The Commutant of L(h) in Its Ultrapower May or May Not Be Trivial

Kirchberg asked in 2004 whether the commutant of L(H) in its (norm) ultrapower is trivial. Assuming the Continnuum Hypothesis, we prove that the answer depends on the choice of the ultrafilter. Let H be a separable infinite dimensional complex Hilbert space, fixed throughout. The purpose of this paper is to prove that, assuming the Continnuum Hypothesis, the commutant of L(H) (the algebra of bounded linear operators on H) in its ultrapower depends on the choice of the ultrafilter. This provides a somewhat surprising—and somewhat incomplete—answer to Question 2.22 of [11]. We follow the convention that 0 ∈ N. Let A be a C*-algebra, and let V be a nonprincipal ultrafilter on N (equivalently, a point in βN \ N). (In the operator algebra literature, nonprincipal ultrafilters are usually denoted ω; since in the set theory literature ω is reserved for the least infinite ordinal we suppress using this symbol altogether.) We denote by `∞(A) the C*-algebra of all bounded functions from N to A, and we denote by AV the ultrapower of A associated with V ([7], [2]). More precisely, AV is the quotient of `∞(A) by the (two-sided, norm-closed, selfadjoint) ideal cV(A) = { (an) ∈ `∞(A) : limn→V ‖an‖ = 0 } . (This algebra is denoted AV in [11] and [12].) We identify A with the subalgebra of AV consisting of all (a, a, a, . . .) + cV(A) for a ∈ A, that is, the image in AV of all constant sequences in `∞(A). Following Kirchberg [11], we define the invariant FV(A) of A to be the quotient of A′ ∩AV by the annihilator Ann ( A,AV ) = {b ∈ AV : ba = 0 for all a ∈ A} Date: August 27, 2008. 1991 Mathematics Subject Classification. 46L05, 03E50. Ilijas Farah and Juris Steprāns were partially supported by NSERC. N. Christopher Phillips was partially supported by NSF grant DMS-0701076, by the Fields Institute for Research in Mathematical Sciences, Toronto, Canada, and by an Elliott Distinguished Visitorship at the Fields Institute. Filename: 2008h24-flat.tex.