On the commutant of $B(H)$ in its ultrapower

TL;DR

This paper investigates the structure of the commutant of the algebra of bounded operators on a Hilbert space within its ultrapower, characterizing ultrafilters that influence its triviality or non-triviality.

Contribution

It provides a characterization of ultrafilters determining when the commutant in the ultrapower is trivial or non-trivial, extending previous understanding.

Findings

Identifies conditions for non-trivial commutant in ultrapower

Extends class of ultrafilters with trivial commutant

Characterizes ultrafilters affecting the algebra's structure

Abstract

Let $B(H)$ be the algebra of bounded linear operators on a separable infinite-dimensional Hilbert space $H$. We study the commutant of $B(H)$ in its ultrapower. We characterize the class of non-principal ultrafilters for which this commutant is non-trivial. Additionally, we extend the class of ultrafilters for which the commutant is trivial.