1997/12/09 by Eberhard Kirchberg, Kirchberg, Eberhard, N. Christopher Phillips +1 · 2 citations
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.funct-an/9712002
We prove that any separable exact C*-algebra is isomorphic to a subalgebra of the Cuntz algebra \cal O2. We further prove that if A is a simple separable unital nuclear C*-algebra, then \cal O2 ⊗ A ≅ \cal O2, and if, in addition, A is purely infinite, then \cal O∞ ⊗ A ≅ A. The embedding of exact C*-algebras in \OA2 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) → \cal O2 such that for every continuous section a of A the function x ↦ ϕx (a (x)) is continuous. Moreover, one can say something about the modulus of continuity of the functions x ↦ ϕx (a (x)) in terms of the structure of the continuous field. In particular, we show that the continuous field θ↦ Aθ of rotation algebras posesses unital embeddings ϕθ in \cal O2 such that the standard generators u (θ) and v (θ) are mapped to Lip1/2 functions.