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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 …

Grothendieck's approach to algebraic geometry in particular tells us to treat all rings as rings of functions on some sort of space. This can also be applied outside of scheme theory (e.g., Gelfand-N …

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 …

Connes defined a noncommutative analog of a closed oriented Riemannian spin^c manifold using spectral triples. Using his definition it is unclear how to separate the smooth structure from the metric. …

The spaces $[0,1]$, $[0,1]^2$, and $S^1$ are all isomorphic as measurable spaces, including their sets of measure 0, as required by the Gelfand-type duality for measurable spaces. For instance, the is …

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}^ …

This result is true and is due to Spitzweck, Berger–Moerdijk, Fresse, and Elmendorf–Mandell. A complete set of references can be found around Proposition 5.7 in the paper https://arxiv.org/abs/1410.56 …

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 …

The traditional mathematical approach to quantum mechanics, as developed by von Neumann, is based on Hilbert spaces and unbounded self-adjoint operators. Another approach, which more closely resembles …

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^ …