3 - The K₀-Group of a Unital C*-Algebra

Summary

An Abelian group K₀(A) is associated to each unital C*-algebra A. The group K₀(A) arises from the Abelian semigroup (D(A), +) (defined in Chapter 2) and the Grothendieck construction (described below). We shall see that K₀ is a functor from the category of unital C*-algebras to the category of Abelian groups, and some of the properties of K₀ will be derived. Some examples of K₀-groups can be found at the end of the chapter.

We extend K₀ to a functor from the category of all C*-algebras (unital or not) in Chapter 4.

Definition of the K₀-group of a unital C*-algebra

The Grothendieck construction. One can associate an Abelian group to every Abelian semigroup in a way analogous to how one obtains the integers from the natural numbers, and in much the same way as one obtains the rational numbers from the integers. We describe here how this works; the proofs of various statements along the way are deferred to the next Paragraph.

Let (S, +) be an Abelian semigroup. Define an equivalence relation ∼ on S × S by (x₁, y₁) ∼ (x₂, y₂) if there exists z in S such that x₁ + y₂ + z = x₂ + y₁ + z. That ∼ is an equivalence relation is proved in Paragraph 3.1.2.

Write G(S) for the quotient (S × S) / ∼, and let 〈 x, y 〉 denote the equiva- lence class in G(S) containing (x, y) in S × S.