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Results tagged with c-star-algebras
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user 25499
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].
7
votes
1
answer
264
views
Can the intersection of a unitary and an irreducibly represented injective $C^*$-algebra be ...
Let $\mathcal{A}$ be an injective $C^*$-algebra irreducibly acting on a Hilbert space $\mathcal{H}$, and let $\phi$ be a completely positive idempotent from $\mathbb{B}(\mathcal{H})$ onto $\mathcal{A} …
- 619
8
votes
Accepted
Projections in the tensor product of von Neumann algebras
The answer is no.
Proof. Let $\mathcal{H}$ and $\mathcal{K}$ be any infinite-dimensional Hilbert spaces, and let $\{\xi_n\}_{n=1}^\infty$ and $\{\eta_n\}_{n=1}^\infty$ be sequences of any orthogo …
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7
votes
1
answer
474
views
Projections in the tensor product of von Neumann algebras
This question seems elementary, but I have already asked an expert who does not know the answer, so I would like to post here.
Let $M$ and $N$ be von Neumann algebras, and let $M\bar{\otimes}N$ be th …
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2
votes
Accepted
Open projections and Murray-von Neumann equivalence
The answer is no.
Proof (Thomas Schick). The idea of the proof is due to Thomas Schick. I thank him for allowing me to reproduce it here. Let $\mathcal{A}:=C([0,1])\otimes\mathbb{M}_2$, where $\m …
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0
votes
Accepted
Is the ideal property of $X^{**}$ inheritable to $X$?
The answer is no.
Proof. Theorem 9 in the reference [3] below implies that there exist a $C^{\star}$-algebra $\mathcal{A}$ and open projections $p,q\in\mathcal{A}^{**}$ such that $p$ and $q$ are …
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3
votes
2
answers
397
views
Is the ideal property of $X^{**}$ inheritable to $X$?
Let $X$ be an operator space such that there is a weak$^*$-continuous complete isometry $\phi$ from its second dual $X^{**}$ into a $W^*$-algebra $M$ in which $\phi(X^{**})$ is a (necessarily weak$^*$ …
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7
votes
1
answer
427
views
Open projections and Murray-von Neumann equivalence
Let $\mathcal{A}$ be a $C^*$-algebra and $p\in\mathcal{A}^{**}$ be an open projection, that is, $p=p^*=p^2$ and $p\in\overline{(p\mathcal{A}^{**}p\cap\hat{\mathcal{A}})}^{\operatorname{w}^*}$, where $ …
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3
votes
1
answer
194
views
Linear independency and compactness of the set of pure states of a $C^*$-algebra
Let $\mathcal{A}$ be a noncommutative $C^*$-algebra and $PS(\mathcal{A})$ be the set of its pure states.
Question 1. Is $PS(\mathcal{A})$ linearly independent (as vectors over $\mathbb{R}$)? (If $\ma …
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4
votes
2
answers
955
views
Do semi-continuous functions generate bounded Borel measurable functions as a $C^*$-algebra?
This question is related to Question 2 of my previous posting.
Question. Let $\mu$ be a Radon measure on a compact Hausdorff space $\Omega$ and $L^{\infty}(\Omega,\mu)$ the set of essentially bounded …
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4
votes
0
answers
282
views
Extensions of completely positive mappings
I would like to ask the following two questions.
Let $1_{\mathcal{H}}\in \mathcal{A}\subset\mathcal{B}\subset\overline{\mathcal{A}}^{SOT}\subset\mathbb{B}(\mathcal{H})$ be a sequence of $C^{\ast}$-sub …
- 619
8
votes
2
answers
946
views
The monotone closure of a $C^*$-algebra
Related to Jon's question, I have two questions. Let $\mathcal{A}$ be a concrete $C^*$-algebra on a Hilbert space $\mathcal{H}$. For any selfadjoint subset $S$ of $\mathbb{B}(\mathcal{H})$, let $S^m$ …
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