In this chapter, we will define the higher If-groups of a Banach algebra and relate them to suspensions in Section 8, and then prove the Bott Periodicity Theorem and establish the fundamental if-theory exact sequence in Section 9.

8. Higher üf-Groups

8.1. Definition of Kx﹛A)

Let A be a local Banach algebra. Recall that GLn(A) = ﹛x ∈ GLn(A+) : x = l n mod Mn(A)﹜. GLn(A) is a closed normal subgroup of GLn(A+). We embed GLn(A) into GLn+1(A) by x —> diag(x, 1). [This embedding is the “exponential” of the embedding of Mn(A) into Mn+1(A) considered in Chapter III. This is the appropriate analog, since the connection between K0 and K1 is given by exponentiation.]

Let GL∞ (A) = lim GLn(A). GL∞ (A) is a topological group with the inductive limit topology. GL∞ (A) can be thought of as the group of invertible infinite matrices which have diagonal elements in 1A + + A, off-diagonal elements in A, and only finitely many entries different from 0 or 1. We will identify elements of GLn(A) with their images in GL∞ (A)0

The embedding of GL∞ (A) into GLn+1(A) maps GLn(A)0 into GLn+1(A)0, and GL∞ (A)0 = lim GLn(A)0.

This is related to, but not the same as, the group Kalg 1of algebraic If-theory: Kalg 1 (A) is the quotient of GL∞ (A) by its commutator subgroup. See [Karoubi 1978, II.6.13] for the relationship.