Quasidiagonal C ∗ -algebras and Nonstandard Analysis”-Arxiv preprint math.OA/0209292, 2002 - arxiv.org

In this paper we use nonstandard analysis ([1], [10], [12]) to investigate injectability into C-algebras B which are infinitesimal hulls of hyperfinite dimensional internal C-algebras B. B is obtained from B by considering the subspace Fin(B) of elements with norm ≪ ∞ and identifying x, y ∈ Fin(B) whenever x − y has infinitesimal norm. Though B is a legitimate standard C-algebra, it is very large, except in the uninteresting case the original B is finite dimensional. We point out that the C-algebras B are exactly ultraproducts of finite-dimensional C-algebras (see [9] or the appendix). A more interesting question from an operator theorist’s viewpoint, is which kinds of separable C-algebras are injectable into B and what kinds of mappings exist from separable C-algebras into B. We show the following: If A is an AF algebra, Proposition 5.3 determines the inner conjugacy classes of C-morphisms for A into a fixed B in terms of certain divisibility properties for sequences. Proposition 5.6 gives a necessary and sufficient condition for injectability of an AF algebra into a fixed B in similar terms. Finally in §8, we give a characterization of nuclear subhyperfinite C-algebras, that is nuclear C-algebras which are injectable into some B. Theorem 8.2 states that for nuclear C-algebras, subhyperfiniteness is equivalent to quasidiagonality. To prove this characterization, we use a lifting theorem for nuclear completely positive contractions reminiscent of that of Choi and Effros [4]. There are numerous open questions we have not dealt with at this time. Of particular interest is the precise relationship between subhyperfinitess and nuclearity. Another question, which is possibly more of a set-theoretic nature, is whether the (enormous) C-algebras B corresponding to distinct hyperfinite dimensional internal C-algebras are non-isomorphic. The structure of the paper is as follows: §2 is devoted to a review of basic ideas of nonstandard analysis. In §3 we review general facts about C-algebras in the internal context. In §4, we prove a number of basic lifting theorems for projections and partial isometries in a context which is unusual, only because the lifting property relates to a map between objects in two different models of a theory.