K-theory for graded Banach algebras. II
type: Item
instance of: scholarly article
title: K-theory for graded Banach algebras. II
zbMATH Open document ID: 0622.46049
published in: Pacific Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let A be a real or complex Banach algebra and assume that A is equipped with a continuous automorphism \(\alpha\) such that \(\alpha ^ 2\) is the identity. In [C. R. Acad. Sci., Paris, Sér. I 304, 267-270 (1987; Zbl 0606.46045)] we have defined an abelian group K(A) for such an algebra. In this paper we use a result of Wood to show that K(A) is isomorphic with K(SA\({\hat \otimes}C)\) where SA is the algebra over continuous functions f: [0,1]\(\to A\) with \(f(0)=f(1)=0\) and equipped with pointwise operations and where SA\({\hat \otimes}C\) denotes the graded tensor product of SA with the Clifford algebra \(C=C^{0,1}\). The periodicity of Clifford algebras is used to show that \(K(S^ 8A)\cong K(A)\) in general and \(K(S^ 2A)\cong K(A)\) in the complex case. All this gives rise to an important periodic exact sequence associated to an algebra A and an invariant closed ideal I with \[ K(I)\quad \to \quad K(A)\quad \to \quad K(A/I)\quad \to \quad K(I{\hat \otimes}C)\quad \to \quad K(A{\hat \otimes}C)\quad \to \quad K(A/I{\hat \otimes}C) \] as its typical part. The usual 6-term periodic exact sequence with \(K_ 0\) and \(K_ 1\) is a special case of this sequence.
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 46M20
zbMATH DE Number: 4009111
zbMATH Keywords: graded Banach algebras
zbMATH Keywords: complex Banach algebra
zbMATH Keywords: continuous automorphism
zbMATH Keywords: periodicity of Clifford algebras
zbMATH Keywords: periodic exact sequence
zbMATH Keywords: invariant closed ideal
author: Alfons Van Daele
MaRDI profile type: Publication
full work available at URL: pjm.1988.134.377
OpenAlex ID: W2153506855
DOI: 10.2140/PJM.1988.134.377
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description: scientific article; zbMATH DE number 4009111