Derivations in algebras of operator-valued functions

TL;DR

This paper investigates derivations in algebras of operator-valued functions, showing all derivations are inner for separable infinite-dimensional Banach spaces, contrasting with the finite-dimensional case where non-inner derivations exist.

Contribution

It establishes conditions under which derivations in algebras of operator-valued functions are inner, particularly for separable infinite-dimensional Banach spaces.

Findings

All derivations are inner for separable infinite-dimensional spaces.

Existence of non-inner derivations in finite-dimensional cases.

Application to derivations in algebras affiliated with von Neumann algebras.

Abstract

In this paper we study derivations in subalgebras of $L_{0}^{wo}(\nu ;% \mathcal{L}(X)) $, the algebra of all weak operator measurable funtions $f:S\to \mathcal{L}(X) $, where $% \mathcal{L}(X) $ is the Banach algebra of all bounded linear operators on a Banach space $X$. It is shown, in particular, that all derivations on $L_{0}^{wo}(\nu ;\mathcal{L}(X)) $ are inner whenever $X$ is separable and infinite dimensional. This contrasts strongly with the fact that $L_{0}^{wo}(\nu ;\mathcal{L}(X)) $ admits non-trivial non-inner derivations whenever $X$ is finite dimensional and the measure $\nu $ is non-atomic. As an application of our approach, we study derivations in various algebras of measurable operators affiliated with von Neumann algebras.