A New Bicommutant Theorem

We prove an analogue of Voiculescu’s theorem: Relative bicommutant of a separable unital subalgebra A of an ultraproduct of simple unital C*-algebras is equal to A. Ultrapowers1 AU of separable algebras are, being subject to well-developed model-theoretic methods, reasonably well-understood (see e.g. [9, Theorem 1.2] and §2). Since the early 1970s and the influential work of McDuff and Connes central sequence algebras A′ ∩ AU play an even more important role than ultrapowers in classification of II1 factors and (more recently) C*-algebras. While they do not have a well-studied abstract analogue, in [9, Theorem 1] it was shown that the central sequence algebra of a strongly self-absorbing algebra ([17]) is isomorphic to its ultrapower. Relative commutants B′ ∩ DU of separable subalgebras of ultrapowers of strongly selfabsorbing C*-algebras play an increasingly important role in classification program for separable C*-algebras ([12, §3], [4]; see also [16], [19]). We make a step towards better understanding of these algebras. Our setting is more general. C*-algebra is primitive if it has representation that is both faithful and irreducible. We prove an analogue of the well-known consequence of Voiculescu’s theorem ([18, Corollary 1.9]) and von Neumann’s bicommutant theorem (A WOT denotes the closure of A in the weak operator topology). Theorem 1. Assume ∏ U Bj is an ultraproduct of unital, primitive C*algebras and A is a separable unital C*-subalgebra. Then (with A WOT computed in the ultraproduct of faithful irreducible representations of Bjs)