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It is enough to show that $z$ commutes to all unitaries. But if $p$ is a minimal projection then $upu^{-1}$ is again a minimal projection.

Every monotone closed $C^*$-algebras has this property. And they are not all von-Neumann algebras. Although the counter-exemple are rather enormous objects. In fact as commutative sub-algebra of $AW^ …

Yes (but maybe there is a more direct argument ? ): $A$ and $K(A) = A \otimes K(H)$ are Morita equivalent so they have equivalente categories of representations, moreover this equivalence is implemen …

All groups algebra are trivial examples because for them the modular time evolution is (can be chosen) trivial: the modular time evolution attached to a state is trivial if and only if the state is a …

Let $A$ be a von Neumann algebra, let $L^2(A)$ be the underlying Hilbert space of the standard form of $A$, and $P \subset L^2(A)$ the canonical positive cone (see for example this paper by Haagerup). …

(I'm going to be a bit informal to be able to go to the point relatively directly, but if you want more details on some specific aspect. I can try to add them) Toposes are closely related to topologic …

By the "category of commutative von Neumann algebras" I mean the category of all commutative von Neumann algebras with normal unital $*$-homomorphisms between them (I don't want to restrict to separab …

Here is one half of an answer, but it was too long for a comment. I'll edit if I manage to finish it... When you have a $C^*$ algebra $C$ included in a von Neumann algebra $B$ ( and you might want to …

The following is an argument for showing that "having direct integral" is definitely not a property nor a "property-like-structure" of $W^*$-categegories, but a real, non-trivial additional structure. …

$L(\Gamma,G)$ is the algebra of endomorphisms of the representation $l^2( \Gamma/G)$ (with $\Gamma$ acting by left multiplication). This answer your second optional question. Also, by classical result …

I think I have an example: Precisely, I will construct an integer function $m$ such that $M$ is bounded and the algebra $\mathcal{M}$ contains a corner which is the von Neuman algebra completion of a …

As the title said, I would like to know if constructive measure theory has been developed somewhere ? I am more precisely interested in the (constructive) theory of completely continuous valuation on …

Assume $A$ and $B$ are two unital $C^*$-algebras which are strongly Morita equivalent. This means that there exists a Hilbert $A$-module $H$ such that $B$ is the algebra of compact operators on $H$. …