ALMOST MULTIPLICATIVE MORPHISMS AND K-THEORY

Abstract

Let X be a compact metric space and A=C(X). Suppose that ℬ is a class of unital C*-algebras satisfying certain conditions, we prove the following: For any ∊>0, finite set F⊂A, there is an integer l such that if ϕ, ψ:A→B(B∈ℬ) are sufficiently multiplicative morphisms (e.g. when both ϕ and ψ are *-homomorphisms) which induce same K-theoretical maps, then there are a unitary u∈Ml+1(B) and a homomorphism σ:A→Ml(B) with finite dimensional image such that [Formula: see text] for all f∈F. In particular, the integer l does not depend on B, ϕ and ψ. This feature has important applications to the classification theory of nuclear C*-algebras.