On Murray-von Neumann algebras -- I: Topological, Order-Theoretic and Analytical Aspects

Abstract

For a countably decomposable finite von Neumann algebra $\mathcal{R}$, we show that any choice of a faithful normal tracial state on $\mathcal{R}$ engenders the same measure topology on $\mathcal{R}$ in the sense of Nelson (J. Func. Anal., 15 (1974), 103--116). Consequently it is justified to speak of `the' measure topology of $\mathcal{R}$. 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 $\ast$-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.

36 pages. Added Remark 2.7. Corrected version of the older Lemma 2.7 appears as Lemma 2.8 (without affecting any other result in the paper.)