The Relative Commutant of Separable C*-algebras of Real Rank Zero

We answer a question of E. Kirchberg (personal communication): does the relative commutant of a separable C*-algebra in its ultrapower depend on the choice of the ultrafilter? All algebras and all subalgebras in this note are C*-algebras and C*subalgebras, respectively, and all ultrafilters are nonprincipal ultrafilters on N. Our C*-terminology is standard (see e.g., [2]). In the following U ranges over nonprincipal ultrafilters on N. With A denoting the (norm, also called C*-) ultrapower of a C*-algebra A associated with U we have FU (A) = A ′ ∩A , the relative commutant of A in its ultrapower. This invariant plays an important role in [8] and [7]. Theorem 1. For every separable infinite-dimensional C*-algebra A of real rank zero the following are equivalent. (1) FU (A) ∼= FV(A) for any two nonprincipal ultrafilters U and V on N. (2) A ∼= A for any two nonprincipal ultrafilters U and V on N. (3) The Continuum Hypothesis. The equivalence of (3) and (2) in Theorem 1 for every infinite-dimensional C*-algebra A of cardinality 20 that has arbitrarily long finite chains in the Murray-von Neumann ordering of projections was proved in [6, Corollary 3.8], using the same Dow’s result from [4] used here. We shall prove (1) implies (3) and (2) implies (3) in Corollary 10 below. The reverse implications are well-known consequences of countable saturatedness of ultrapowers associated with nonprincipal ultrafilters on N (see [1, Proposition 7.6]). The implication from (3) to (1) holds for every separable C*-algebra A and the implication from (3) to (2) holds for every C*-algebra A of size 20 . The point is that if A is separable then the isomorphism between diagonal copies of A extends to an isomorphism between Date: October 8, 2008. Partially supported by NSERC. I would like to thank N. Christopher Phillips for many useful comments on the first draft of this paper. In this version Theorem 1 was proved only for UHF algebras, and Chris’s suggestion to use of ≤ instead of helped me extend the result to its present form. Filename: 2008i15-non-unique-commutant.tex.