The K-theory of a C*-algebra is defined in terms of equivalence classes of its projections and equivalence classes of its unitary elements — possibly after adjoining a unit and forming matrix algebras. We shall in this chapter derive the facts needed about projections and unitary elements with emphasis on the equivalence relation defined by homotopy and also — for projections — Murray–von Neumann equivalence and unitary equivalence.

Homotopy classes of unitary elements

Homotopy. Let X be a topological space. Say that two points a, b in X are homotopic in X, written a ∼hb in X, if there is a continuous function ν: [0,1] → X such that ν(0) = a and ν(1) = b. The relation ∼h is an equivalence relation on X. The continuous function ν above is called a continuous path from a to b, and it is often denoted by t ν(t) or t → νt, with or without specifying explicitly that t belongs to the interval [0,1].

Needless to say, the reference to the space X is crucial. For example, any two elements a, b in a C*-algebra A are homotopic in A. Indeed, take the continuous path t ↦ (1 − t)a + tb. But, as we shall see, two projections in A need not be homotopic in the set of all projections in A. We shall nevertheless sometimes omit the reference to the space X and just write a ∼hb instead of a ∼hb in X, when it is clear from the context in which space the homotopy should be realized.