Crossed Module Actions on Continuous Trace C*-Algebras

We discuss some recent developments that illustrate the interplay between the theory of crossed products of continuous trace C*-algebras and algebraic topology, summarizing results relating topological invariants coming from the theory of fiber bundles to continuous trace C*-algebras and their automorphism groups and the structure of the associated crossed product C*-algebras. This survey article starts from the classical theory of Dixmier, Douady, and Fell, and discusses the more recent work of Echterhoff, Phillips, Raeburn, Rosenberg, and Williams, among others. The topological invariants involved are Čech cohomology, the cohomology of locally compact groups with Borel cochains of C. Moore, and the recently introduced equivariant cohomology theory of Crocker, Kumjian, Raeburn and Williams.

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Continuous trace C*-algebras with given Dixmer-Douady class

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700023661 ◽

1985 ◽

Vol 38 (3) ◽

pp. 394-407 ◽

Cited By ~ 3

Author(s):

Iain Raeburn ◽

Joseph L. Taylor

Keyword(s):

Automorphism Group ◽

Outer Automorphism ◽

Explicit Construction ◽

Irreducible Representations ◽

Outer Automorphism Group ◽

C Algebras ◽

Finite Dimensional ◽

Continuous Trace ◽

Separable Algebras

AbstractWe give an explicit construction of a continuous trace C*algebra with prescribed Dixmier-Douady class, and with only finite-dimensional irreducible representations. These algebras often have non-trivial automorphisms, and we show how a recent description of the outer automorphism group of a stable continuous trace C*algebra follows easily from our main result. Since our motivation came from work on a new notion of central separable algebras, we explore the connections between this purely algebraic subject and C*a1gebras.

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CONTINUOUS–TRACE C*-ALGEBRAS NOT ISOMORPHIC TO THEIR OPPOSITE ALGEBRAS

International Journal of Mathematics ◽

10.1142/s0129167x01000642 ◽

2001 ◽

Vol 12 (03) ◽

pp. 263-275 ◽

Cited By ~ 7

Author(s):

N. CHRISTOPHER PHILLIPS

Keyword(s):

Homotopy Equivalent ◽

C Algebra ◽

C Algebras ◽

Continuous Trace

We give examples of locally trivial continuous-trace C *-algebra not isomorphic to their opposite algebras. Our examples include a unital C *-algebra which is both stably isomorphic to and homotopy equivalent to its opposite algebra, a unital C *-algebra which is homotopy equivalent to but not stably isomorphic to its opposite algebra, and a unital C *-algebra which is not even stably homotopy equivalent to its opposite algebra.

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