A double commutant theorem for operator algebras

TL;DR

This paper extends the classical double commutant theorem to unital nonselfadjoint operator algebras, establishing canonical faithful representations that satisfy a generalized double commutant property.

Contribution

It introduces a generalized double commutant theorem for unital nonselfadjoint operator algebras with canonical faithful representations.

Findings

Existence of canonical faithful Hilbert space representations

Generalization of von Neumann's double commutant theorem

Examples and complementary results provided

Abstract

Every unital nonselfadjoint operator algebra possesses canonical and functorial classes of faithful (even completely isometric) Hilbert space representations satisfying a double commutant theorem generalizing von Neumann's classical result. Examples and complementary results are given.