On a certain class of operator algebras and their derivations

Given a von Neumann algebra $M$ with a faithful normal finite trace, we introduce the so-called finite tracial algebra $M_f$ as the intersection of $L_p$-spaces $L_p(M,\mu)$ over all $p\geqslant1$ 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.