On the commutant of B(H) in its ultrapower

Abstract

Let $${\cal B}\left( H \right)$$ be the algebra of bounded linear operators on a separable infinite-dimensional Hilbert space H. In 2004 Kirchberg asked whether the relative commutant of $${\cal B}\left( H \right)$$ in its ultrapower is trivial. In [13] the authors have shown that under the Continuum Hypothesis the commutant of $${\cal B}\left( H \right)$$ in its ultrapower depends on the choice of the ultrafilter. We here give a combinatorial characterization of the class of non-principal ultrafilters for which this commutant is non-trivial, answering Question 5.2 of [13]. This reduces Kirchberg’s question to a purely set-theoretic question: Can the existence of non-flat ultrafilters be proven in ZFC? In addition, we introduce the notion of quasi P-points and show that for such ultrafilters, and for ultrafilters satisfying the three functions property, the relative commutant of $${\cal B}\left( H \right)$$ in its ultrapower is trivial.