Morita Equivalence and Continuous-Trace 𝐶*-Algebras

Raeburn, Iain; Williams, Dana P.

doi:10.1090/surv/060

Cited by 459 publications

(703 citation statements)

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“…[45,Theorem C.26]) given by σ(I) = x if and only if I contains the ideal generated by functions in C 0 (G (0) ) that vanish at x. Since V is an K(V)-A imprimitivity bimodule, the Rieffel correspondence restricts to a homeomorphism h : Prim K(V) → Prim A (see, for example, [33,Proposition 3.3.3]). Therefore one obtains by composition a continuous map…”

Section: Main Results and Applicationsmentioning

confidence: 99%

“…If V is a right Hilbert module over a C * -algebra A [20,24,33,39], then there is a left A-module V * with a conjugate linear isomorphism from V to V * , written v → v * such that av * = (va * ) * and A u * , v * = u, v A . Rankone operators on V are defined via θ u,v (w) = u · v, w .…”

Section: Background About Fell Bundlesmentioning

confidence: 99%

“…Since V is a K(V)-A-imprimitivity bimodule, it follows that K(V) and A are Morita equivalent. Let h : Prim A → Prim K(V) be the Rieffel correspondence (see, for example, [33,Corollary 3.3]). Then, from the definition of V and K(V) and [17, Formula (1) on page 1247] it follows that if P ∈ Prim K(V) and g ∈ G then g · P = h(g · h −1 (P )).…”

Section: Corollary 39 Let G Be a Locally Compact Hausdorff Groupoidmentioning

confidence: 99%

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A stabilization theorem for Fell bundles over groupoids

Ionescu¹,

Kumjian²,

Sims³

et al. 2017

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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ABSTRACT. We study the C * -algebras associated to upper-semicontinuous Fell bundles over second-countable Hausdorff groupoids. Based on ideas going back to the Packer-Raeburn "Stabilization Trick," we construct from each such bundle a groupoid dynamical system whose associated Fell bundle is equivalent to the original bundle. The upshot is that the full and reduced C * -algebras of any saturated upper-semicontinuous Fell bundle are stably isomorphic to the full and reduced crossed products of an associated dynamical system. We apply our results to describe the lattice of ideals of the C * -algebra of a continuous Fell bundle by applying Renault's results about the ideals of the C * -algebras of groupoid crossed products. In particular, we discuss simplicity of the Fell-bundle C * -algebra of a bundle over G in terms of an action, described by the first and last named authors, of G on the primitive-ideal space of the C * -algebra of the part of the bundle sitting over the unit space. We finish with some applications to twisted k-graph algebras, where the components of our results become more concrete.

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Section: Main Results and Applicationsmentioning

confidence: 99%

Section: Background About Fell Bundlesmentioning

confidence: 99%

Section: Corollary 39 Let G Be a Locally Compact Hausdorff Groupoidmentioning

confidence: 99%

See 1 more Smart Citation

A stabilization theorem for Fell bundles over groupoids

Ionescu¹,

Kumjian²,

Sims³

et al. 2017

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Let J be the ideal of C * (E) generated by φ(I). By Theorem 5.22 the homomorphism φ induces an isomorphism φ I : C * (G)/I → Q(C * (E)/J)Q where Q ∈ M(C * (E)/J) is the image of Q ∈ M(C * (E)) under the extension of the quotient map to multiplier algebras (see [16,Corollary 2.51]). In particular, the projection Q is full.…”

Section: Quotients By Gauge-invariant Idealsmentioning

confidence: 99%

Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence

Katsura¹,

Muhly²,

Sims³

et al. 2010

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We prove that the classes of graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence. This result answers the long-standing open question of whether every Exel-Laca algebra is Morita equivalent to a graph algebra. Given an ultragraph G we construct a directed graph E such that C * (G) is isomorphic to a full corner of C * (E). As applications, we characterize real rank zero for ultragraph algebras and describe quotients of ultragraph algebras by gauge-invariant ideals.

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“…A version of Morita's theory appropriate for C*-algebras was developed by Rieffel (see Rieffel, 1974 andRaeburn andWilliams, 1998) and Blecher (see Blecher, 2001). It is also well known, for example (see Landsman, 2001a andLandsman, 2001b), that C*-algebras form a bicategory and two C*-algebras A, B are isomorphic in this bicategory iff they are Morita equivalent.…”

Section: Introductionmentioning

confidence: 99%