Embedding of Exact

We prove that any separable exact C*-algebra is isomorphic to a subalgebra of the Cuntz algebra O2. We further prove that if A is a simple separable unital nuclear C*-algebra, then O2 ⊗ A ∼= O2, and if, in addition, A is purely infinite, then O∞ ⊗ A ∼= A. The embedding of exact C*-algebras in O2 is continuous in the following sense. If A is a continuous field of C*-algebras over a compact manifold or finite CW complex X with fiber A(x) over x ∈ X, such that the algebra of continuous sections of A is separable and exact, then there is a family of injective homomorphisms φx : A(x) → O2 such that for every continuous section a of A the function x 7→ φx(a(x)) is continuous. Moreover, one can say something about the modulus of continuity of the functions x 7→ φx(a(x)) in terms of the structure of the continuous field. In particular, we show that the continuous field θ 7→ Aθ of rotation algebras posesses unital embeddings φθ in O2 such that the standard generators u(θ) and v(θ) satisfy max(‖φθ1 (u(θ1)) − φθ2 (u(θ2))‖, ‖φθ1 (v(θ1))− φθ2 (v(θ2))‖) < C|θ1 − θ2| 1/2 for some constant C. 0. Introduction It has recently become clear that the exact C*-algebras form an important class of C*-algebras more general than the nuclear C*-algebras. (A C*-algebra A is called exact if the functor A ⊗min − preserves short exact sequences.) For example, every C*-subalgebra of a nuclear C*-algebra is exact. The class of exact C*-algebras has a number of good functorial properties (see Section 7 of [Kr4]); in particular, unlike the class of nuclear C*-algebras, it is closed under passage to subalgebras. (Unfortunately, though, it is not closed under arbitrary extensions, only under “locally liftable” ones. See [Kr2] and Section 7 of [Kr4].) The reduced C*-algebras of some discrete groups (including free groups), and perhaps all discrete groups, are exact. (See Remark 7.8 of [Kr2].) Separable exact C*-algebras can be characterized as exactly those C*-algebras which occur as subquotients of the (nuclear) CAR (or 2 UHF) algebra ([Kr3]). In this paper, we show that every separable exact C*-algebra is isomorphic to a subalgebra of the Cuntz algebra O2. Thus, a separable C*-algebra is exact if and only if it is isomorphic to a subalgebra of of a nuclear C*-algebra, if and only if it is isomorphic to a subalgebra of the particular nuclear C*-algebra O2. The methods used to prove the embedding in O2 show that separable exact C*-algebras which are “close” in a certain sense have nearby embeddings in O2.We prove that if X is a compact metric space which is sufficiently nice (certainly including all compact manifolds and all finite CW complexes), and if A is a continuous field over X in the sense of Dixmier (Chapter 10 of [Dx]) such that the algebra of continuous sections of A is separable and exact, than A has a continuous representation in O2. That is, there is a collection of injective homomorphisms φx from the fibers A(x) of A to O2 such that, for every continuous section a of A, the function x 7→ φx(a(x)) is continuous. Moreover, one can say something about the “smoothness” of these functions. For example, we show that there are injective homomorphisms φθ from the rotation algebras Aθ (rational and irrational) to O2 Date: February 7, 2008. 1991 Mathematics Subject Classification. Primary: 46L35; Secondary: 46L05.