K-theory for graded Banach algebras. II
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.
scientific article; zbMATH DE number 4009111
scientific article