Almost multiplicative morphisms and K-theory

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 epsilon > 0, finite set F subset of A, there is an integer l such that if phi, psi: A --> B(B is an element of B) are sufficiently multiplicative morphisms (e.g. when both phi and psi are *-homomorphisms) which induce same K-theoretical maps, then there are a unitary u is an element of Ml+1(B) and a homomorphism sigma : A --> M-l(B) with finite dimensional image such that parallel tou* diag(phi (f), sigma (f))u - diag(psi (f), sigma (f))parallel to < <epsilon> for all f is an element of F. In particular, the integer l does not depend on B, phi and psi. This feature has important applications to the classification theory of nuclear C*-algebras.