On a certain class of operator algebras and their derivations

Abstract

Given a von Neumann algebra M with a faithful normal finite trace, we introduce the so-called finite tracial algebra \( \mathcal{M}_f \) as the intersection of \( L_p\text{-spaces} \) \( L_p(M, \mu) \) over all \( p \geq 1 \) and over all faithful normal finite traces \( \mu \) on M. Basic algebraic and topological properties of finite tracial algebras are studied. We prove that all derivations on these algebras are inner.