An extra structure is added to the Abelian group K0(A) of a C*-algebra A by specifying a certain subset of it, called K0(A)+. The set K0(A)+ consists of all elements in K0(A) of the form [p]0, where p is a projection in P∞(A). When A is a unital, stably finite C*-algebra, then (K0(A), K0(A)+) has the pleasant structure of an ordered Abelian group. We shall for this purpose also discuss finiteness properties of C*-algebras and of projections.

The ordered K0-group of stably finite C*-algebras

An element a in a unital C*-algebra A is called left-invertible if there exists an element b in A such that ba = 1, and a is called right-invertible if ab = 1 for some b in A. If b1a = ab1 = 1, then b1 = b1ab2 = b2 and this shows that a is invertible if and only if a is both left- and right-invertible. Moreover, a is left-invertible if and only if a*a is invertible, and, similarly, a is right-invertible if and only if aa* is invertible. (See Exercise 5.1).

Definition 5.1.1. A projection p in a C*-algebra A is said to be infinite if it is equivalent to a proper subprojection of itself, i.e., if there is a projection q in A such that p ∼ q < p. If p is not infinite, then p is said to be finite.

A unital C*-algebra A is said to be finite if its unit 1A is a finite projection. Otherwise A is called infinite. If Mn(A) is finite for all positive integers n, then A is stably finite.