This chapter is devoted to a development of the basic facts about equivalence of idempotents and projections, inductive limits, and quotients. All readers are urged to at least browse through the chapter, since much of our basic notation is established here.

3. Local Banach Algebras and Inductive Limits

3.1. Local Banach Algebras

While we will be primarily concerned with the study of Banach algebras, in fact almost entirely with C*-algebras, it is sometimes useful to deal with a dense *-subalgebra which is “complete enough” to make the theory work. DEFINITION 3.1.1. A local Banach algebra is a normed algebra A which is closed under holomorphic functional calculus (i.e. if x ∈ A and f is an analytic function on a neighborhood of the spectrum of x in the completion of A, with f(0) = 0 if A is nonunital, then f(x) ∈ A.) For technical reasons we will also require that all matrix algebras over A have the same property. If A is a *-algebra, it will be called a local Banach * -algebra; if the norm is a pre-C*-norm, A will be called a local C*-algebra.

Note that our definition of a local C*-algebra does not agree with the definitions in [Behncke and Cuntz 1976] and [Blackadar and Handelman 1982].

EXAMPLES 3.1.2.

• (a) If A is any C*-algebra, then the Pedersen ideal [Pedersen 1979, 5.6] P(A) is a local C*-algebra. In particular, if X is a locally compact Hausdorff space, then CC(X) is a local C*-algebra.

• (b) Generalizing (a) [Pedersen 1979, 5.6.1], an algebraic direct limit of Banach algebras (see Theorem 3.3.2) is a local Banach algebra.