The double commutation theorem for selfdual Hilbert right W*-modules

TL;DR

This paper extends the classical double commutation theorem to the setting of selfdual Hilbert right W*-modules over W*-algebras, providing a unified proof for both real and complex cases.

Contribution

It proves that the double commutant of an involutive unital subalgebra of adjointable operators on a selfdual Hilbert W*-module equals the W*-subalgebra generated by it, generalizing the classical theorem.

Findings

Double commutant equals generated W*-subalgebra in this setting

Unified proof for real and complex cases

Extension of classical theorem to W*-modules

Abstract

Let E be a W*-algebra, H a selfdual Hilbert right E-module, L(H) the W*-algebra of adjointable operators on H, and F an involutive unital subalgebra of L(H). We prove that the double commutant of F is the W*-subalgebra of L(H) generated by F. The proofs work simultaneously for the real and for the complex case. If E is the field of real or complex numbers then H is a Hilbert space and this result becomes the well-known classical double commutation theorem.