A Unitary as a Product of Symmetries

Abstract

It was proved by Fillmore that a unitary of a properly infinite von Neumann algebra A can be expressed as a product of at most four symmetries. In this paper we introduce an axiom (ENCP) for Baer $^ \ast$-rings and prove that Fillmore’s result is true if A is a properly infinite Baer $^ \ast$-ring satisfying (ENCP) and $LP \sim RP$. This also affirmatively answers the open problem on $A{W^ \ast }$-algebras posed by Berberian.