A new bicommutant theorem

TL;DR

This paper proves a bicommutant theorem analogue for separable unital subalgebras within ultraproducts of simple unital C*-algebras, extending classical results in operator algebra theory.

Contribution

It establishes that the relative bicommutant of such subalgebras equals the subalgebra itself in the ultraproduct setting, generalizing Voiculescu's theorem.

Findings

Bicommutant of subalgebra equals the subalgebra in ultraproducts

Extension of Voiculescu's theorem to new algebraic context

Advances understanding of structure of ultraproducts in C*-algebras

Abstract

We prove an analogue of Voiculescu's theorem: Relative bicommutant of a separable unital subalgebra $A$ of an ultraproduct of simple unital C*-algebras is equal to $A$.