Homotopy of a pair of approximately commuting unitaries in a simple \(C^*\)-algebra
type: Item
instance of: scholarly article
title: Homotopy of a pair of approximately commuting unitaries in a simple \(C^*\)-algebra
zbMATH Open document ID: 0939.46033
author: Ola Bratteli
author: George A. Elliott
author: David E. Evans
author: Akitaka Kishimoto
published in: Journal of Functional Analysis
publication date: 1999-04-11T00:00:00Z
review text: Let A be a \(K_{1}\)-simple real rank zero C*-algebra, as defined in the article. This class of algebras includes unital purely infinite simple C*-algebras [\textit{J. Cuntz}, Ann. Math., II. Ser. 113, 181-197 (1981; Zbl 0446.46052)], unital real rank zero simple A{\textbf{T}}-algebras [\textit{G. A. Elliot}, J. Reine Angew. Math. 443, 179-219 (1993; Zbl 0809.46067)] and (not necessarily simple) unital AF-algebras. The main results are homotopy lemmas like the following: given \(\varepsilon > 0\) there exists a \(\delta > 0\) such that if u and v are unitaries in A, almost commuting in that \(\|vu - uv \|< \delta\), the spectrum of \(u\), except for a single gap larger than \(\delta\), is \(\delta\)-dense in \({\mathbb T}\), i.e. spectral projections corresponding to spectral intervals of length greater than or equal to \(\delta\) are non-zero, \([v]_{1} = 0\) and a certain \(K_{0}\)-valued obstruction, the isospectral obstruction, is zero, then \(v\) can be deformed to \(1\) along a path in \({\mathcal{U}}(A)\) of length less than \(5\pi + 1\). The authors prove also a `super-homotopy' lemma of a more symmetric form in that there are two pairs of almost commuting unitaries (u,v); one pair can be deformed into the other along a path of pairs of almost commuting unitaries in the algebra, the length of the path being bounded by a larger constant than for the homotopy lemma. The Bott obstruction for almost commuting unitaries in a unital C*-algebra seemed [in \textit{T. A. Loring}, Can. J. Math. 40, No. 1, 197-216 (1988; Zbl 0647.18007)] to come out of the blue. When \(A\) is a \(K_1\)-simple real rank zero C*-algebra and \(u\) has finite spectrum, or even if \(u \in {\mathcal{U}}_{0}(A)\), the authors define the more concrete isospectral obstruction. They generalize the role of \(v\) by considering a unital endomorphism \(\lambda\) of \(A\). Both \(Q\) and \(E\) denote spectral projections of intersecting half-open intervals with a large half-open gap. They assume \(\…
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47B15
zbMATH DE Number: 1250656
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: homotopy
zbMATH Keywords: approximately commuting unitaries
zbMATH Keywords: \(K_1\)-simple real rank zero \(C^*\)-algebra
zbMATH Keywords: homotopy lemmas
zbMATH Keywords: isospectral obstruction
zbMATH Keywords: super-homotopy
zbMATH Keywords: Bott obstruction
zbMATH Keywords: spectral gap
reviewed by: Aubrey Wulfsohn
MaRDI profile type: Publication
full work available at URL: jfan.1998.3261
OpenAlex ID: W1984261923
cites work: Q4249866
cites work: \(C^*\)-algebras of real rank zero
cites work: Stable isomorphism of hereditary subalgebras of \(C^*\)-algebras
cites work: Simple \(C^*\)-algebras generated by isometries
cites work: K-theory for certain C*-algebras
cites work: Approximately unitarily equivalent morphisms and inductive limit \(C^*\)- algebras
cites work: Q3931913
cites work: Q4311499
cites work: Derivations of matroid C\(^*\)-algebras. II
cites work: On the classification of C*-algebras of real rank zero.
cites work: Classification of certain infinite simple \(C^*\)-algebras. II
cites work: Invariants of almost commuting unitaries
cites work: Exponential rank of \(C^{\ast}\)-algebras with real rank zero and the Brown-Pedersen conjectures
cites work: Almost commuting unitary elements in purely infinite simple \(C^*\)- algebras
cites work: K-Theory and Asymptotically Commuting Matrices
cites work: Classification of direct limits of even Cuntz-circle algebras
cites work: Extensions of Inductive Limits of Circle Algebras
cites work: Classification of direct limits of generalized Toeplitz algebras
cites work: Q3324492
cites work: A Property of Purely Infinite Simple C ∗ -Algebras
cites work: Q4012787
DOI: 10.1006/JFAN.1998.3261
description: scientific article; zbMATH DE number 1250656