The commutant of L(H) in its ultrapower may or may not be trivial

TL;DR

This paper investigates whether the commutant of the algebra of bounded operators on a Hilbert space in its ultrapower is trivial, showing that the answer varies depending on the ultrafilter under the Continuum Hypothesis.

Contribution

It demonstrates that the triviality of the commutant in ultrapowers of L(H) depends on the choice of ultrafilter under the Continuum Hypothesis.

Findings

The commutant may be non-trivial for some ultrafilters.

The triviality of the commutant depends on the ultrafilter choice.

Results are established assuming the Continuum Hypothesis.

Abstract

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.