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Spectra for Infinite Tensor Product Type Actions of Compact Groups

Published online by Cambridge University Press:  20 November 2018

Elliot C. Gootman  and
Aldo J. Lazar
Elliot C. Gootman
Affiliation:
Department of Mathematics University of Georgia Athens, Georgia 30602 U.S.A.gootman@alpha.math.uga.edu
Aldo J. Lazar
Affiliation:
Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences Tel Aviv University Tel Aviv 69978 Israelaldo@math.tau.ac.il
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Abstract

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We present explicit calculations of the Arveson spectrum, the strong Arveson spectrum, the Connes spectrum, and the strong Connes spectrum, for an infinite tensor product type action of a compact group. Using these calculations and earlier results (of the authors and C. Peligrad) relating the various spectra to the ideal structure of the crossed product algebra, we prove that the topology of G influences the ideal structure of the crossed product algebra, in the following sense: if G contains a nontrivial connected group as a direct summand, then the crossed product algebra may be prime, but it is never simple; while if G is discrete, the crossed product algebra is simple if and only if it is prime. These results extend to compact groups analogous results of Bratteli for abelian groups. In addition, we exhibit a class of examples illustrating that for compact groups, unlike the case for abelian groups, the Connes spectrum and strong Connes spectrum need not be stable.

Keywords

46L5522D25
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 2 , 01 April 1996 , pp. 330 - 342
Copyright
Copyright © Canadian Mathematical Society 1996

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