An Equivariant Brauer Group and Actions of Groups onC*-Algebras

✍ Scribed by David Crocker; Alexander Kumjian; Iain Raeburn; Dana P. Williams

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 524 KB
• Volume: 146
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.3010

✦ Synopsis

Suppose that (G, T ) is a second countable locally compact transformation group given by a homomorphism l: G Ä Homeo(T ), and that A is a separable continuous-trace C*-algebra with spectrum T. An action :: G Ä Aut(A) is said to cover l if the induced action of G on T coincides with the original one. We prove that the set Br G (T ) of Morita equivalence classes of such systems forms a group with multiplication given by the balanced tensor product: [A, :][B, ;]=[A C 0 (T ) B, : ;], and we refer to Br G (T ) as the Equivariant Brauer Group. We give a detailed analysis of the structure of Br G (T ) in terms of the Moore cohomology of the group G and the integral cohomology of the space T. Using this, we can characterize the stable continuous-trace C*-algebras with spectrum T which admit actions covering l. In particular, we prove that if G=R, then every stable continuous-trace C*-algebra admits an (essentially unique) action covering l, thereby substantially improving results of Raeburn and Rosenberg.