C^*-Algebra Homomorphisms and KK-Theory

Let A be a simple C^*-algebra which can be written as an inductive limit of P₁M_n1(C(X₁))P₁ → P₂M_n2(C(X₂))P₂ → …, where X_n are finite CW complexes with sup dim(X_n < + ∞ and P_i ∈ M_ni(C(X_i)) are projections. Let X be a finite CW complex. In this paper, we will give a necessary and sufficient condition for a KK-element α ∈ KK(C(X),A) to be realized by a C^*- algebra homomorphism Φ: C(X) → A. If we further suppose that A has a unique trace, then the set of all injective homomorphisms from C(X) to A ⊗ 𝒦 can be characterized up to modulo approximately unitary equivalence.