Quasi-Product Actions of a Compact Group on a C*-Algebra

✍ Scribed by O. Bratteli; G.A. Elliott; A. Kishimoto

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 946 KB
• Volume: 115
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1993.1093

✦ Synopsis

Let (\alpha) be an action of a compact group on a separable prime (C^{*})-algebra (A). Several conditions on (\alpha) are shown to be equivalent, among which are the following two:

there exists a faithful irreducible representation of (A) which is also irreducible on (A^{x});

there exists an (\chi)-invariant pure state (\omega) of (A) such that (\pi_{\omega|| A^{x}}) is faithful (it follows also that (\pi_{\omega}) is faithful).

These two conditions mean, roughly speaking, that there exist, respectively, free orbits and fixed points in the space of equivalence classes of irreducible representations of (A), acted upon by (G). 1993 Academic Press. Inc.