Commutator estimates in the operator $L^p$-spaces

Abstract

We consider commutator estimates in non-commutative (operator) $L^p$-spaces associated with general semi-finite von Neumann algebra. We discuss the difficulties which appear when one considers commutators with an unbounded operator in non-commutative $L^p$-spaces with $p \neq \infty$. We explain those difficulties using the example of the classical differentiation operator.