 ˜ µ n = Z ∂E n X s ( e )= r ( α ) f ( ρ n +1 ( eα )) d ˜ µ n ( α )= Z ∂E n X s ( e )= r ( ρ n ( α )) f ( eρ n ( α )) ˜ µ n ( α )= Z ∂E n g ◦ ρ n d ˜ µ n (7) = Z ∂E n − g d ˜ µ n − = Z ∂E n − X s ( e )= r ( α ) f ( eα ) d ˜ µ n − = Z ∂E n \ E f dσ ∗ n ˜ µ n − . This shows(8) Z ∂E n +1 \ E f ◦ ρ n +1 dσ ∗ n +1 ˜ µ n +1 = Z ∂E n \ E f dσ ∗ n ˜ µ n − for all f ∈ C c ( ∂E n \ E ) and hence for all f ∈ C b ( ∂E n \ E ) by the Monotone ConvergeTheorem using that ∂E n \ E is an open subset of ∂E n .Now, assuming n ≥ n −
1, let f ∈ C c ( ∂E n ). Using (8) and the factthat ρ n +1 restricts to the identity map on E , we have Z ∂E n +1 f ◦ ρ n +1 d ˜ µ n +1(6) = Z ∂E n +1 \ E f ◦ ρ n +1 σ ∗ n +1 ˜ µ n + Z E f ◦ ρ n +1 dµ − Z r − ( E ) f ◦ ρ n +1 ◦ r dµ (8) = Z ∂E n \ E f dσ ∗ n ˜ µ n − + Z E f dµ − Z r − ( E ) f ◦ r dµ (6) = Z ∂E n f d ˜ µ n , which verifies (7).Note that (7) holds for all continuous bounded functions f on ∂E n +1 , and in particular,taking f = 1 in (7) shows ˜ µ n is a probability measure for each n ≥ µ = µ is. Now,using (7), Proposition 3.3, and Gelfand Duality, the measures ˜ µ n on ∂E n induce a Radonprobability measure ˜ µ on ∂E such that(9) Z ∂E f ◦ ρ ∞ ,n d ˜ µ = Z ∂E n f d ˜ µ n for every f ∈ C c ( ∂E n ) and n ∈ N .It is immediate from the construction that (4) holds. To show (5), it suffices to show(10) Z ∂E \ E f ◦ ρ n, ∞ dσ ∗ ˜ µ = Z ∂E f ◦ ρ n, ∞ d ˜ µ RACES ON TOPOLOGICAL GRAPH ALGEBRAS 13 for each n ≥ f ∈ C c ( ∂E n \ E ). Calculations similar to those above yield Z ∂E \ E f ◦ ρ n, ∞ dσ ∗ ˜ µ = Z ∂E n \ E f dσ ∗ n ˜ µ n − . By the definition of ˜ µ n , if f ∈ C c ( ∂E \ E ), then Z ∂E n \ E f dσ ∗ n ˜ µ n − = Z ∂E f d ˜ µ n (9) = Z ∂E f ◦ ρ n, ∞ d ˜ µ. The last two equalities verify (10) and hence (5) holds. (cid:3)
For a vertex v ∈ E , write E k v for the set of paths α ∈ E k such that s ( α ) = v . Applying(5) by induction yields the following corollary. Corollary 4.5.
Let µ be an invariant measure on E and let ˜ µ denote the induced measureon ∂E as in Proposition 4.4. For any integer k ≥ and f ∈ C c ( ∂E \ ∂E k − ) , (11) Z ∂E X β ∈ E k r ( α ) f ( βα ) d ˜ µ ( α ) = Z ∂E f d ˜ µ The Boundary Representation
In this section, we construct a Hilbert C ( ∂E )-module L ( E ) and a faithful repre-sentation π : C ∗ ( E ) → B ( L ( E )). Roughly, this can be thought of as the left regularrepresentation of the graph on it’s boundary path space. Moreover, if µ is an invariantmeasure on E , there is an induced representation π µ of C ∗ ( E ) on a Hilbert space L ( E, µ )(see Definition 5.11). This representation will be used in the next section to show µ inducesa tracial state on C ∗ ( E ).The following is a standard construction in the groupoid literature (see, for exampleDefinition 2.4 in [14]). Define the subset X ⊆ ∂E × Z × ∂E by X = { ( α, k − ℓ, β ) : k, ℓ ∈ N , α, β ∈ ∂E, | α | ≥ k, | β | ≥ ℓ, σ k ( α ) = σ ℓ ( β ) } . Since σ : ∂E \ E → ∂E is a local homeomorphism, the sets Z ( U, k, ℓ, V ) := { ( α, k − ℓ, β ) ∈ X : α ∈ U, β ∈ V, σ k ( α ) = σ ℓ ( β ) } where k, ℓ ∈ N , U ⊆ ∂E \ ∂E k − and V ⊆ ∂E \ ∂E ℓ − are open, and σ k (resp. σ ℓ ) is ahomeomorphisms on U (resp. V ) form a basis for a locally compact Hausdorff topologyon X . Moreover, the map d : X → ∂E given by d ( α, n, β ) = β is a local homeomorphism. Remark 5.1.
After identifying ∂E with the subspace { ( β, , β ) : β ∈ ∂E } ⊆ X , thespace X forms a locally compact, Hausdorff, amenable, `etale groupoid with unit space ∂E . Moreover the C ∗ -algebra generated by X is precisely C ∗ ( E ) as was shown by Yeendin [18] (see also [12] where the twisted analogue of this result is proven).We recall the from Definition 6.4 in [12] the characterization of convergent nets in X . Lemma 5.2.
Consider a net ( α i , n i , β i ) ⊆ X and a point ( α, n, β ) ∈ X . Find integers k, ℓ ≥ such that (1) n = k − ℓ , | α | ≥ k , | β | ≥ ℓ , and σ k ( α ) = σ ℓ ( β ) , and (2) If k ′ , ℓ ′ ≥ satisfy (1) with k ′ ≤ k and ℓ ′ ≤ ℓ , then k = k ′ and ℓ = ℓ ′ . Then ( α i , n i , β i ) converges to ( α, n, β ) if and only if α i → α , β i → β , and for all sufficientlylarge i , we have n i = n , | α i | ≥ k , | β i | ≥ ℓ , and σ k ( α i ) = σ ℓ ( β i ) . Definition 5.3.
The space C c ( X ) is a right C ( ∂E )-module with the action( f · b )( α, n, β ) = f ( α, n, β ) b ( β )for f ∈ C c ( X ), b ∈ C ( ∂E ), and ( α, n, β ) ∈ X . Define a C ( ∂E )-valued inner product on C c ( X ) by h f, g i ( β ) = X d ( x )= β f ( x ) g ( x )for each f, g ∈ C c ( X ) and β ∈ ∂E . Let L ( E ) denote the Hilbert C ( ∂E )-module obtainedby completing C c ( X ) with respect to the inner product above.We will construct a covariant Toeplitz representation of E on L ( E ). First we isolate afew technical lemmas which will be used throughout the section. Define X := { ( α, n, β ) ∈ X : | α | ≥ } and let d : X → ∂E denote the restriction of d ; that is, d ( α, n, β ) = β . Lemma 5.4.
For any β ∈ ∂E , d − ( β ) = { ( eα, n + 1 , β ) : ( α, n, β ) ∈ X, e ∈ E , and s ( e ) = r ( α ) } In particular, given a funtion f : X → C and β ∈ ∂E , (12) X x ∈ d − ( β ) f ( x ) = X ( α,n,β ) ∈ X X s ( e )= r ( α ) f ( eα, n + 1 , β ) whenever one (and hence both) sums exist. Lemma 5.5.
The map ˜ σ : X → X given by ˜ σ ( α, n, β ) = ( σ ( α ) , n − , β ) is a local homeomorphism.Proof. Given a net ( α i , n i , β i ) ∈ X converging to ( α, n, β ) ∈ X , it is easy to show( σ ( α i ) , n − , β i ) converges to ( σ ( α ) , n − , β ) using Lemma 5.2. Hence ˜ σ is continuous.Given a basic open set Z ( U, k, ℓ, V ) in X as in the definition of the topology on X , wehave ˜ σ ( Z (( U ) , k, ℓ, V ) ∩ X ) = Z ( σ ( U \ E ) , k, ℓ, V ) . As σ is open, this shows ˜ σ is open. That ˜ σ is locally injective follows follows from the factthat σ is locally injective. (cid:3) Lemma 5.6.
Given f ∈ C c ( X ) and ξ ∈ C c ( E ) , the function g : X → C given by g ( α, n, β ) = ( ξ ( α ) f ( σ ( α ) , n − , β ) | α | ≥ | α | = 0 is continuous and supp( g ) ⊆ X is compact. RACES ON TOPOLOGICAL GRAPH ALGEBRAS 15
Proof.
First we show g : X → C is continuous. Suppose ( α i , n i , β i ) → ( α, n, β ) in X . Then,in particular, α i → α in ∂E . Assuming | α | ≥
1, we have | α i | ≥ i since ∂E \ E is open in ∂E . By the continuity of ˜ σ (Lemma 5.5), ( σ ( α i ) , n i − , β i ) → ( σ ( α ) , n − , β ). Also, α i → α and by the continuity of ρ (Lemma 3.3), α i → α . Henceby the continuity of ξ and f , we have g ( α i , n i , β i ) → g ( α, n, β ) . Now suppose | α | = 0. Since supp( ξ ) ⊆ E is compact, the set U := ∂E \ Z (supp( ξ )) isopen. Since α ∈ U , α i ∈ U for all sufficiently large i . Thus for large i , either | α i | = 0 or | α i | ≥ α i / ∈ supp( ξ ). In either case, for large i , g ( α i , n i , β i ) = g ( α, n, β ) = 0 . This proves the continuity of g . Note that this also shows supp( g ) ⊆ X .To show g has compact support, suppose (( α i , n i , β i )) i ∈ I is a net in the support of g .Then for each i , | α i | ≥ α i ∈ supp( ξ ), and ( σ ( α i ) , n i − , β i ) ∈ supp( f ). Hence thereis a subnet (( α i ( j ) , n i ( j ) , β i ( j ) )) j ∈ J and points e ∈ supp( ξ ) and ( α ′ , n, β ) ∈ supp( f ) suchthat α i ( j )1 → e and ( σ ( α i ( j ) ) , n i ( j ) − , β i ( j ) ) → ( α ′ , n, β ). Using the characterization ofconvergent nets in X given in Lemma 5.2, it is easy to show ( α i ( j ) , n i ( j ) , β i ( j ) ) convergesto ( eα ′ , n + 1 , β ) ∈ X . Hence g has compact support. (cid:3) We now construct the desired representation of C ∗ ( E ). Recall from Definition 2.6, H ( E )is the completion of C c ( E ) with respect to the C ( E )-valued inner product induced bythe local homeomorphism s : E → E . It is easily verified that there is a ∗ -homomorphism π : C ( E ) → B ( L ( E )) such that π ( a )( f )( α, n, β ) = a ( r ( α )) f ( α, n, β )for each a ∈ C ( E ), f ∈ C c ( X ), and ( α, n, β ) ∈ X . The following proposition describedthe linear map π : H ( E ) → B ( L ( E )). The covariance is checked in Proposition 5.8. Therepresentation is shown to be faithful and gauge invariant in Theorem 5.10. Proposition 5.7.
There is a contractive, linear map π : H ( E ) → B ( L ( E )) such that π ( ξ )( f )( α, n, β ) = ( ξ ( α ) f ( σ ( α ) , n − , β ) | α | ≥ | α | = 0 for each ξ ∈ C c ( E ) , f ∈ C c ( X ) , and ( α, n, β ) ∈ X . Moreover, we have π ( ξ ) ∗ ( f )( α, n, β ) = X s ( e )= r ( α ) ξ ( e ) f ( eα, n + 1 , β ) for each ξ ∈ C c ( E ) , f ∈ C c ( X ) , and ( α, n, β ) ∈ X . Proof.
Lemma 5.6 shows π ( ξ )( f ) ∈ C c ( X ). It is clear that π ( ξ ) : C c ( X ) → C c ( X ) islinear. To show π ( ξ ) is bounded, let f ∈ C c ( X ) be given and note that k π ( ξ )( f ) k = sup β ∈ ∂E X x ∈ d − ( β ) | π ( ξ )( f )( x ) | = sup β X ( α,n,β ) , | α |≥ | π ( ξ )( f )( α, n, β ) | = sup β X ( α,n,β ) X s ( e )= r ( α ) | π ( ξ )( f )( eα, n + 1 , β ) | = sup β X ( α,n,β ) X s ( e )= r ( α ) | ξ ( e ) | | f ( α, n, β ) | = sup β X ( α,n,β ) h ξ, ξ i ( r ( α )) | f ( α, n, β ) | ≤ k ξ k sup β X ( α,n,β ) | f ( α, n, β ) | = k ξ k k f k Hence π ( ξ ) is contractive and extends to a contractive linear map L ( E ) → L ( E ).It remains to show π ( ξ ) is adjointable with the adjoint given as above. To this end,fix ξ ∈ C c ( E ) and define T : C c ( X ) → C c ( X ) by T ( f )( α, n, β ) = X s ( e )= r ( α ) ξ ( e ) f ( eα, n + 1 , β ) . we claim T ( f ) ∈ C c ( X ). Define g : X → C by g ( α, n, β ) = ξ ( e ) f ( σ ( α ) , n − , β ) and notethat g ∈ C c ( X ) by Lemma 5.6. Now, for x ∈ X , T ( f )( x ) = X ˜ σ ( y )= x g ( y )and T ( f ) ∈ C c ( X ) by Lemma 2.5 since ˜ σ : X → X is a local homeomorphism by Lemma5.5. RACES ON TOPOLOGICAL GRAPH ALGEBRAS 17
It is clear that T ( f ) is linear. We show T ( f ) is bounded and hence extends to a linearmap L ( E ) → L ( E ). Fix f ∈ C c ( X ) and note k T ( f ) k = sup β ∈ ∂E X d ( x )= β | T ( f )( x ) | = sup β X ( α,n,β ) X s ( e )= r ( α ) | ξ ( e ) | | f ( eα, n + 1 , β ) | = sup β X ( α,n,β ) X s ( e )= r ( α ) h ξ, ξ i ( r ( α )) | f ( eα, n + 1 , β ) | = sup β X ( α,n,β ) , | α |≥ h ξ, ξ i ( r ( α )) | f ( α, n, β ) | ≤ k ξ k sup β X ( α,n,β ) | f ( α, n, β ) | = k ξ k k f k So T is contractive as claimed.Finally, note that for each f, g ∈ C c ( X ) and β ∈ ∂E , h π ( ξ )( f ) , g i ( β ) = X d ( x )= β π ( ξ )( f )( x ) g ( x )= X ( α,n,β ) , | α |≥ ξ ( α ) f ( σ ( α ) , n − , β ) g ( α, n, β ) (12) = X ( α,n,β ) X s ( e )= r ( α ) ξ ( e ) f ( α, n, β ) g ( eα, n + 1 , β )= X ( α,n,β ) f ( α, n, β ) X s ( e )= r ( α ) ξ ( e ) g ( eα, n + 1 , β )= X ( α,n,β ) f ( α, n, β ) T ( g )( α, n, β )= h f, T g i ( β ) . So π ( ξ ) is adjointable and the adjoint is as claimed in the statement of the proposition.It is clear that π : C c ( E ) → B ( L ( E )) is linear and π is contractive by the calculationsabove. So π extends to a linear contractive map on H ( E ) completing the proof. (cid:3) Proposition 5.8.
The pair ( π , π ) constructed above is a covariant representation of E and hence induces a ∗ -homomorphisms π : C ∗ ( E ) → B ( L ( E )) .Proof. First note that if a ∈ C c ( E ), ξ ∈ C c ( E ), and f ∈ C c ( X ), then π ( a ) π ( ξ )( f )( α, n, β ) = a ( r ( α )) π ( ξ )( α, n, β ) = a ( r ( α )) ξ ( α ) f ( σ ( α ) , n − , β )= ( aξ )( α ) f ( σ ( α ) , n − , β ) = π ( aξ )( f )( α, n, β ) for all ( α, n, β ) ∈ X with | α | ≥
1. Also, when ( α, n, β ) ∈ X with | α | = 0, π ( a ) π ( ξ )( f )( α, n, β ) = π ( aξ )( f )( α, n, β ) = 0 . Hence π ( a ) π ( ξ ) = π ( aξ ) for all a ∈ C ( E ) and ξ ∈ H ( E ).Now, for ξ, η ∈ C c ( E ), f ∈ C c ( E ), and ( α, n, β ) ∈ X , we have π ( ξ ) ∗ π ( η )( f )( α, n, β ) = X s ( e )= r ( α ) ξ ( e ) π ( η )( f )( eα, n + 1 , β ) = X s ( e )= r ( α ) ξ ( e ) η ( e ) f ( α, n, β )= h ξ, η i ( r ( α )) f ( α, n, β ) = π ( h ξ, η i )( f )( α, n, β ) . So π ( ξ ) ∗ π ( η ) = π ( h ξ, η i ) for all ξ, η ∈ H ( E ).It remains to verify covariance. The pair ( π , π ) induces a map ϕ : K ( H ( E )) → B ( L ( E )) such that ϕ ( ξ ⊗ η ∗ ) = π ( ξ ) π ( η ) ∗ , where ξ ⊗ η ∗ : H ( E ) → H ( E ) is the rankone operator given by ( ξ ⊗ η ∗ )( ζ ) = ξ h η, ζ i . Let λ : C b ( E ) → B ( L ( E )) denote themap given by pointwise multiplication; that is, λ ( ξ )( η )( x ) = ξ ( x ) η ( x ). We need to show ϕ ( λ ( a ◦ r )) = π ( a ) for each a ∈ C ( E ).Suppose a ∈ C c ( E ). Then a ◦ r ∈ C c ( E ) and by Lemma 2.11, there are n ∈ N and ξ i , η i ∈ C c ( E ) such that λ ( a ◦ r ) = P ni =1 ξ i ⊗ η ∗ i , a ◦ r = P ni =1 ξ i η i , and if e, e ′ ∈ E aredistinct edges with s ( e ) = s ( e ′ ), then ξ i ( e ) η i ( e ′ ) = 0. For f ∈ C c ( X ) and ( α, n, β ) ∈ X with | α | ≥ ϕ ( λ ( a ◦ r ))( f )( α, n, β ) = m X i =1 π ( ξ ) π ( η ) ∗ ( f )( α, n, β )= m X i =1 ξ ( α ) π ( η ) ∗ ( f )( σ ( α ) , n − , β )= m X i =1 ξ ( α ) X s ( e )= s ( α ) η ( e ) f ( eσ ( α ) , n, β )= m X i =1 X s ( e )= s ( α ) ξ ( α ) η ( e ) f ( eσ ( α ) , n, β )= m X i =1 ξ ( α ) η ( α ) f ( α, n, β )= a ( r ( α )) f ( α, n, β ) = π ( a )( f )( α, n, β ) . When | α | = 0, ϕ ( λ ( a ◦ r ))( f )( α, n, β ) = 0. Also, if | α | = 0, then α = r ( α ) ∈ E andhence a ( r ( α )) = 0. So π ( a )( f )( α, n, β ) = 0. Therefore, ϕ ( λ ( a )) = π ( a ).This shows the pair ( π , π ) is covariant and hence induces a representation π : C ∗ ( E ) → B ( L ( E )). (cid:3) We next show π is faithful. There is a natural Z -grading on L ( E ) which induces anaction of T on B ( L ( E )). From here, the faithfulness of π will follow from the GaugeInvariant Uniqueness Theorem. We first describe the action on B ( L ( E )). RACES ON TOPOLOGICAL GRAPH ALGEBRAS 19
Definition 5.9.
Define unitaries U z : L ( E ) → L ( E ) by U z ( f )( α, n, β ) = z n f ( α, n, β )for f ∈ C c ( X ) and ( α, n, β ) ∈ X . Now define a strongly continuous action γ ′ of T on B ( L ( E )) by γ ′ z ( T ) = U z T U ∗ z for T ∈ B ( L ( E )) and z ∈ T . The action γ ′ will be calledthe gauge action on B ( L ( E )). Theorem 5.10.
The representation π : C ∗ ( E ) → B ( L ( E )) constructed above is a faithfuland preserves the gauge actions.Proof. The gauge invariance is easily verified. Now, by the Gauge Invariance UniquenessTheorem, to show π is faithful, it’s enough to show π is faithful. Fix a non-zero a ∈ C ( E )and fix a vertex v ∈ E with a ( v ) = 0. There is an α ∈ ∂E such that r ( α ) = v . Let f ∈ C c ( X ) be such that f ( α, , α ) = 0. Then π ( a )( f )( α, , α ) = a ( r ( α )) f ( α, , α ) = 0 . In particular, π ( a ) = 0. So π is injective and hence π is injective. (cid:3) For each invariant measure µ on E , we construct a Hilbert space L ( E, µ ) and a repre-sentation π µ on C ∗ ( E ) on L ( E, µ ). This representation is the key to lifting the measure µ to a tracial state on C ∗ ( E ). Definition 5.11.
Let µ denote an invariant measure on E . Let ˜ µ denote the probabilitymeasure on ∂E given by Proposition 4.4. There is a representation C ( ∂E ) → L ( ∂E, ˜ µ )given by multiplication. Now define L ( E, µ ) := L ( E ) ⊗ C ( ∂E ) L ( ∂E, ˜ µ ) . Then π induces a representation π µ : C ∗ ( E ) → B ( L ( E )) given by π µ ( x ) = π ( x ) ⊗ id foreach x ∈ C ∗ ( E ).Note that the gauge action γ ′ of T on B ( L ( E )) induces an action, also denoted γ ′ of T on B ( L ( E, µ )). Also, the representation π µ is gauge invariant. Remark 5.12.
As mentioned in the introduction, the spaces L ( E ) and L ( E, µ ) can beviewed as fibred objects over ∂E . This will not be needed in the paper, but we give anoutline of how this can be done.For each β ∈ ∂E , define H ( β ) := ℓ ( d − ( β )) and let Q β ∈ ∂E H ( β ) denote the ℓ ∞ -productof the spaces H ( β ). The map C c ( ∂E ) → Y β ∈ ∂E H ( β ) f ( f | d − ( β )) β ∈ ∂E is linear and isometric. Hence it extends to a linear isometry on L ( E ). There is a uniquecontinuous field of Hilbert spaces H over ∂E in the sense of Chapter 10 of [4] such thatthe fibre over β is H ( β ) for every β ∈ ∂E . Moreover, given a vector field ξ ∈ H , ξ ∈ L ( E )if and only if the function β
7→ k ξ ( β ) k vanishes at infinity.If µ is an invariant measure on E and ˜ µ is the induced measure on ∂E , there is a uniquemeasurable field of Hilbert spaces β H ( β ) containing H . Moreover, there is a unitary L ( E, µ ) → Z ⊕ ∂E H ( β ) d ˜ µ ( β ) given by restricting f ∈ L ( E, µ ) to the fibres of d : X → ∂E as above. Conjugating π µ by the unitary also yields a direct integral decomposition for the representation π µ .6. Traces on Topological Graph Algebras
We now show every invariant measure µ on a topological graph E induces a gaugeinvariant tracial state on C ∗ ( E ). The strategy is to construct of vector ˆ u ∈ L ( E, µ ) suchthat h ˆ u, π µ ( · )ˆ u i is a tracial state on C ∗ ( E ).Define a function ι : ∂E → X by ι ( α ) = ( α, , α ) and note that ι is a homeomorphismonto a closed and open subset of X , as follows easily from the characterization of convergentnets in ∂E given in Lemma 5.2. We will identify ∂E as a subset of X through the map ι .Fix an increasing net of compact sets K i ⊆ ∂E with union ∂E . Let f i ∈ C c ( ∂E ) ⊆ L ( E )be such that 0 ≤ f i ≤ f i | K i = 1. Defineˆ u i = f i ⊗ ∈ L ( E ) ⊗ C ( ∂E ) L ( ∂E, ˜ µ ) = L ( E, µ ) . Fix ε >
0. Since ˜ µ is a Radon measure on ∂E , there is i such that ˜ µ ( ∂E \ K i ) < ε . Now,for i, j ≥ i , we have k ˆ u i − ˆ u j k = Z ∂E | f i − f j | d ˜ µ ≤ ˜ µ ( ∂E \ K i ) < ε. So ˆ u i is a Cauchy net in L ( E, µ ) and hence converges to ˆ u ∈ L ( E, µ ).Note that ˆ u is a unit vector. Hence composing the vector state defined by ˆ u with therepresentation π µ : C ∗ ( E ) → B ( L ( E, µ )) yields a state on C ∗ ( E ) given by(13) h ˆ u, π µ ( x )ˆ u i = lim i Z ∂E h f i , π ( x ) f i i ( α ) d ˜ µ ( α )for any x ∈ C ∗ ( E ), for any net ( f i ) ⊆ C c ( ∂E ) increasing pointwise to 1.We will show that this state is tracial in Theorem 6.3. First we need a couple lemmas. Lemma 6.1.
Let H be a C*-correspondence over a C*-algebra A . Let ( π , π ) denotethe universal covariant representation on the Cuntz-Pimsner algebra O A ( H ) , and let π k denote the induced maps H ⊗ k → O A ( H ) . If τ is a gauge invariant state on O A ( H ) suchthat τ ( π k ( ξ ) ∗ π k ( η )) = τ ( π k ( η ) π k ( ξ ) ∗ ) for all ξ, η ∈ H ⊗ k and k ≥ , then τ is a tracialstate on O A ( H ) .Proof. The elements of the form π k ( ξ ) π ℓ ( η ) ∗ for k, ℓ ∈ N , ξ ∈ H ⊗ k and η ∈ H ⊗ ℓ span adense subset of O A ( H ). Hence to show τ is a tracial state on O A ( H ), it’s enough to show τ ( xπ ( ζ )) = τ ( π ( ζ ) x ) and τ ( xπ ( ζ ) ∗ ) = τ ( π ( ζ ) ∗ x )for all x ∈ O A ( H ) and ζ ∈ H . In fact, the second equation follows from the first by takingadjoints. So it’s enough to verify the first equation.Fix ζ ∈ H . Suppose k, ℓ ∈ N , ξ ∈ H ⊗ k , and η ∈ H ⊗ ℓ . If ℓ = k + 1, then τ ( π k ( ξ ) π ℓ ( η ) ∗ π ( ζ )) = τ ( π ( ζ ) π k ( ξ ) π ℓ ( η ) ∗ ) = 0since τ is gauge invariant. Hence to verify τ is a tracial state, it’s enough to show τ ( π k ( ξ ) π k +1 ( η ⊗ η ′ ) ∗ π ( ζ )) = τ ( π ( ζ ) π k ( ξ ) π k +1 ( η ⊗ η ′ ) ∗ ) RACES ON TOPOLOGICAL GRAPH ALGEBRAS 21 for all k ∈ N , ξ, η ′ ∈ H ⊗ k , and η ∈ H . We now verify this equality. τ ( π k ( ξ ) π k +1 ( η ⊗ η ′ ) ∗ π ( ζ ))) = τ ( π k ( ξ ) π k ( η ′ ) ∗ π ( η ) ∗ π ( ζ ))= τ ( π k ( ξ ) π k ( η ′ ) ∗ π ( h η, ζ i )) = τ ( π k ( ξ ) π k ( h ζ, η i η ′ ) ∗ )= τ ( π k ( h ζ, η i η ′ ) ∗ π k ( ξ )) = τ ( π k ( η ′ ) ∗ π ( h η, ζ i ) π k ( ξ ))= τ ( π k ( η ′ ) ∗ π ( η ) ∗ π ( ζ ) π k ( ξ )) = τ ( π k +1 ( η ⊗ η ′ ) ∗ π k +1 ( ζ ⊗ ξ ))= τ ( π k +1 ( ζ ⊗ ξ ) π k +1 ( η ⊗ η ′ ) ∗ ) = τ ( π ( ζ ) π k ( ξ ) π k +1 ( η ⊗ η ′ ) ∗ ) . Hence τ is a tracial state. (cid:3) Lemma 6.2.
Let H be a C*-correspondence over a C*-algebra A . Let ( π , π ) denote theuniversal covariant representation on the Cuntz-Pimsner algebra O A ( H ) . If ( τ i ) is a netof gauge invariant tracial states on O A ( H ) and τ is a gauge invariant tracial on O A ( H ) ,then τ i → τ weak* if and only if τ i ( π ( a )) → τ ( π ( a )) for every a ∈ A .Proof. The forward implication is trivial. Assume τ i ◦ π → τ ◦ π weak*. If k, ℓ ∈ N with k = ℓ , ξ ∈ H ⊗ k and ℓ ∈ H ⊗ ℓ , then τ i ( π k ( ξ ) π ℓ ( η ) ∗ ) = τ ( π k ( ξ ) π ℓ ( η ) ∗ ) = 0 . If k ∈ N and ξ, η ∈ H ⊗ k , then τ i ( π k ( ξ ) π k ( η ) ∗ ) = τ i ( π ( h η, ξ i )) → τ ( π ( h η, ξ i )) = τ ( π k ( ξ ) π k ( η ) ∗ ) . Since the elements of the form π k ( ξ ) π ℓ ( η ) ∗ for k, ℓ ∈ N , ξ ∈ H ⊗ k , and η ∈ H ⊗ ℓ , span adense subset of O A ( H ), it follows that τ i → τ weak*. (cid:3) Let T (C ∗ ( E )) denote the space of tracial states on C ∗ ( E ) equipped with the weak*topology and let T (C ∗ ( E ) γ denote the subspace consisting of tracial states invariant underthe gauge action γ : T y C ∗ ( E ). Also, let T ( E ) denote the space of invariant measureson E equipped with the weak* topology. Theorem 6.3.
Given τ ∈ T (C ∗ ( E )) , there is a unique µ ∈ T ( E ) such that (14) τ ( a ) = Z E a ( v ) dµ ( v ) for all a ∈ C ( E ) ⊆ C ∗ ( E ) . Conversely, given µ ∈ T ( E ) , there is a unique τ ∈ T (C ∗ ( E )) γ such that (15) τ ( x ) = h ˆ u, π µ ( x )ˆ u i L ( E,µ ) for each x ∈ C ∗ ( E ) . These maps define inverse affine homeomorphisms between T ( E ) and T (C ∗ ( E )) γ .Proof. Let ( π u , π u ) denote the universal covariant representation of E on C ∗ ( E ).Given τ ∈ T (C ∗ ( E )), the composition C ( E ) π u −→ C ∗ ( E ) τ −→ C is a state on C ( E )and hence there is a unique Radon probability measure µ on E satisfying (14). We needto show the invariance of µ . Let ϕ : K ( H ( E )) → C ∗ ( E ) denote the ∗ -homomorphism induced by ( π u , π u ); that is, ϕ ( ξ ⊗ η ∗ ) = π u ( ξ ) π u ( η ) ∗ for ξ, η ∈ H ( E ). Note that since τ is a tracial state, for all ξ, η ∈ H ( E ) we have(16) τ ( π u ( h η, ξ i )) = τ ( π u ( η ) ∗ π ( ξ )) = τ ( π u ( ξ ) π ( η ∗ )) = τ ( ϕ ( ξ ⊗ η ∗ )) . Suppose a ∈ C c ( E ) and fix ζ ∈ C c ( E ) with 0 ≤ ζ ≤ a ◦ r and let λ : C b ( E ) → B ( H ( E ))be the multiplication map as in Proposition 2.8. By Lemma 2.11, there are ξ i , η i ∈ C c ( E )for i = 1 , . . . , n with P i ξ i ⊗ η ∗ i = λ ( ζ ) and P i ξ i η i = ζ . By Lemma 2.12, ϕ ( λ ( ζ )) ≤ π u ( a )and now we compute Z E ζ ds ∗ µ = Z E X s ( e )= v ζ ( e ) dµ ( v ) = n X i =1 Z E X s ( e )= v ξ ( e ) η ( e ) dµ ( v )= n X i =1 Z E h η, ξ i dµ = n X i =1 τ ( π u ( h η, ξ i )) (16) = n X i =1 τ ( ϕ ( ξ i ⊗ η ∗ i ))= τ ( ϕ ( λ ( ζ ))) ≤ τ ( π u ( a )) = Z E a dµ Now the Monotone Convergence Theorem implies Z E a ◦ r ds ∗ µ ≤ Z E a dµ. Moreover, when a ∈ C c ( E ), we may take ζ = a ◦ r in the calculation above. Then ϕ ( λ ( ζ )) = π u ( a ) by the covariance condition in the definition of C ∗ ( E ). Repeating thecalculation above yields Z E a ◦ r ds ∗ µ = Z E a dµ which proves µ is an invariant measure on E .Conversely, suppose µ ∈ T ( E ) is an invariant measure. Then µ defines a state τ onC ∗ ( E ) via the formula given in (15). The representation π : C ∗ ( E ) → B ( L ( E )) is gaugeinvariant by Theorem 5.10. Moreover, with the unitaries U z on L ( E ) as in Definition 5.9,for every f ∈ C c ( ∂E ), U z f = f . Hence for x ∈ C ∗ ( E ), z ∈ T , and f ∈ C c ( ∂E ), h f, π ( γ z ( x )) f i = h f, U z π ( x ) U ∗ z f i = h f, π ( x ) f i . It follows from (13) that τ ( γ z ( x )) = τ ( x ) for every x ∈ C ∗ ( E ) and z ∈ T ; that is, τ isgauge invariant.We prove τ is tracial. Fix an integer k ≥ ξ, η ∈ C c ( E k ). Define K := { α ∈ ∂E : r ( α ) ∈ s (supp( ξ )) } ∪ { α ∈ ∂E \ ∂E k − : α . . . α k ∈ supp( ξ ) } and note that K is compact by Proposition 3.3. Let f ∈ C c ( ∂E ) be such that 0 ≤ f ≤ f | K = 1 and fix f ∈ C c ( ∂E ) with f ≤ f ≤ RACES ON TOPOLOGICAL GRAPH ALGEBRAS 23
Given a vertex v ∈ E , write E k v for all paths α ∈ E k with s ( α ) = v . Fix β ∈ ∂E .Since f is supported on ∂E ⊆ X , we have h f, π k ( ξ ) ∗ π k ( η )( f ) i ( β ) = X d ( x )= β f ( x ) π k ( ξ ) ∗ π k ( η )( f )( x )= f ( β, , β ) π k ( ξ ) ∗ π k ( η )( f )( β, , β )= X α ∈ E k r ( β ) ξ ( α ) η ( α ) f ( β, , β ) = X α ∈ E k r ( β ) ξ ( α ) η ( α ) . Similarly, if | β | ≥ k , h f, π k ( η ) π k ( ξ ) ∗ ( f ) i ( β ) = f ( β, , β ) η ( β . . . β k ) X α ∈ E k s ( β k ) ξ ( α ) f ( ασ ( β ) , , β )= η ( β · · · β k ) ξ ( β · · · β k ) f ( β, , β ) = ξ ( β · · · β k ) η ( β · · · β k ) , and if | β | < k , then h f, π k ( η ) π k ( ξ ) ∗ ( f ) i ( β ) = 0.Now, by the invariance of the measure ˜ µ given in Corollary 4.5, we have Z ∂E h f,π k ( η ) π k ( ξ ) ∗ ( f ) i d ˜ µ = Z ∂E \ ∂E k − ξ ( β · · · β k ) η ( β · · · β k ) d ˜ µ (11) = Z ∂E X α ∈ E k r ( β ) ξ ( α ) η ( α ) d ˜ µ ( β ) = Z ∂E h f, π k ( ξ ) ∗ π k ( η )( f ) i ( β ) . Now by (13), τ ( π ku ( η ) π ku ( ξ ) ∗ ) = τ ( π ku ( ξ ) ∗ π ku ( η )) and Lemma 6.1 implies τ is a tracial state.Hence τ ∈ T (C ∗ ( E )) γ .We now show the maps constructed above are inverses. Starting with an invariantmeasure µ on E , let τ denote the tracial state on C ∗ ( E ) defined by (15) and let µ ′ denotethe invariant measure induced by τ as in (14). Fix a ∈ C c ( E ) and let f ∈ C c ( ∂E ) besuch that 0 ≤ f ≤ f ( β ) = 1 for all β ∈ ∂E with r ( β ) ∈ supp( a ). Then Z E a dµ ′ (14) = τ ( π u ( a )) (15) = Z ∂E h ˆ u, π ( a )ˆ u i d ˜ µ (13) = Z ∂E h f, π ( a ) f i d ˜ µ = Z ∂E f ( α ) a ( r ( α )) f ( α ) d ˜ µ ( α ) = Z ∂E a ◦ r d ˜ µ (4) = Z E a dµ. Hence µ = µ ′ .Now suppose τ is a gauge invariant tracial state on C ∗ ( E ). Let µ be the invariantmeasure on E defined by (14) and let τ ′ be the tracial state on C ∗ ( E ) defined by (15).Arguing as in the previous paragraph, we have τ ( π u ( a )) = R E a dµ = τ ′ ( π u ( a )) for each a ∈ C ( E ). Now, for k ∈ N and ξ, η ∈ C c ( E k ) ⊆ H ( E ) ⊗ k , we have τ ( π ku ( ξ ) π ku ( η ) ∗ ) = τ ( π ku ( η ) ∗ π ku ( ξ )) = τ ( π u ( h η, ξ i )) = τ ′ ( π u ( h ξ, η i )) = τ ′ ( π ku ( ξ ) π ku ( η ) ∗ ) , since τ and τ ′ are both tracial states. Moreover, since τ and τ ′ are both gauge invariant,if k, ℓ ∈ N are distinct integers, ξ ∈ C c ( E k ) and η ∈ C c ( E ℓ ), τ ( π ku ( ξ ) π ℓu ( η ) ∗ ) = 0 = τ ( π ku ( ξ ) π ℓu ( η ) ∗ ) . Hence τ = τ ′ .It is clear from the construction that the map T (C ∗ ( E )) γ → T ( E ) is affine and hence theinverse map is affine. Suppose τ i , τ are gauge invariant tracial states on C ∗ ( E ) and µ i , µ arethe corresponding invariant measure on C ∗ ( E ). Then µ i → µ if and only if τ i ◦ π → τ ◦ π .By Lemma 6.2, µ i → µ if and only if τ i → τ . Hence the map T (C ∗ ( E )) γ → T ( E ) is ahomeomorphism. (cid:3) The following result shows the representation associated to an invariant measure µ on E constructed in 5.11 is precisely the GNS representation of the gauge invariant trace τ on C ∗ ( E ) corresponding to µ . Theorem 6.4. If τ is a gauge invariant trace on C ∗ ( E ) and µ is the correspondinginvariant measure on E given by (14) , the then representations π τ : C ∗ ( E ) → B ( L (C ∗ ( E ) , τ )) and π µ : C ∗ ( E ) → B ( L ( E, µ )) are unitarily equivalent.Proof. Fix an increasing net K i of compact subsets of ∂E ⊆ X whose interiors cover ∂E .Let f i ∈ C c ( ∂E ) be such that 0 ≤ f i ≤ f | K i = 1. Given k, ℓ ∈ N , ξ ∈ C c ( E k ), and η ∈ C c ( E ℓ ), the net π k ( ξ ) π ℓ ( η ) ∗ ( f i ) in C c ( X ) is eventually constant. Abusing notationslightly, we let π k ( ξ ) π ℓ ( η ) ∗ ( χ ∂E ) denote the limit in C c ( X ).Consider the set A = span { π k ( ξ ) π ℓ ( η ) ∗ ( χ ∂E ) : k, ℓ ∈ N , ξ ∈ C c ( E k ) , η ∈ C c ( E ℓ ) } ⊆ C c ( X ) . We will show A is uniformly dense in C c ( X ). Fix k, ℓ ∈ N , ξ ∈ C c ( E k ), η ∈ C c ( E ℓ ), and( α, n, β ) ∈ X . If k − ℓ = n and there are α ′ ∈ E k , β ′ ∈ E ℓ , and ω ∈ ∂E with α ′ ω = α and β ′ ω = β , then π k ( ξ ) π ℓ ( η ) ∗ ( χ ∂E ) = ξ ( α ′ ) η ( β ′ );otherwise, π k ( ξ ) π ℓ ( η ) ∗ ( χ ∂E ) = 0. It follows that for k, k ′ , ℓ, ℓ ′ ∈ N , ξ ∈ C c ( E k ) , ξ ′ ∈ C c ( E k ′ ), η ∈ C c ( E ℓ ), and η ′ ∈ C c ( E ℓ ′ ), π k ( ξ ) π ℓ ( η ) ∗ ( χ ∂E ) π k ′ ( ξ ′ ) π ℓ ′ ( η ′ ) ∗ ( χ ∂E ) = ( π k ( ξξ ′ ) π ℓ ( ηη ′ ) ∗ ( χ ∂E ) k = k ′ , ℓ = ℓ ′ π k ( ξ ) π ℓ ( η ) ∗ ( χ ∂E ) = π k ( ξ ) π ℓ ( η ) ∗ ( χ ∂E ) . Hence A is a ∗ -subalgebra of C c ( X ).Given ( α, n, β ) ∈ X , there are k, ℓ ∈ N , α ′ ∈ E k , β ′ ∈ E ℓ , and ω ∈ ∂E such that k − ℓ = n , α = α ′ ω , and β = β ′ ω . Fix ξ ∈ C c ( E k ) and η ∈ C c ( E ℓ ) such that ξ ( α ′ ) = 1 and η ( β ′ ) = 1. Then π k ( ξ ) π ℓ ( η ) ∗ ( χ ∂E )( α, n, β ) = 1 . RACES ON TOPOLOGICAL GRAPH ALGEBRAS 25
So the algebra A does not vanish at any point in X . Similar considerations show A separates points in X . Hence the Stone-Weierstrass Theorem implies A is uniformly densein C c ( X ).Since C c ( X ) is dense in L ( E ), we have A is dense in L ( E ) and in particular, thevectors π kµ ( ξ ) π ℓµ ( η ) ∗ ˆ u span a dense set of subspace of L ( E, µ ). This shows ˆ u is a cyclicvector for the representation π µ . The result follows from (15) and the uniqueness of theGNS representation. (cid:3) We would like to mention two special cases of Theorem 6.3. If X is a locally compactHausdorff space and σ is a homeomorphisms of X , Theorem 6.3 reduces to the classicalresult that every invariant probability measure on ( X, σ ) induces a tracial state on thecrossed product C ( X ) ⋊ σ Z . When E is a discrete graph, our notion of invariant measureson E agrees with Tomforde’s notion of graph traces on E introduced in [16] and we recoverthe results of Section 3.3 in [16].For crossed products, if σ is a free action of Z on a locally compact Hausdorff space X ,then every tracial state on C ( X ) ⋊ Z is gauge invariant. If E is a discrete graph satisfyingcondition (K), then every tracial state on C ∗ ( E ) is gauge invariant as noted in Section 3.3of [16]. It is very likely that these results have have an analogue for topological graphs,but we were unable to prove this. Freeness of topological graphs was defined by Katsurain [10, Definition 7.2] and gives a simultaneous generalization of the notions of freeness forhomeomorphisms and condition (K) for discrete graphs. Conjecture 6.5. If E is a free topological graph, then every tracial on C ∗ ( E ) is gaugeinvariant. In particular, if E is a topological graph such that C ∗ ( E ) is simple, then E is free by [10,Theorem 8.12]. Hence if the conjecture above holds, Theorem 6.3 would yield a completedescription of the tracial state simplex for simple topological graph C ∗ -algebras. We provea special case of Conjecture 6.5 in Corollary 6.8 below. Lemma 6.6.
Let H be a C*-correspondence over a C*-algebra A , and let π k : H ⊗ k → C ∗ ( E ) denote the canonical map for k ≥ . If τ is a tracial state on O A ( H ) and τ ( π k ( ξ )) =0 for all ξ ∈ H ⊗ k and k ≥ , then τ is gauge invariant.Proof. Suppose k, ℓ ∈ N with k > ℓ . Let ξ, η ∈ H ⊗ k and ξ ′ ∈ H ⊗ ( k − ℓ ) be given. Then τ ( π k ( ξ ⊗ ξ ′ ) π ℓ ( η ) ∗ ) = τ ( π k ( η ) ∗ π k ( ξ ) π k − ℓ ( ξ ′ )) = τ ( π k − ℓ ( h η, ξ i ξ ′ )) = 0 . It follows that if k, ℓ ∈ N are distinct, ξ ∈ H ⊗ k and η ∈ H ⊗ ℓ , then τ ( π k ( ξ ) π ℓ ( η ) ∗ ) = 0.Indeed, when k > ℓ , this is an immediate consequence of the calculation above, and when ℓ > k , the result follows since by taking adjoints.Now let γ : T y O A ( H ) denote the gauge action. For k, ℓ ∈ N , ξ ∈ H ⊗ k , and η ∈ H ⊗ ℓ ,and z ∈ T , τ ( γ z ( π k ( ξ ) π ℓ ( η ) ∗ )) = z k − ℓ τ ( π k ( ξ ) π ℓ ( η ) ∗ ) = τ ( π k ( ξ ) π ℓ ( η ) ∗ ) . Since the elements of the form π k ( ξ ) π ℓ ( η ) ∗ span a dense subspace of O A ( H ), τ is gaugeinvariant. (cid:3) A path α ∈ E ∗ is called a cycle if | α | ≥ s ( α ) = r ( α ). Proposition 6.7.
Suppose τ is a tracial state on a topological graph algebra C ∗ ( E ) and µ is the invariant measure on E induced by τ as in (14) . If for every cycle α ∈ E ∗ , s ( α ) / ∈ supp( µ ) , then τ is gauge invariant.Proof. Let τ be a tracial state on C ∗ ( E ). Fix k ≥ ξ ∈ C c ( E k ) be given. ByLemma 6.6, it’s enough to show τ ( π k ( ξ )) = 0.If α ∈ E k with s ( α ) = r ( α ), then there are disjoint open neighborhoods U α , V α ⊆ E of s ( α ) and r ( α ), respectively. Now, s − ( U α ) ∩ r − ( V α ) ⊆ E k is an open neighborhood of α and hence contains a compact neighborhood K α of α . If α ∈ E k and s ( α ) = r ( α ), then s ( α ) / ∈ supp( µ ). As supp( µ ) is closed, there is a compact neighborhood K α ⊆ E k of α with s ( K α ) ∩ supp( µ ) = ∅ .The interiors of the sets K α , α ∈ E k , form an open cover of E k . Fix α (1) , . . . , α ( n ) ∈ E k such that the interiors of the K α ( i ) form an open cover of the compact set supp( ξ ) ⊆ E k .There are functions ξ i ∈ C c ( E k ) such that P i ξ i = ξ and each ξ i is supported in theinterior on K α ( i ) . To show τ ( π k ( ξ )) = 0, it’s enough to show τ ( π k ( ξ i )) = 0 for all i .If s ( α ( i )) = r ( α ( i )), the sets s ( K α ( i ) ) and r ( K β ( i ) ) are disjoint compact subsets of E .Hence there is an a ∈ C c ( E ) with a i | s ( K α ( i ) ) = 1 and a | r ( K α ( i ) ) = 0. Now, ξ i a = ξ i and aξ i = 0. Thus τ ( π k ( ξ i )) = τ ( π k ( ξ i ) π ( a i )) = τ ( π ( a i ) π k ( ξ i )) = 0 . If s ( α ( i )) = r ( α ( i )), then s ( K α ( i ) ) is a compact subset of E disjoint from supp( µ ).There is a positive function a ∈ C c ( E ) with a i | s ( K α ( i ) ) = 1 and a | supp( µ ) = 0. Then ξ i a / = ξ i and τ ( π ( a )) = R a dµ = 0. Now by the Cauchy-Schwartz inequality, | τ ( π k ( ξ i )) | = | τ ( π k ( ξ i ) π ( a ) / ) | = τ ( π k ( ξ i ) ∗ π k ( ξ i )) / τ ( π ( a )) / = 0 . Hence τ ( π k ( ξ i )) = 0.We have shown τ ( π k ( ξ i )) = 0 for all i = 1 , . . . , n . Hence τ ( π k ( ξ )) = 0. By Lemma 6.6, τ is gauge invariant. (cid:3) Corollary 6.8. If E has no cycles, every tracial state on C ∗ ( E ) is gauge invariant. To prove Conjecture 6.5, it is enough to show if µ is an invariant measure on E , α is acycle in E , and s ( α ) ∈ supp( µ ), then s ( α ) is periodic in the sense of [10, Definition 7.1].Indeed, by definition, a free topological graph has no periodic vertices and hence therewould be no cycles with source in supp( µ ). Then Proposition 6.7 would prove Conjecture6.5. For a discrete graphs, there is an easy direct proof of this is given below in Corollary6.10. For topological graphs, a similar approach should work, but we could not overcomethe topological technicalities.Using our result Proposition 6.7, we can recover the two special cases of Conjecture 6.5mentioned above. Recall a cycle α = α . . . α n ∈ E n is called simple if α i = α n for each1 ≤ i < n . A discrete graph E satisfies condition (K), if there is no vertex v ∈ E whichis the source of a unique simple cycle. Corollary 6.9. If σ : Z y X is a free action on a locally compact Hausdorff space X ,then every tracial state on C ( X ) ⋊ σ Z is gauge invariant. RACES ON TOPOLOGICAL GRAPH ALGEBRAS 27
Proof.
Viewing (
X, σ ) as a topological graph E with E = E = X , s = σ , and r = id, avertex v ∈ E is the source of a cycle if and only if v is a periodic point for the action σ .Since the action is free, E has no cycles. (cid:3) Corollary 6.10. If E is a discrete graph satisfying condition (K), then every tracial stateon C ∗ ( E ) is gauge invariant.Proof. For α ∈ E k , k ≥
1, let δ α ∈ C c ( E k ) denote the indicator function of { α } and let s α = π k ( δ α ) ∈ C ∗ ( E ). Similarly for v ∈ E , we let p v = π ( δ v ) ∈ C ∗ ( E ).Suppose τ is a tracial state on C ∗ ( E ) and let µ denote the induced invariant measureon E . Assume v ∈ E is the source of a cycle. We claim there is an n ≥ α, β ∈ E n with s ( α ) = s ( β ) = v . There are distinct simple cycles α ′ ∈ E k and β ′ ∈ E ℓ with s ( α ′ ) = s ( β ′ ). If k = ℓ , set n = k = ℓ , α = α ′ , and β = β ′ . Otherwise, if k = ℓ , define n := kℓ , α := ( α ′ ) ℓ ∈ E n and β := ( β ′ ) k ∈ E n . If α n = β n , then α and β aredistinct. If α n = β n , then the edge α n = α ′ k = β ′ ℓ occurs exactly ℓ times in the path α and k times in the path β since α ′ and β ′ are simple. As k = ℓ , we have α = β .Now since α, β ∈ E n are distinct cycles, s ∗ α s β = 0. It follows that s α s ∗ α + s β s ∗ β ≤ p v and s ∗ α s α = s ∗ β s β = p v . As τ is a tracial state, 0 ≤ τ ( p v ) ≤ τ ( p v ) and τ ( p v ) = 0. So v / ∈ supp( µ ) and the result follows Proposition 6.7. (cid:3) Totally Disconnected Graphs and K-Theory
In this section, we will show the gauge invariant tracial states on C ∗ ( E ) can be detectedin the K-theory of C ∗ ( E ) when E is totally disconnected. First we recall the Pimsner-Voiculescu sequence for topological graphs given in [8]. For a topological graph E , viewthe Hilbert module H ( E ) as a C ( E )– C ( E ) C ∗ -correspondence by restricting scalarson the left. Then taking the Kasparov product with the C ∗ -correspondence H ( E ) inducesa morphism [ E ] : K ∗ ( E ) → K ∗ ( E ) on topological K-theory. There is a natural six termexact sequence(17) K ( E ) K ( E ) K (C ∗ ( E ))K (C ∗ ( E )) K ( E ) K ( E ) ι − [ E ] π ∗ π ∗ ι − [ E ] When E is a totally disconnected topological graph, the six term exact sequence takes asimpler form (Proposition 7.1). First we introduce some notation.For a totally disconnected space X , we view K ( X ) as the Grothendieck group of thefinitely generated Hilbert modules over C ( X ). Given a compact open set U ⊆ X , the ideal C ( U ) E C ( X ) is a finitely generated Hilbert module over C ( X ). The Hilbert modules ofthe form C ( U ) generate K ( X ) as an abelian group. Moreover, there is an isomorphism ρ X : K ( X ) → C ( X, Z ) given by C ( U ) χ U for each compact open set U ⊆ X , where χ U is the indicator function of the set U .If X and Y are totally disconnected spaces, H is a Hilbert C ( X )-module, and a rep-resentation C ( Y ) → K ( H ) is given, then the C ∗ -correspondence H induces a morphism[ H ] : K ( Y ) → K ( X ) given by [ K ] [ K ⊗ C ( Y ) H ] for every finitely generated Hilbert C ( Y )-module K . Specializing to a totally disconnected topological graph E , there is an faithful represen-tation C ( E ) → K ( H ( E )). The morphism [ E ] : K ( E ) → K ( E ) is the morphisminduced by the C ∗ -correspondence H ( E ). Proposition 7.1.
For a totally disconnected topological graph E , there is an exact se-quence (C ∗ ( E )) C ( E , Z ) C ( E , Z ) K (C ∗ ( E )) 0 . ι − ψ where the map ψ : C ( E , Z ) → C ( E , Z ) is given by ψ ( f )( v ) = X s ( e )= v f ( r ( e )) for all f ∈ C ( E , Z ) and v ∈ E .Proof. Since E and E are totally disconnected, K ( E ) = K ( E ) = 0. Hence in viewof (17), it is enough to show(18) K ( E ) K ( E ) C ( E , Z ) C ( E , Z ) [ E ] ρ E ρ E ψ commutes.Fix a compact open set U ⊆ E . Then C ( U ) is a singly generated Hilbert C ( E )-module with generator 1 C ( U ) . Moreover, r − ( U ) is a compact open subset of E and henceis a Hilbert submodule of H ( E ). The map C ( U ) ⊗ C ( E ) H ( E ) → C ( r − ( U )) a ⊗ f ( a ◦ r ) f and the map C ( r − ( U )) → C ( U ) ⊗ C ( E ) H ( E ) g C ( U ) ⊗ g ◦ r are inverse isomorphisms of Hilbert C ( E )-modules.Let V , . . . , V n ⊆ E be a partition of U into compact open sets such that s | V i is ahomeomorphism onto s ( V i ) for each i = 1 , . . . , n , and let σ i : V i → s ( V i ) denote thehomeomorphism obtained by restricting s . Then the map C ( r − ( U )) → n M i =1 C ( s ( V i )) f ( f ◦ σ − i ) ni =1 and the map n M i =1 C ( s ( V i )) → C ( r − ( U )) ( g i ) ni =1 n X i =1 g ◦ σ i are inverse isomorphisms of Hilbert C ( E )-modules. RACES ON TOPOLOGICAL GRAPH ALGEBRAS 29
Combining the last two paragraphs, we have[ E ]( C ( U )) = [ C ( U ) ⊗ H ( E )] = [ C ( r − ( U ))] = n X i =1 [ C ( s ( V i ))]in K ( E ). Hence for v ∈ E , ρ E ([ E ]( C ( U )))( v ) = n X i =1 χ s ( V i ) ( v ) = r − ( U ) ∩ s − ( v ))where S denotes the cardinality of a set S . Similarly, for v ∈ E , ψ ( ρ E ( C ( U )))( v ) = ψ ( χ U )( v ) = X s ( e )= v χ U ( r ( e )) = r − ( U ) ∩ s − ( v )) . Hence the diagram (18) commutes and this completes the proof. (cid:3)
Given any C ∗ -algebra A , the group K ( A ) has a natural order structure determined bythe semigroup K ( A ) + ⊆ K ( A ) consisting of all elements in K ( A ) defined by a matrixprojection over A . There is also a distinguished subset Σ( A ) ⊆ K ( A ) + called the scale consisting of all elements in K ( A ) defined by projections in A . If A has an approximateunit consisting a projections, a state on K ( A ) is a group morphism τ : K ( A ) → R suchthat τ ( x ) ≥ x ∈ K ( A ) + and sup x ∈ Σ( A ) τ ( x ) = 1. The collection of states onK ( A ) is denoted S (K ( A )). Note that S (K ( A )) is convex and weak* compact.For any C ∗ -algebra A with an approximate unit consisting of projections, there is anatural continuous affine map T ( A ) → S (K ( A )) induced by restricting a tracial state tothe set of projections. Moreover, when A is unital, Blackadar and Rørdam have shownevery state on T ( A ) → S (K ( A )) is induced by a quasitrace on A (see [3]). This resultalso holds when A has an approximate unit consisting of projections as can be shown byapplying the unital result to the corners defined by these projections. Moreover, when A isexact, every quasitrace on C ∗ ( E ) is a trace as was shown by Haagerup in [6] in the unitalcase and was extended to non-unital C ∗ -algebras by Kirchberg in [7]. Combining theseresults, when A is a exact C ∗ -algebra with an approximate unit consisting of projections,the canonical map T ( A ) → S (K ( A )) is always surjective. In particular this holds for A = C ∗ ( E ) when E is a totally disconnected topological graph.There is no known method for computing the order structure of K (C ∗ ( E )) for a topo-logical graph E , even when E is totally disconnected. However, there are special cases.When E is discrete, the ordered K -group is computed in [16], and when C ∗ ( E ) is givenby a dynamical system σ : Z y X with X totally disconnected, the ordered K -group isdescribed in Theorem 5.2 of [1]. In both cases, the order structure on K is determinedby the Pimsner-Voiculescu sequence above. It may be posible to give a similar descriptionof the order structure for general topological graph algebras in the totally disconnectedsetting. Computing the states on K-theory can be viewed as a first step towards such aresult. Indeed, for a simple, unital C ∗ -algebra A , the order structure on the K -group iscompletely determined, at least up to perforation, by the states.Composting the map T (C ∗ ( E )) → S (K (C ∗ ( E ))) with the map obtained in 6.3 yields acontinuous affine map T ( E ) → S (K (C ∗ ( E ))). We will show this map is in fact bijective. Theorem 7.2.
Let E be a topological graph with E totally disconnected. The canonicalmap T (C ∗ ( E )) → S (K (C ∗ ( E ))) restricts to an affine homeomorphism on T (C ∗ ( E )) γ .Proof. It is clear that the map is continuous and affine. As noted above, every state f on S (K (C ∗ ( E ))) lifts to a tracial state τ on C ∗ ( E ). Define a tracial state τ on C ∗ ( E ) by τ ( a ) = Z T τ ( γ z ( a )) dz and note that τ is gauge invariant. Fix a projection p ∈ M n (C ∗ ( E )). For each z ∈ T ,there is a path of projections in M n (C ∗ ( E )) connecting p and γ z ( p ) since γ is a stronglycontinuous action and T is connected. Hence p and γ z ( p ) are unitarily equivalent in theunitization of M n (C ∗ ( E )) (see Proposition 4.3.3 in [2], for example) and in particular, τ ( p ) = τ ( γ z ( p )). It follows that τ ( p ) = τ ( p ) for every projection p ∈ M n (C ∗ ( E )). Inparticular, τ also induces the state f on K (C ∗ ( E )).Now suppose τ and τ are two gauge invariant tracial states on C ∗ ( E ) which agree onK (C ∗ ( E )). Then the tracial states τ ◦ π and τ ◦ π induce the same state K ( C ( E )).Since E is totally disconnected, C ( E ) is spanned by projections and hence τ ◦ π = τ ◦ π . Now for k ∈ N and ξ, η ∈ C c ( E k ), τ ( π k ( ξ ) π k ( η ) ∗ ) = τ ( π ( h η, ξ i )) = τ ( π ( h η, ξ i )) = τ ( π k ( ξ ) π k ( η ) ∗ )since both τ and τ are tracial states. For distinct k, ℓ ∈ N , ξ ∈ C c ( E k ), and η ∈ C c ( E ℓ ), τ ( π k ( ξ ) π ℓ ( η ) ∗ ) = τ ( π k ( ξ ) π ℓ ( η ) ∗ ) = 0since both τ and τ are gauge invariant. So τ = τ .We have shown the canonical map T (C ∗ ( E )) γ → S (K (C ∗ ( E ))) is continuous, affine,and bijective. To show the inverse in continuous, suppose ( τ i ) is a net of gauge invarianttracial states on C ∗ ( E ) and τ is a gauge invariant tracial state on C ∗ ( E ). Let f i and f denote the states on K (C ∗ ( E )) induced by the τ i and τ . If f i → f , then f i ◦ π ∗ → f ◦ π ∗ on K ( C ( E )). Since C ( E ) is spanned by its projections, τ i ◦ π → τ ◦ π weak*. ByLemma 6.2, τ i → τ weak* and this completes the proof. (cid:3) For an simple, weakly unperforated ordered group K with a distinguished order unit,positive cone is determined by the states on K ; for x ∈ K , x > f ( x ) > f ∈ S ( K ) (see Theorem 6.8.5 in [2]). We were not able to determine when K (C ∗ ( E ))is weakly unperforated for a minimal, totally disconnected topological graph E . However,we can still describe the positive cone in K-theory up to perforation. Corollary 7.3.
Suppose a minimal topological graph in the sense of [10, Definition 8.8] such that E is totally disconnected and compact. Let x ∈ K (C ∗ ( E )) be given and let a ∈ C ( E , Z ) be such that π ( a ) = x . Then nx ≥ for some n ≥ if and only if R a dµ ≥ for all µ ∈ T ( E ) .Proof. Since E is compact, C ∗ ( E ) is unital. By Theorem 8.12 in [10], either C ∗ ( E )is simple or E is generated by a cycle. If E is generated by a cycle, then C ∗ ( E ) isMorita equivalent to C ( T ). In either case, the ordered abelian group K (C ∗ ( E )) is simple.Theorem 7.2 implies f ( x ) ≥ f ∈ S (K (C ∗ ( E ))) if and only if R a dµ ≥ µ ∈ T ( E ). The result follows from (the proof of) Theorem 6.8.5 in [2]. (cid:3) RACES ON TOPOLOGICAL GRAPH ALGEBRAS 31
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