Non-commutative Arens algebras and their derivations

Abstract

Given a von Neumann algebra M with a faithful normal semi-finite trace τ , we consider the non-commutative Arens algebra L ω (M, τ ) = p 1 L p (M, τ ) and the related algebras L ω 2 (M, τ ) = p 2 L p (M, τ ) and M +L ω 2 (M, τ ) which are proved to be complete metrizable locally convex *-algebras. The main purpose of the present paper is to prove that any derivation of the algebra M + L ω 2 (M, τ ) is inner and all derivations of the algebras L ω (M, τ ) and L ω 2 (M, τ ) are spatial and implemented by elements of M + L ω 2 (M, τ ). In particular we obtain that if the trace τ is finite then any derivation on the noncommutative Arens algebra L ω (M, τ ) is inner.

Innerness of derivations on subalgebras of measurable operators

“…It should be noted also that a complete description of derivations on general L ω (M, τ ) was obtain in [2]. Namely.…”

“…Therefore, given by a non commutative Arens algebra L ω (M, τ ) in the case of a finite trace τ (see [2]). …”

Local and 2-local derivations on noncommutative Arens algebras

“…In the papers [2], [5] S. Albeverio and the first two authors have proved the spatiality of derivations on the non commutative Arens algebra L ω (M, τ ) associated with an arbitrary von Neumann algebra M and a faithful normal semi-finite trace τ . Moreover if the trace τ is finite then every derivation on L ω (M, τ ) is inner.…”

Local Derivations on Subalgebras of τ-Measurable Operators with Respect to Semi-finite von Neumann Algebras

“…It is proved in [1] that L ω (M, τ ) is a locally convex complete metrizable * -algebra with respect to the topology t generated by the family of norms [2,Theorem 3.7] that if M is a von Neumann algebra with a faithful normal semi-finite trace τ then any derivation D on L ω (M, τ ) is spatial, moreover it is implemented by an element of…”