5. Basic K0-Theory

Throughout this section, A will denote a local Banach algebra. (In fact, for much of the section A could just be any ring.)

5.1. Basic Definitions

The definition of K0(A) requires simultaneous consideration of all of the matrix algebras over A. The most elegant way to do this is the following:

DEFINITION 5.1.1. M ∞(A) is the algebraic direct limit of Mn(A) under the embeddings a —> diag(a, 0).

M∞ (A) may be thought of as the algebra of all infinite matrices over A with only finitely many nonzero entries. Whenever it is convenient, we will identify Mn(A) with its image in the upper left-hand corner of Mn+k(A) or M∞A). When it is necessary to topologize M∞(A), any topology inducing the natural topologies on Mn(A) will do. For example, M∞ (A) may be given the inductive limit topology if desired. Better yet, one may choose the norms on Mn(A) so that the embeddings are isometries; M∞A) is a local Banach algebra with the induced norm.

If A is a (local) C*-algebra, the embeddings are isometries, so M∞(A) has a natural norm. The completion is called the stable algebra of A, denoted A ⊗ K (it is the C*-tensor product of A and K). DEFINITION 5.1.2. Proj(A) is the set of algebraic equivalence classes of idempotents in A. We set V(A) = Proj(M∞ (A)).