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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 …

Here is my precise question: Let $A$ and $B$ be two $C^*$ algebras. Let $f: A \rightarrow B$ a homomorphism of $C^*$ algebras which is isometric (on its image). Is $f$ necessary a $*$-homomorphism ? …

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 …

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) …

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 $C^*$ …

You already mentioned one possible extension, but there is one even simpler: Just forget both the topology and the $*$ and take the algebraic K-theory. Indeed for a $C^*$-algebra the algebraic $K_0$- …

This paper proves that there is no functor from the category of C* algebra (and morphisms) to the category of locales/topological spaces/a lot of other things that extend the gelfand duality and send …

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 …

If $G$ is normal (and you don't care it has compact resolvent or not), then $G_0 = G +P$ is $f(G)$ where $f$ is the measurable function that send $0$ to $1$ and is the identity on other value. If I r …

I am looking for a reference for the following result: If $A$ and $B$ are C* algebras, $H$ is a right Hilbert $A$-modules, $\phi :A \rightarrow B$ is a morphism, and assume that there is a map $\eta …

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 …

I don't know what are the motivation for the formulaton in terms of $C^*$ algebras, but you are essentially asking for hausdorff compact/locally compact spaces such all their compact/locally compact q …

Let $A$ and $B$ be two $C^{*}$ algebras. Then the category of $*$-representation of $A$ on Hilbert space is equivalent to that of $B$ if and only if their enveloping von Neumann algebra are morita equ …

It depends if you use the monodromy groupoid or the holonomy groupoid (Is there a more canonical choice for this construction ? ). Basically, the monodromy groupoid is exactly the same as the action …

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 …