K-theory for graded Banach algebras II

Abstract

Let A be a real or complex Banach algebra and assume that A is equipped with a continuous automorphism α such that α2 is the identity. In "K-theory for graded Banach algebras I" we have associated a group K(A) to such a pair (A, α). In this paper we prove that this group K(A) is isomorphic with K(SA⊗C) where SA is the algebra of continuous functions f : [0,1] → A with f(0) = f(1) = 0 and equipped with pointwise operations and where SA⊗C denotes the graded tensor product of SA with the Clifford algebra C = C0,1. The periodicity of Clifford algebras is used to show that K(S8A) = K(A) in general and K(S2A) = 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 as its typical part. The usual 6-term periodic exact sequence with K0 and K1 is a special case of this sequence. © 1988 by Pacific Journal of Mathematics.