  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 $p \geq 1$ 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.
