Commutation Theorem - Hilbert Algebras

Hilbert Algebras

The theory of Hilbert algebras was introduced by Godement (under the name "unitary algebras"), Segal and Dixmier to formalize the classical method of defining the trace for trace class operators starting from Hilbert-Schmidt operators. Applications in the representation theory of groups naturally lead to examples of Hilbert algebras. Every von Neumann algebra endowed with a semifinite trace has a canonical "completed" or "full" Hilbert algebra associated with it; and conversely a completed Hilbert algebra of exactly this form can be canonically associated with every Hilbert algebra. The theory of Hilbert algebras can be used to deduce the commutation theorems of Murray and von Neumann; equally well the main results on Hilbert algebras can also be deduced directly from the commutation theorems for traces. The theory of Hilbert algebras was generalised by Takesaki as a tool for proving commutation theorems for semifinite weights in Tomita–Takesaki theory; they can be dispensed with when dealing with states.