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Given two locally compact spaces $X$ and $Y$ then the product $X \times Y$ is an open subset inside $\overline{X} \times \overline{Y}$, where $\overline{X}$ and $\overline{X}$ are the one point compac …

I would like to know if the following is true : Let $\mathcal{H}$ be the complex Hilbert space $L^2([0,1])$ for the Lebesgue measure. Let $q$ be the orthogonal projection on the subspace of $\mathcal{ …

No that's not possible (except the trivial case). Any $*$-homomorphism between $C^*$-algebras is automatically contractive, and if it is injective then it is isometric. You can apply this to the ident …

After more thought, I think the correct statement is the following: Theorem: Let $\pi : X \to Y$ be a continuous map between compact Hausdorff spaces. Then the following condition are equivalents: (a) …

I fix $C$ a symmetric monoidal model category (with a cofibrant unit if it helps). I'm assuming that it is closed, or at least that the tensor product commutes to colimits in each variable. If $X$ is …

The main obstruction to this kind of duality is not so much that not every $C^*$-algebra is a convolution algebra (though, at least if we don't use twisted convolution algebra, there are known obstruc …

(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 …

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. …

$\pi$ is an irreducible representation. So because of Schur lemma and the double commutant theorem the map $\pi:A \rightarrow B(H)$ is surjective. Now the state $\phi$ is defined as a normale state o …

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 …

I would strongly recommend to have a look (and to always keep available) to the books by Takesaki "The theory of Operator algebras" I,II and III You might not need to know everything that is in these …

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 …

One can find in the literature two notions of $C^*$-algebra over a topological space $X$. The first is as the data of an open surjection $ \pi: B \rightarrow X$ together with the structure of a $C^*$ …

It is easy to see that $\int \psi d\delta_t$ where $\delta_t$ is the Dirac mass at $t$ is $\psi(\{t\})$ So you are starting from a $T \in C([0,1])^{**}$ and you are attaching to it the function $t \m …

If $O$ is a (strongly ?) self absorbing $C^*$-algebra, one has an equivalence relation on (separable) $C^*$-algebras: "being $O$-stably isomorphic" i.e. $A$ and $B$ are $O$-stably isomorphic if and on …