*-algebras of unbounded operators affiliated with a von Neumann algebra. (Russian, English) Zbl 1098.47034

Muratov, M. A.; Chilin, V. I.

Zap. Nauchn. Semin. POMI 326, 183-197 (2005); translation in J. Math. Sci., New York 140, No. 3, 445-451 (2007).

Let \(S(M)\) denote the *-algebra of measurable operators affiliated to a von Neumann algebra \(M\). Let further \(S(M,\tau )\subset S(M)\) be the *-subalgebra of all \(\tau\)-measurable operators, where \(\tau\) is a faithful normal semi-finite trace on \(M\). \(S(M)\) and \(S(M,\tau )\) are then *-algebras of closed and densely defined operators acting on the same Hilbert space \({\mathcal H}\) as the von Neumann algebra \(M\). Moreover, \(M\) is a *-subalgebra of \(S(M,\tau )\) (and \(S(M)\)) and coincides with the set of all bounded operators of \(S(M,\tau )\) and \(S(M)\). Another interesting unital *-algebra of closed operators acting on \({\mathcal H}\) and affiliated to \(M\) is \(LS(M)\), which is given by the set of all linear operators \(T\) that are locally measurable with respect to \(M\) (i.e., \(T\,\eta\,M\), and there exists a sequence \(\{ Z_n\}_{n=1}^\infty\) of central projections of \(M\) such that \(Z_n\uparrow I\) (unity) and \(TZ_n\in S(M)\), \(n=1,2,\dots)\). Thus, we have the chain \(M\subset S(M,\tau )\subset S(M)\subset LS(M)\) of *-algebras of operators acting on the same Hilbert space \({\mathcal H}\).
The paper under review considers the above *-algebras and proves necessary and sufficient conditions such that some of these algebras coincide. More precisely, the validity of (i) \(S(M)=M,\) (ii) \(LS(M)=S(M),\) and (iii) \(LS(M)=M\) is characterized.

Reviewer: Gerald Hofmann (Leipzig)

MSC:

47C10	Linear operators in \({}^*\)-algebras
47L60	Algebras of unbounded operators; partial algebras of operators
47C15	Linear operators in \(C^*\)- or von Neumann algebras

Keywords:

von Neumann algebras; *-algebras of unbounded operators affiliated with a von Neumann algebra; *-algebras of (locally) measurable and \(\tau\)-measurable operators