Asymptotic unitary equivalence in C*-algebras

Let C = C(X) be the unital C*-algebra of all continuous functions on a finite CW complex X and let A be a unital simple C*-algebra with tracial rank at most one. We show that two unital monomorphisms φ,ψ: C → A are asymptotically unitarily equivalent, i.e., there exists a continuous path of unitaries {u t : t ∈ [0, 1)} ⊂ A such that lim t→1 u* t φ(f)u t = ψ(f) for all f ∈ C(X) if and only if [φ] = [ψ] in KK(C,A), τ ◦φ = τ ◦ψ for all τ ∈ T(A), and φ † = ψ †, where T(A) is the simplex of tracial states of A and φ †, ψ †: U ∞(C)/DU ∞(C) → U ∞(A)/DU ∞(A) are the induced homomorphisms and where U ∞(A) = ∪ =1 ∞ U(M k (A)) and U ∞(C) = ∪ =1 ∞ U(M k (C)) are usual infinite unitary groups, respectively, and DU ∞(A) and DU ∞(C) are the commutator subgroups of U ∞(A) and U ∞(C), respectively. We actually prove a more general result for the case in which C is any general unital AH-algebra.

Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.