A relative bicommutant theorem: the stable case of Pedersen's question

TL;DR

This paper extends Voiculescu's bicommutant theorem to the stable case of Pedersen's question, showing that certain subalgebras of a stable corona algebra are equal to their bicommutants.

Contribution

It provides an affirmative answer to Pedersen's question for stable σ-unital C*-algebras, expanding the understanding of bicommutant properties in corona algebras.

Findings

Confirmed Pedersen's question for stable σ-unital C*-algebras

Extended Voiculescu's theorem to a new class of corona algebras

Clarified the structure of subalgebras in stable corona algebras

Abstract

In 1976, D. Voiculescu proved that every separable unital sub-C*-algebra of the Calkin algebra is equal to its (relative) bicommutant. In his minicourse (see reference), G. Pedersen asked in 1988 if Voiculescu's theorem can be extended to a simple corona algebra of a $\sigma$-unital C*-algebra. In this note, we answer Pedersen's question for a stable $\sigma$-unital C*-algebra.