Approximate Double Commutants and Distance Formulas

TL;DR

This paper extends the theory of approximate double commutants in unital C*-algebras, providing new metric results and generalizations for various subalgebras, including nonselfadjoint cases.

Contribution

It introduces new metric estimates for approximate double commutants and generalizes classical theorems to nonselfadjoint subalgebras within C*-algebra frameworks.

Findings

Proved metric results for AH subalgebras of von Neumann algebras.

Established distance formulas for AF subalgebras of primitive C*-algebras.

Extended classical approximation theorems to nonselfadjoint subalgebras.

Abstract

We extend work of the first author concering relative double commutants and approximate double commutants of unital subalgebras of unital C*-algebras, including metric versions involving distance estimates. We prove metric results for AH subalgebras of von Neumann algebras or AF subalgebras of primitive C*-algebras. We prove other general results, including some for nonselfadjoint commutative subalgebras, using C*-algebraic versions of the Stone-Weierstrass and Bishop-Stone-Weierstrass theorems.