scholarly journals The representation theory of C*-algebras associated to groupoids

2012 ◽  
Vol 153 (1) ◽  
pp. 167-191 ◽  
Author(s):  
LISA ORLOFF CLARK ◽  
ASTRID AN HUEF
Keyword(s):  
Representation Theory ◽  
Haar System ◽  
Directed Graphs ◽  
Orbit Space ◽  
Locally Compact ◽  
C Algebras ◽  
Bounded Trace ◽  
Equivariant Functions

AbstractLet E be a second-countable, locally compact, Hausdorff groupoid equipped with an action of such that G: = E/ is a principal groupoid with Haar system λ. The twisted groupoid C*-algebra C*(E; G, λ) is a quotient of the C*-algebra of E obtained by completing the space of -equivariant functions on E. We show that C*(E; G, λ) is postliminal if and only if the orbit space of G is T0 and that C*(E; G, λ) is liminal if and only if the orbit space is T1. We also show that C*(E; G, λ) has bounded trace if and only if G is integrable and that C*(E; G, λ) is a Fell algebra if and only if G is Cartan.Let be a second-countable, locally compact, Hausdorff groupoid with Haar system λ and continuously varying, abelian isotropy groups. Let be the isotropy groupoid and : = /. Using the results about twisted groupoid C*-algebras, we show that the C*-algebra C*(, λ) has bounded trace if and only if is integrable and that C*(, λ) is a Fell algebra if and only if is Cartan. We illustrate our theorems with examples of groupoids associated to directed graphs.

TOPOLOGICAL QUIVERS

2005 ◽  
Vol 16 (07) ◽  
pp. 693-755 ◽  
Author(s):  
PAUL S. MUHLY ◽  
MARK TOMFORDE
Keyword(s):  
Uniqueness Theorem ◽  
Directed Graphs ◽  
Algebra Structure ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Gauge Invariant ◽  
C Algebras ◽  
Natural Analogues ◽  
General Perspective ◽  
The Ideal

Topological quivers are generalizations of directed graphs in which the sets of vertices and edges are locally compact Hausdorff spaces. Associated to such a topological quiver [Formula: see text] is a C*-correspondence, and from this correspondence one may construct a Cuntz–Pimsner algebra [Formula: see text]. In this paper we develop the general theory of topological quiver C*-algebras and show how certain C*-algebras found in the literature may be viewed from this general perspective. In particular, we show that C*-algebras of topological quivers generalize the well-studied class of graph C*-algebras and in analogy with that theory much of the operator algebra structure of [Formula: see text] can be determined from [Formula: see text]. We also show that many fundamental results from the theory of graph C*-algebras have natural analogues in the context of topological quivers (often with more involved proofs). These include the gauge-invariant uniqueness theorem, the Cuntz–Krieger uniqueness theorem, descriptions of the ideal structure, and conditions for simplicity.

scholarly journals GROUPOID -ALGEBRAS WITH HAUSDORFF SPECTRUM

2013 ◽  
Vol 88 (2) ◽  
pp. 232-242 ◽  
Author(s):  
GEOFF GOEHLE
Keyword(s):  
Haar System ◽  
Sufficient Conditions ◽  
Orbit Space ◽  
Necessary And Sufficient Conditions ◽  
Locally Compact ◽  
Fell Topology ◽  
Necessary And Sufficient ◽  
Convergent Sequences

AbstractSuppose that $G$ is a second countable, locally compact Hausdorff groupoid with abelian stabiliser subgroups and a Haar system. We provide necessary and sufficient conditions for the groupoid ${C}^{\ast } $-algebra to have Hausdorff spectrum. In particular, we show that the spectrum of ${C}^{\ast } (G)$ is Hausdorff if and only if the stabilisers vary continuously with respect to the Fell topology, the orbit space ${G}^{(0)} / G$ is Hausdorff, and, given convergent sequences ${\chi }_{i} \rightarrow \chi $ and ${\gamma }_{i} \cdot {\chi }_{i} \rightarrow \omega $ in the dual stabiliser groupoid $\widehat{S}$ where the ${\gamma }_{i} \in G$ act via conjugation, if $\chi $ and $\omega $ are elements of the same fibre then $\chi = \omega $.

scholarly journals Topological Amenability Is a Borel Property

2015 ◽  
Vol 117 (1) ◽  
pp. 5 ◽  
Author(s):  
Jean Renault
Keyword(s):  
Haar System ◽  
Locally Compact

We establish that a $\sigma$-compact locally compact groupoid possessing a continuous Haar system is topologically amenable if and only if it is Borel amenable. We give some examples and applications.

Moore Cohomology and Central Twisted Crossed Product C*-Algebras

1996 ◽  
Vol 48 (1) ◽  
pp. 159-174 ◽  
Author(s):  
Judith A. Packer
Keyword(s):  
Cohomology Group ◽  
Hausdorff Space ◽  
Countable Group ◽  
Equivalence Classes ◽  
Locally Compact ◽  
C Algebras ◽  
Integer Coefficients ◽  
Direct Sum ◽  
Cech Cohomology ◽  
The Relationship

AbstractLet G be a locally compact second countable group, let X be a locally compact second countable Hausdorff space, and view C(X, T) as a trivial G-module. For G countable discrete abelian, we construct an isomorphism between the Moore cohomology group Hn(G, C(X, T)) and the direct sum Ext(Hn-1(G), Ȟl(βX, Ζ)) ⊕ C(X, Hn(G, T)); here Ȟ1 (βX, Ζ) denotes the first Čech cohomology group of the Stone-Čech compactification of X, βX, with integer coefficients. For more general locally compact second countable groups G, we discuss the relationship between the Moore group H2(G, C(X, T)), the set of exterior equivalence classes of element-wise inner actions of G on the stable continuous trace C*-algebra C0(X) ⊗ 𝒦, and the equivariant Brauer group BrG(X) of Crocker, Kumjian, Raeburn, and Williams. For countable discrete abelian G acting trivially on X, we construct an isomorphism is the group of equivalence classes of principal Ĝ bundles over X first considered by Raeburn and Williams.

scholarly journals Extension problems and non-Abelian duality for C*-algebras

2007 ◽  
Vol 75 (2) ◽  
pp. 229-238 ◽  
Author(s):  
Astrid an Huef ◽  
S. Kaliszewski ◽  
Iain Raeburn
Keyword(s):  
Compact Group ◽  
Unitary Representation ◽  
Closed Subgroup ◽  
Locally Compact Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Test Question ◽  
Locally Compact ◽  
Dual Representation ◽  
C Algebras

Suppose that H is a closed subgroup of a locally compact group G. We show that a unitary representation U of H is the restriction of a unitary representation of G if and only if a dual representation Û of a crossed product C*(G) ⋊ (G/H) is regular in an appropriate sense. We then discuss the problem of deciding whether a given representation is regular; we believe that this problem will prove to be an interesting test question in non-Abelian duality for crossed products of C*-algebras.

scholarly journals On $C*$-algebras associated with locally compact groups

1996 ◽  
Vol 124 (10) ◽  
pp. 3151-3158 ◽  
Author(s):  
M. B. Bekka ◽  
E. Kaniuth ◽  
A. T. Lau ◽  
G. Schlichting
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
C Algebras

scholarly journals Equivariant KK-theory for inverse limits of G-C*-algebras

1994 ◽  
Vol 56 (2) ◽  
pp. 183-211 ◽  
Author(s):  
Claude Schochet
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Inverse Limits ◽  
Locally Compact ◽  
C Algebras

AbstractThe Kasparov groups are extended to the setting of inverse limits of G-C*-algebras, where G is assumed to be a locally compact group. The K K-product and other important features of the theory are generalized to this setting.

scholarly journals 1-Algebra of a Locally Compact Groupoid

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-17
Author(s):  
Massoud Amini ◽  
Alireza Medghalchi ◽  
Ahmad Shirinkalam
Keyword(s):  
Invariant Measure ◽  
Banach Algebra ◽  
Haar System ◽  
Locally Compact ◽  
Radon Measures ◽  
Integrable Functions

For a locally compact groupoid with a fixed Haar system and quasi-invariant measure , we introduce the notion of -measurability and construct the space 1(, , ) of absolutely integrable functions on and show that it is a Banach -algebra and a two-sided ideal in the algebra () of complex Radon measures on . We find correspondences between representations of on Hilbert bundles and certain class of nondegenerate representations of 1(, , ).

C*-algebras of Clifford semigroups

1989 ◽  
Vol 111 (1-2) ◽  
pp. 129-145 ◽  
Author(s):  
John Duncan ◽  
A.L.T. Paterson
Keyword(s):  
Representation Theory ◽  
Clifford Semigroup ◽  
Sheaf Theory ◽  
C Algebras ◽  
Clifford Semigroups

SynopsisWe investigate algebras associated with a (discrete) Clifford semigroup S =∪ {Ge: e ∈ E{. We show that the representation theory for S is determined by an enveloping Clifford semigroup UC(S) =∪ {Gx: x ∈ X} where X is the filter completion of the semilattice E. We describe the representation theory in terms of both disintegration theory and sheaf theory.

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