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Results tagged with c-star-algebras
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user 50614
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].
2
votes
Extending a $ * $-Representation of $ ({C_{c}}(G,\mathscr{A}),\star,^{*}) $ to a $ * $-Repre...
Here is a complete argument proving automatic continuity for any $ C^{\ast} $-dynamical system $ (G,A,\alpha) $ where $ G $ is discrete.
Haar’s Theorem essentially states that every Haar measure on …
- 1,396
5
votes
2
answers
215
views
On the coincidence (or non-coincidence) of two norms defined on the quotient of a given Hilb...
Let $ A $ be a $ C^{\ast} $-algebra, $ I $ a closed two-sided ideal of $ A $, and $ \mathcal{E} $ a Hilbert $ A $-module. Let
$$
\mathcal{E}_{I}
\stackrel{\text{df}}{=}
\{ x \in \mathcal{E} \mid \lang …
- 1,396
1
vote
0
answers
117
views
Finding the infimum of the range of a certain non-negative function associated to a $ C^{*} ...
Let $ A $ be a non-trivial $ C^{*} $-algebra and $ n \in \mathbb{N} $. Setting $ \mathcal{D} \stackrel{\text{df}}{=} A^{n} \setminus \{ (0_{A},\ldots,0_{A}) \} $, we can define a function $ f: \mathca …
- 1,396
9
votes
0
answers
283
views
Verifying Woronowicz’s proof that all $ {\text{SU}_{q}}(2) $’s are isomorphic as $ C^{*} $-a...
This question is related to one that I asked some time ago.
Definition 1. Let $ q \in (-1,1) $. Let $ A $ be a unital $ C^{*} $-algebra and $ (x,y) $ a pair of elements of $ A $. Then define the …
- 1,396
8
votes
1
answer
365
views
Does the following $ C^{*} $-algebraic result have a purely algebraic proof?
While studying the proof of Bott periodicity for operator $ K $-theory in this set of notes, I learned this fact:
Theorem. Let $ A $ and $ B $ be $ C^{*} $-algebras. Let $ f,g: A \to B $ be $ * $- …
- 1,396
4
votes
1
answer
412
views
A question on an argument in Woronowicz’s paper on the compact quantum group $ {\text{SU}_{q...
Let $ q \in [0,1) $. The compact quantum group $ {\text{SU}_{q}}(2) $ is defined to be the universal unital $ C^{*} $-algebra that is generated by two elements $ \alpha $ and $ \beta $ satisfying the …
- 1,396
2
votes
Accepted
A question about correlations between $ C^{*} $-algebras
It turns out that I have figured it out. Here is a full solution.
Suppose that the hypotheses on $ R $ in the second definition hold. Let
$$
I_{A} \stackrel{\text{df}}{=} \{ a \in A \mid (a,0_{B}) …
- 1,396
4
votes
1
answer
182
views
A question about correlations between $ C^{*} $-algebras
I was studying J. M. G. Fell’s paper The Structure of Algebras of Operator Fields when I encountered the concept of a correlation between two $ C^{*} $-algebras.
Definition. Let $ A $ and $ B $ be …
- 1,396
14
votes
Accepted
Realisation of the noncommutative torus as a universal $ C^{*} $-algebra
According to what I have seen in the literature so far, the standard procedure consists of two main steps:
Prove the existence of a universal $ C^{*} $-algebra $ A_{\theta} $ generated by two unitar …
- 1,396
4
votes
Accepted
Proving a certain $ C^{*} $-algebraic inequality
The following argument seems easier, but there might be a still more fundamental one.
Notice that $ \phi: A^{\sim} \to \mathbb{C} $ above is also a $ C^{*} $-algebraic homomorphism. As $ C^{*} $-alge …
- 1,396
8
votes
1
answer
352
views
Proving a certain $ C^{*} $-algebraic inequality
Let $ A $ be a non-unital $ C^{*} $-algebra. Is there an ‘elementary’ way to prove, for all $ (a,\lambda) \in A \times \mathbb{C} $, the inequality
$$
|\lambda| \leq \sup_{b \in A, ~ \| b \| \leq 1} \ …
- 1,396
3
votes
2
answers
457
views
If $ F(x,\bullet) \in {L^{2}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $...
This question is related to something that I asked yesterday: If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?
Pietro Majer provi …
- 1,396
4
votes
1
answer
380
views
A Hilbert-space completion of a Hilbert $ C^{*} $-module over a separable $ C^{*} $-algebra
Let $ B $ be a separable $ C^{*} $-algebra and $ \mathcal{E} $ a Hilbert $ B $-module. We know that $ B $ has a faithful state $ \phi $. Using $ \phi $, we can construct a $ \mathbb{C} $-valued pre-in …
- 1,396