A Characterization of $(\sigma,\tau)-$ derivations on von Neumann algebras

TL;DR

This paper characterizes $(\sigma, au)$-derivations on von Neumann algebras, showing that certain bounded linear maps satisfying specific algebraic properties are indeed $(\sigma, au)$-derivations, expanding understanding of their structure.

Contribution

It provides new conditions under which bounded linear maps on von Neumann algebras are identified as $(\sigma, au)$-derivations, including properties related to projections and invertibility.

Findings

Maps satisfying projection property are $(\sigma, au)$-derivations.

Maps satisfying product property with invertibility condition are $(\sigma, au)$-derivations.

Extends the theory of derivations in von Neumann algebras.

Abstract

Let $\mathcal A$ be a von Neumann algebra and $\mathcal M$ be a Banach $\mathcal A-$module. It is shown that for every homomorphisms $\sigma, \tau$ on $\mathcal A$, every bounded linear map $f:\mathcal A\to \mathcal M$ with property that $f(p^2)=\sigma(p)f(p)+f(p)\tau(p)$ for every projection $p$ in $\mathcal A$ is a $(\sigma,\tau)-$derivation. Also, it is shown that a bounded linear map $f:\mathcal A \to \mathcal M $ which satisfies $f(ab)= \sigma(a)f(b)+f(a)\tau(b)$ for all $a,b\in \mathcal A$ with $ab=S$, is a $(\sigma,\tau)-$ derivation if $\tau(S)$ is left invertible for fixed $S$.