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.

对于一个可数可分解有限冯·诺依曼代数\（{\ mathscr {R}} \） ，我们表明，一个忠实的正常tracial状态对任何选择\（{\ mathscr {R}} \）滋生在同一测量拓扑\ （{\ mathscr {R}} \）在纳尔逊的意义上（J Funct Anal 15：103–116，1974）。因此，有理由说\（{\ mathscr {R}} \）的'the'度量拓扑。观察到这一点之后，我们将测度拓扑的概念扩展到一般的有限冯·诺伊曼代数，并命名为\（{\ mathfrak {m}} \）-拓扑。我们注意到\（{\ mathfrak {m}} \）的过程-完成以函数的形式产生Murray-von Neumann代数，并为它们提供了一个内在的描述，即单位有序复杂拓扑\（* \）-代数。这样就可以研究抽象的Murray-von Neumann代数，而避免引用希尔伯特空间。此外，它使Murray-von Neumann子代数的适当概念以及元素的光谱和点谱的固有性质变得显而易见，而与元素的周围Murray-von Neumann代数无关。在这种情况下，我们展示了Borel函数演算对于正常元素的良好定义，并在\（{\ mathfrak {m}} \）中将其与近似技术一起使用-拓扑将许多有界自伴算子的标准算子不等式转移到Murray-von Neumann代数中（无界）自伴算子的设置。在代数方面，Murray-von Neumann代数已被描述为关于它们的非零除数的乘积子集的有限von Neumann代数的Ore局部化。我们的讨论表明，此外，它们的描述还具有基本的拓扑，阶理论和分析方面。