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Making no claims of originality, one possible proof can be obtained by combining Example 4.60, Lemma 5.11, and Lemma 3.14 in arXiv:2005.05284. This shows that for any Radon measure its algebra of equi …

Is it true that π maps clopen sets into clopen sets? This is true if and only if the inclusion $V_1→V_2$ is a morphism of von Neumann algebras, i.e., its image is closed in the ultraweak topology. S …

I am surprised that nobody mentioned the book by Sinclair and Smith “Hochschild Cohomology of von Neumann algebras” so far. If I remember it correctly, they claim that nobody knows whether there is a …

Given any von Neumann algebra $M$, we can define its noncommutative $\def\L{{\cal L}} \L^p$-spaces $\L^p(M)$ for any $\def\C{{\bf C}} p∈\C$ such that $\Re p≥0$. Here I use the notation $\L^p:={\rm L}^ …

I am not sure how far you want to go, but some basics are explained in this answer: Is there an introduction to probability theory from a structuralist/categorical perspective? In particular, you have …

The opposite category of commutative von Neumann algebras is not a topos because categorical products with a fixed object do not always preserve small colimits. See Theorem 6.4 in Andre Kornell's Quan …

One possible interpretation of Connes's statement is that up to an isomorphism, there is a unique faithful indecomposable representation of any commutative von Neumann algebra on a Hilbert space. Inde …

The Gelfand duality theorem for commutative von Neumann algebras states that the following three categories are equivalent: (1) The opposite category of the category of commutative von Neumann algebra …

As shown in the paper Gelfand-type duality for commutative von Neumann algebras, the following categories are equivalent. The category CSLEMS of compact strictly localizable enhanced measurable spac …

Suppose M is a von Neumann algebra. Consider a monoidal category of bimodules over M. Here a bimodule is a Hilbert space with two normal representations of M. The monoidal structure is given by Connes …

This is false. Consider, for example, the case of M being the von Neumann algebra of bounded complex-valued functions on an infinite countable set I. It acts on the Hilbert space of square-summable fu …

How the norm on L^{P} space related to weight φ? The L^p-spaces and their norms are independent of the choice of the weight φ. See, for instance, the exposition by Yamagami in “Algebraic Aspects …

A von Neumann algebra is a $C^*$-algebra $A$ that admits a predual, i.e., a Banach space $A_*$ such that there is an isomorphism $A\to(A_*)^*$. A morphism of von Neumann algebras is a morphism of $C^ …

Such t_0 is unique if its support is at most p, where p is the support of ϕ. Note that we can replace t_0 by pt_0p and the support of pt_0p is at most p. Without this additional condition t_0 is high …

No. Take e=0 and 0 < f < 1 such that both f and 1−f are infinite, with (1−f)~1. Then e≾f because 0≾f for any projection f. Also 1−e≾1−f because 1≾1−f, which holds by definition of f.