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Results tagged with c-star-algebras
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user 22131
A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying (a b)* = b* a* and the C*-identity ‖a* a‖ = ‖a‖². Related tags: [banach-algebras], [von-neumann-algebras], [operator-algebras], [spectral-theory].
5
votes
Accepted
A coproduct of $C^\ast$-algebras
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 …
- 42.4k
2
votes
1
answer
220
views
Non-perfect type one C^*-algebra, and a lemma in Fourier analysis
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{ …
- 63.9k
10
votes
Accepted
A C*-algebra enjoying some different C*-norms
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 …
- 42.4k
18
votes
Accepted
Non-commutative duality I: Which C*-algebras are (isomorphic to a) convolution algebra?
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 …
- 42.4k
6
votes
What does it mean for a category to admit direct integrals?
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. …
- 42.4k
2
votes
Homomorphism to multiplier algebra of groupoid $C^\ast$-algebra
This is not a complete answer, but that was way too long for a comment:
First I started almost sure that this sort of things would be in the literature, but it seems harder than I thought to find som …
- 42.4k
3
votes
$C^*$-algebra which is not von Neumann, but satisfies the property that ever self-adjoint el...
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^ …
- 42.4k
5
votes
Accepted
Counterexample to Riesz representation for Hilbert modules
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)= …
- 42.4k
5
votes
Relation between norm of any element of $C^*$-algebra in terms of self adjoint elements
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 $c$ …
- 42.4k
8
votes
Accepted
How the modular theory of von Neumann algebras, deal with generating C*-algebras?
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 …
- 42.4k
1
vote
Accepted
Certain groupoid and its $C^{*}$ algebra
I will explain how I ended up with the description given in my comment above.
Let $\{1, \dots ,n \}$ be your finite subset of $\mathbb{R}$ (only their order is important, we do not really care about …
- 42.4k
5
votes
unitization-process of unital- and non-unital $C^*$-algebras
The question is already answer but there is a point I want to add:
Some time ago I wrote a paper about the Gelfand duality for non-unital algebra within constructive mathematics, my proof goes throug …
- 42.4k
1
vote
Accepted
On the second dual of $C[0,1]$
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 …
- 42.4k
4
votes
0
answers
134
views
References for a lemma about compact operators on a Hilbert module
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 …
- 42.4k
4
votes
Accepted
$H^{*}$ algebras as a generalization of $C^{*}$ algebras
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 …
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