Extending traces on fixed point \(C^ *\) algebras under Xerox product type actions of compact Lie groups
type: Item
instance of: scholarly article
title: Extending traces on fixed point \(C^ *\) algebras under Xerox product type actions of compact Lie groups
zbMATH Open document ID: 0623.46027
DOI: 10.1016/0022-1236(87)90080-2
published in: Journal of Functional Analysis
publication date: 1987-01-01T00:00:00Z
review text: On the UHF \(C^*\)-algebra \(A=\otimes M_ n(C)\) of the type \(n^{\infty}\), the (Xerox) product action \(\alpha (g)=\otimes Ad \pi (g)\) of a compact connected Lie group G via a unitary representation \(\pi\) : \(G\to M_ n(C)\) is considered and the fixed point algebras \(A^ G\) and \(A^ T\) are studied, where T is the maximal torus in G. The principal result (Theorem) asserts that every (pure) trace on \(A^ G\) extends to a (pure) trace on \(A^ T\). In the course of the proof, it is shown (Proposition II.2) that \(K_ 0(A^ G)\) is finitely generated as a ring and \(K_ 0(A^ T)\) is finitely generated as a \(K_ 0(A^ G)\)-module.
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 46L30
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 4011211
zbMATH Keywords: Xerox product action
zbMATH Keywords: UHF \(C^ *\)-algebra
zbMATH Keywords: compact connected Lie group
zbMATH Keywords: unitary representation
zbMATH Keywords: fixed point algebras
zbMATH Keywords: maximal torus
zbMATH Keywords: trace
author: David Handelman
reviewed by: Huzihiro Araki
MaRDI profile type: Publication
full work available at URL: 0022-1236(87)90080-2
OpenAlex ID: W2020252911
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description: scientific article; zbMATH DE number 4011211