Algebraic \(K\)-theory of normed algebras
type: Item
instance of: scholarly article
title: Algebraic \(K\)-theory of normed algebras
zbMATH Open document ID: 0981.46057
DOI: 10.1023/A:1007885125771
author: Hvedri Inassaridze
published in: \(K\)-Theory
publication date: 2002-03-18T00:00:00Z
review text: In order to extend Quillen's \(K\)-theory of discrete rings [\textit{D. Quillen}, Higher algebraic \(K\)-theory'', Lect. Notes Math. 341, 85-147 (1973; Zbl 0292.18004)] to the category of normed \(A\)-algebras, the author gives in the first part of the paper a Quillen's type construction for topological exact categories, namely for the category of finitely generated projective modules over a normed \(k\)-algebra \(A\), where \(k\) is a commutative Banach ring with unit, and their bounded \(A\)-homomorphisms. It is proved that Quillen's \(K\)-theory on the category of special normed real algebras is isomorphic to topological \(K\)-theory. The second section deals with the relation between algebraic and topological \(K\)-functors on the category of real normed algebras. So, the natural sufficient conditions are obtained for the isomorphism of algebraic and topological \(K\)-functors on this category (Theorems 10 and 22 of this paper). These results allow the author to introduce the notion of \(K\)-stability of real normed algebras (Definition 12) which is closely related with stability of \(C^*\)-algebras and with \(K_*\)-regularity of discrete rings. It is shown that a wide class of complex normed algebras including polynomial algebras \(A[x_1,\dots, x_n]\) over stable \(C^*\)-algebras \(A\), Laurent polynomial algebras \(A[t, t^{-1}]\) over stable \(C^*\)-algebras \(A\), \(C^*\)-algebras of continuous maps from compact spaces to the Calkin algebra, retracts of \(C^*\)-algebras are \(K_i\) stable for \(i\geq 1\). Most of them are not complete and on the category of these complex normed algebras one has the equality of algebraic and topological \(K\)-functors (Theorems 14 and 18, Remark 19, Corollary 13, Propositions 20 and 21).
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 19K99
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 1563217
zbMATH Keywords: category of normed \(A\)-algebras
zbMATH Keywords: topological exact categories
zbMATH Keywords: category of finitely generated projective modules over a normed \(k\)-algebra
zbMATH Keywords: commutative Banach ring
zbMATH Keywords: isomorphism of algebraic and topological \(K\)-functors
zbMATH Keywords: stability of \(C^*\)-algebras
zbMATH Keywords: \(K_*\)-regularity of discrete rings
zbMATH Keywords: Laurent polynomial algebras
zbMATH Keywords: Calkin algebra
zbMATH Keywords: retracts of \(C^*\)-algebras
zbMATH Keywords: \(K\)-functors
reviewed by: Liliana Răileanu
MaRDI profile type: Publication
full work available at URL: a:1007885125771
OpenAlex ID: W2075479612
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description: scientific article; zbMATH DE number 1563217