On Murray-von Neumann algebras—I: topological, order-theoretic and analytical aspects

Abstract

For a countably decomposable finite von Neumann algebra \({\mathscr {R}}\), we show that any choice of a faithful normal tracial state on \({\mathscr {R}}\) engenders the same measure topology on \({\mathscr {R}}\) in the sense of Nelson (J Funct Anal 15:103–116, 1974). Consequently it is justified to speak of ‘the’ measure topology of \({\mathscr {R}}\). Having made this observation, we extend the notion of measure topology to general finite von Neumann algebras and denominate it the \({\mathfrak {m}}\)-topology. We note that the procedure of \({\mathfrak {m}}\)-completion yields Murray-von Neumann algebras in a functorial manner and provides them with an intrinsic description as unital ordered complex topological \(*\)-algebras. This enables the study of abstract Murray-von Neumann algebras avoiding reference to a Hilbert space. Furthermore, it makes apparent the appropriate notion of Murray-von Neumann subalgebras, and the intrinsic nature of the spectrum and point spectrum of elements, independent of their ambient Murray-von Neumann algebra. In this context, we show the well-definedness of the Borel function calculus for normal elements and use it along with approximation techniques in the \({\mathfrak {m}}\)-topology to transfer many standard operator inequalities involving bounded self-adjoint operators to the setting of (unbounded) self-adjoint operators in Murray-von Neumann algebras. On the algebraic side, Murray-von Neumann algebras have been described as the Ore localization of finite von Neumann algebras with respect to their corresponding multiplicative subset of non-zero-divisors. Our discussion reveals that, in addition, there are fundamental topological, order-theoretic and analytical facets to their description.