Relative commutant of an unbounded operator affiliated with a finite von Neumann algebra (Q2835233)

The article deals with some modifications of the classical Fuglede-Putnam theorem for unbounded operators affiliated with finite von Neumann algebras. Let \({\mathfrak A}\) be a finite von Neumann algebra, \({\mathfrak A}'\) its commutant, and assume that the center \({\mathfrak A} \cap {\mathfrak A}'\) of \({\mathfrak A}\) is trivial. The set of densely defined closed operators \(T\) affiliated with \({\mathfrak A}\) (\(TU= UT\) for any unitar operator \(U \in {\mathfrak A}'\)) and with natural algebraic operations form a \(*\)-algebra \({\mathcal A}_{\mathfrak J} ({\mathfrak J})\). The main result is the following: if \(T \in {\mathcal A}_{\mathfrak J}({\mathfrak A})\) and \(N,M\) are normal operators in \({\mathfrak A}\) with \(TN = MT\), then \(TN^* = M^*T\). Further, the authors construct some examples of closed operators affiliated with a finite von Neumann algebra with trivial relative commutant.