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.
