The Pommiez operator (∆ f )(z) = ( f (z)− f (0))/z is considered in the space (G) of the holomorphic functions in an arbitrary finite Runge domain G. A new proof of a representation formula of Linchuk of the commutant of ∆ in (G) is given. The main result is a representation formula of the commutant of the Pommiez operator in an arbitrary invariant hyperplane of (G). It uses an explicit convolu...

The category of von Neumann correspondences from B to C (or von Neumann B–C–modules) is dual to the category of von Neumann correspondences from C to B via a functor that generalizes naturally the functor that sends a von Neumann algebra to its commutant and back. We show that under this duality, called commutant, Rieffel’s Eilenberg-Watts theorem (on functors between the categories of represen...

Simple associative superalgebra with 2 independent supertraces is presented. Its commutant is a simple Lie superalgebra which has at least 2 independent invariant bilinear forms.

Murray-von Neumann algebras are algebras of operators affiliated with finite von Neumann algebras. In this article, we study commutativity and affiliation of self-adjoint operators (possibly unbounded). We show that a maximal abelian self-adjoint subalgebra A of the Murray-von Neumann algebra A(f)(R) associated with a finite von Neumann algebra R is the Murray-von Neumann algebra A(f)(A(0)), wh...