Narrow Results

The present paper is a survey of recent results concerning derivations on various algebras of measurable operators affiliated with von Neumann algebras. A complete description of derivation is obtained in the case of type I von Neumann algebras. A special section is devoted to the Abelian case, namely to the existence of nontrivial derivations on algebras of measurable function. Local derivations on the above algebras are also considered.

In this paper, we prove that regular paramedial division groupoids and binary regular paramedial division algebras are linear over an abelian group. Using this results we prove that every finitely generated paramedial division groupoid is a quasigroup and that every paramedial division groupoid is a homomorphic image of a paramedial quasigroup.