Algebraic \(K\)-theory of stable \(C^*\)-algebras
type: Item
instance of: scholarly article
title: Algebraic \(K\)-theory of stable \(C^*\)-algebras
zbMATH Open document ID: 0635.46061
DOI: 10.1016/0001-8708(88)90034-5
author: Nigel Higson
published in: Advances in Mathematics
publication date: 1988-01-01T00:00:00Z
review text: A \(C^*\)-algebra \(A\) is said to be stable if it is *-isomorphic to the tensor product \({\mathcal K}\otimes A\), where \({\mathcal K}\) is the \(C^*\)-algebra of compact operators on a separable Hilbert space. Most of the paper is devoted to the study of these algebras. The main theorem of this paper is as follows: If \(A\) is a stable \(C^*\)- algebra then the algebraic \(K\)-theory of \(A\) is equal to the topological \(K\)-theory of \(A\), i.e. \(K_*(A)\sim K_*^{top}(A)\). Besides this theorem, the author presents a number of other results, mostly on the same theme of comparing algebraic and topological K-theory, but occasionally as minor digressions from it. This paper is divided into six sections. I. Multiplier algebras and tensor products. II. \(K\)-theory of \(C^*\)-algebras. III. A homotopy invariance theorem. IV. Algebraic \(K_ 2\) of stable \(C^*\)-algebras. V. Algebraic \(K\)-theory of Calkin algebras. (This is the main section. The author proves the following conjecture of M. Karoubi: \(K_*(A\otimes C)\sim K_*^{top}(A\otimes C)\), where \(C\) is the Calkin algebra. The algebra \(A\otimes C\) may be regarded as a suspension of the \(A\otimes {\mathcal K}\); thus Karoubi's conjecture asserts that the algebraic and topological \(K\)-theories of stable \(C^*\)-algebras agree.) VI. Further results (There are two main topics: the study of the group \(\mathrm{GL}_ 1A\) instead of the stable version \(\mathrm{GL}_{\infty}A\) and the Karoubi-Villamayor algebraic \(K\)-theory).
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 4034437
zbMATH Keywords: tensor product
zbMATH Keywords: stable \(C^*\)-algebra
zbMATH Keywords: comparing algebraic and topological \(K\)-theory
zbMATH Keywords: multiplier algebras
zbMATH Keywords: homotopy invariance
zbMATH Keywords: Calkin algebras
zbMATH Keywords: suspension
zbMATH Keywords: Karoubi's conjecture
zbMATH Keywords: Karoubi-Villamayor algebraic \(K\)-theory
reviewed by: V. M. Deundjak
MaRDI profile type: Publication
full work available at URL: 0001-8708(88)90034-5
OpenAlex ID: W2081386899
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description: scientific article; zbMATH DE number 4034437