Commutativity Conditions in Algebras with $C^{\ast }$-equalities

Abstract

Departing from Fuglede-Putnam-Rosenblum's theorem, we examine several commutativity conditions in involutive algebras with $C^{\ast }$-equalities. Among questions considered are Ogasawara's theorem on operator algebras and Radjavi-Rosenthal's result on an algebra of normal operators. In the frame of $C^{\ast }$-algebras, conditions of apparently different natures turn out to be equivalent. Also, remarks are made about Hirshfeld-Zelazko's problem.