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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 …

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 …

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 …

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^ …

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.

This follows from the reduction theory for von Neumann algebras (alias direct integral decomposition). Any von Neumann algebra is a direct integral of factors (i.e., von Neumann algebras with a trivia …

First of all, one should mention that not every triple (X,B,μ) (i.e., what is often called a measure space) satisfies the property that its C*-algebra of bounded functions is a von Neumann algebra (= …

The category of commutative von Neumann algebras is contravariantly equivalent to the category of measurable spaces. Assuming the axiom of choice, isomorphism classes of objects in the above two cate …

Yes. The kernel of F is an ultraweakly closed *-ideal of M generated by some central projection z. M splits as a direct sum of zM and (1-z)M. As a 2x2 matrix F has only two nonzero entries, one that …

The unitary group of any purely infinite von Neumann algebra is contractible (this is a generalization of Kuiper's theorem due to Brüning and Willgerodt, “Eine Verallgemeinerung eines Satzes von N. Ku …