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This notion of convergence is not often refered to; I think mostly because it does not come from a topology. But this an excellent notion of convergence, probably the best we can put on the space of ( …

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 …

I like to call this result the localic Baire category theorem, and it plays essentially the same role as Baire category theorem: it lets you "construct" object by showing that some spaces are non-empt …

I always had the impression that there was a duality (i.e. a contravariant equivalence of categories) between Banach spaces and certain notion of pointed compact convex set (something like algebras fo …

The image of $f$ is an interval $[a,b] \subset [0,1]$. $T_f$ induce an isometric inclusion of $C([a,b])$ in $C([0,1])$. If $T_f$ was compact then weak convergence of a bounded sequence in $C([a,b])$ …

The answer is No. Rougly, because it is not a good idea to look at continuous $\mathbb{C}$ valued function on a space which is not completely Haussdorff as completely haussdorf is exactly the hypothes …

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) …

Take $A= \mathcal{C}([0,1])$ and $H$ the ideal of $A$ of functions that vanish at $0$. $H$ is a Hilbert $A$ module (as any ideal, with the natural multiplication of $A$ and the scalar product $(x,y)= …

The norm of the self adjoint part is not sufficient: Even if $b$ and $c$ commutes (which should be the easiest case) then it corresponds (by looking at the commutative algebra generated by $b$ and $c$ …

Nam-Kiu Tsing argument indeed become way much simpler for Hilbert spaces. In fact I wouldn't be surprise if his argument was inspired from the (very easy) case of Hilbert spaces: If you have a counta …

I assume that $A^* A$ denotes the algebra generated by product $a^* b$ with $a, b \in A$, but the following argument also applies to some others interpretation of $A^* A$. If $A^* A =A$ then for any …

AS I said, it depends way to much on your framework to give a definitive answer ! here are some exemples that works in some cases: Take $E$ to be the free $\mathbb{Q}$-vector space on a set $S$, and …

Hello, I have recently read about the construction of the descent map in Kasparov KK theory, which, for a group $G$ and two $G$-equivariant $C^*$ algebra $A$ and $B$ send $KK_i^G(A,B)$ to $KK_i(A \rt …

Let assume that you consider unital algebra only (one can still study non unital algebra by unitarizing them, but notion of spectrum is always a little annoying when one want to consider non unital al …

The limit always exists whether or not $H$ is closed: The inductive limit in the category of hausdorff lc vector space will be the quotient of $F$ by the closure of of $H$: a cone for the inductive li …