Imitation product-type actions of UHF algebras
type: Item
instance of: scholarly article
title: Imitation product-type actions of UHF algebras
zbMATH Open document ID: 0607.46037
DOI: 10.1016/0021-8693(86)90050-5
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let A be an algebra given as an infinite tensor product of matrix algebras, say \(A=\otimes_ kM_{n(k)}C\), and let g be an automorphism of A of the form \(g=\otimes Ad u_ k\), where each \(u_ k\) belongs to \(M_{n(k)}C\) and \(u^ 2_ k=1\), \(u_ k=u^*_ k\). Then the fixed point algebra \(A^ g=\{a\in A|\) \(a^ g=a\}\) has at most 2 rationally independent equivalence classes of idempotents (compactly expressed, the \({\mathbb{Z}}\)-rank of \(K_ 0(A^ g)\) is at most 2). In 1977, George Elliott asked me whether the converse holds. (Quoted from the paper.) Specifically, if g is an order 2 \((^*\)-) automorphism of A such that rank \(K_ 0(A^ g)\leq 2\), is g of the form \(\otimes Ad u_ k\) (which may correspond to a different infinite factorization of A), that is, is it of product type ? The problem is solved with counterexamples, and with two positive results, the second of which is: Corollary. Let A be a UHF algebra, and g an order two automorphism that is locally finite dimensional. Suppose the \({\mathbb{Z}}\)-rank of \(K_ 0(A^ g)\) (or of \(K_ 0(A\times <g>))\) is 1 or 2. (a) If A unitally contains a UHF algebra of type \(2^{\infty}q^{\infty}\) (for some odd prime q), then g is conjugate to a product type action. (b) If A contains the UHF algebra \(\otimes M_ pC\) (p varying over all primes), then g is conjugate to a product-type action.
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 46L40
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 46M05
zbMATH DE Number: 3981835
zbMATH Keywords: infinite tensor product of matrix algebras
zbMATH Keywords: automorphism
zbMATH Keywords: fixed point algebra
zbMATH Keywords: UHF algebra
zbMATH Keywords: order two automorphism
zbMATH Keywords: product type action
author: David Handelman
reviewed by: Palle Jorgensen
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90050-5
OpenAlex ID: W2047048301
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description: scientific article; zbMATH DE number 3981835