Non-stable \(K\)-theory and extremally rich \(C^\ast\)-algebras
type: Item
instance of: scholarly article
title: Non-stable \(K\)-theory and extremally rich \(C^\ast\)-algebras
zbMATH Open document ID: 1303.46062
author: Lawrence G. Brown
author: Gert Kjaergård Pedersen
published in: Journal of Functional Analysis
publication date: 2014-07-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/0708.3078
review text: A unital \(C^\ast\)-algebra \(A\) has stable rank one if the set \(A^{-1}\) of invertible elements is dense in \(A\), is isometrically rich if the set \(A^{-1}_l \cup A^{-1}_r\) of one-sided invertible elements is dense, and is extremally rich if the set \(A^{-1}_q\) of quasi-invertible elements is dense. A \(C^\ast\)-algebra \(A\) has weak cancellation if any pair of projections \(p, q\) in \(A\) that generate the same closed ideal \(I\) of \(A\) and have the same image in \(K_0(I)\) must be Murray-von Neumann equivalent in \(A\). If the matrix algebra \(\mathbb M_n(A)\) with coefficients from \(A\) has weak cancellation for every \(n\), or equivalently, if \(A \otimes \mathcal K\) has weak cancellation, one says that \(A\) has stable weak cancellation. ``A main question in this paper is whether every extremally rich \(C^\ast\)-algebra has weak cancellation. Some answers to this question are in Section 7.3 and Corollary 3.6, which state that every isometrically rich \(C^\ast\)-algebra has weak cancellation.'' One says that \(A\) has \(K_1\)-surjectivity if the map from the ``classifying space'' \(\mathcal U( A)/\mathcal U_0(A)\) to \(K_1(A)\) is surjective, \(K_1\)-injectivity if this map is injective, and \(K_1\)-bijectivity if it is bijective. Theorem 4.4 states that every extremally rich \(C^\ast\)-algebra with weak cancellation has \(K_1\)-surjectivity. In Corollary 5.3, the authors show that if \(A\) is extremally rich with weak cancellation, then \(K_e(A) = \varinjlim_n (\mathcal E(\mathbb M_n ( A))/\mathrm{homotopy})\) and one says that \(A\) has \(K_e\)-surjectivity, \(K_e\)-injectivity, or \(K_e\)-bijectivity if the map from \(\mathcal E(A)/\mathrm{homotopy}\) to \(K_e(A)\) is respectively surjective, injective, or bijective. These properties actually imply the corresponding \(K_1\)-properties. In Theorem 4.7, the authors show that every extremally rich \(C^*\)-algebra with weak cancellation has \(K_e\)-surjectivity. One says that a (non-unital) \(C^\ast\)…
reviewed by: Do Ngoc Diep
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 19K14
zbMATH DE Number: 6320963
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: \(C^\ast\)-algebras
zbMATH Keywords: extremally rich \(C^\ast\)-algebra
zbMATH Keywords: weak cancellation
MaRDI profile type: Publication
OpenAlex ID: W2964199334
arXiv ID: 0708.3078
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DOI: 10.1016/J.JFA.2014.04.003
description: scientific article