Residually Finite Dimensional C*-algebras

A C*-algebra is called residually finite dimensional (RFD for brevity) if it has a separating family of finite dimensional representations. A C*-algebra A is said to be AF embeddable if there is an AF algebra B and a ∗-monomorphisms α : A→ B. In this note we discuss the question of AF embeddability of RFD algebras. Since a C*-subalgebra of a nuclear C*-algebra must be exact [Ki], the nonexact RFD algebras ( such that the C*-algebra of the free group on two generators) are not AF embeddable. In this note we show that the cone over any nuclear RFD algebra is AF embeddable (see Theorem 6). Using a result of Spielberg [S] we obtain that the AF embeddability of a nuclear RFD algebra A ( with all ideals in the bootstrap category of [RS]) depends only on the homotopy type of A. The question whether all the exact or even nuclear RFD algebras are AF embeddable is open. The main ingredient of the proof is Theorem 5, which shows that if two ∗-homomorphisms from a nuclear RFD algebra A are asymptotically homotopic (in the sense of [CH]), then they are stably approximately unitarily equivalent. The case A = C(X) with X a compact metric space was proved in [D1]. The case when A is homogeneous is treated in [L1]. Very general related results appear in [L2]. We hope that the result given in Theorem 5 will be useful in the classification problem of simple nuclear C*-algebras. Indeed, by a result of Blackadar and Kirchberg [BK1,2] any separable nuclear C*-algebra having a separating family of quasidiagonal representations, is an inductive limit of nuclear RFD algebras. The reader is refered to [GoMe], [ExL] and [D2] for other results on RFD algebras.