All Questions
16 questions
1
vote
1
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252
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GNS Representation — A theorem from Thirring’s book
After the GNS representation for $C^{*}$-algebras is presented in Thirring's book Quantum mathematical physics, the author states the following theorem.
The Spectral Theorem: For any given Hermitian (...
- 1,305
0
votes
1
answer
163
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Regarding socle of a C* algebra
I wanted to know if the socle of a complex C*-algebra is essential?
Can anyone suggest a text where the socle is studied in detail. I tried reading it from the book by Bernard Aupetit, A Primer in ...
- 149
4
votes
0
answers
123
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Restricting a function defined on an étale groupoid to an isotropy group
Let $\mathcal G$ be an étale groupoid, let $x$ be a point in the unit space of $\mathcal G$, and let $\mathcal G(x)$
be the isotropy group of $x$.
If $f$ is a continuous, complex valued, compactly ...
- 2,263
6
votes
2
answers
1k
views
Vector-Valued Stone-Weierstrass Theorem?
The standard statement of the Stone-Weierstrass theorem is:
Let $X$ be compact Hausdorff topological space, and $\mathcal{A}$ a subalgebra of the continuous functions from $X$ to $\mathbb{R}$ which ...
- 263
3
votes
0
answers
83
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Reference request for representation theory of TRO
Let $H$ and $K$ be Hilbert spaces. Recall that a Ternary ring of operator(TRO) $V$ is a closed subspace of $B(H,K)$ such that $xy^{\ast}z \in V$ for all $x,y,z \in V$. I have recently started reading ...
- 1,115
2
votes
1
answer
124
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Twisted canonical commutation relations
I am dealing with universal C*-algebra generated by $x,y$ with the following relations: $xy = qyx$, $x^{*}y = qyx^{*}$, $y^{*}x = qxy^{*}$, $x^{*}x = q^2xx^{*} - (1-q^2)yy^{*}$, $y^{*}y = q^2yy^{*} - (...
- 161
7
votes
1
answer
394
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Inverse limit in the category of $C^{\ast}$-algebras or operator spaces
Does the inverse limits (projective limits) exist in the category of $C^{\ast}$-algebras or operator spaces?
I tried to search but could not find a proper reference. Any reference or comments about ...
- 1,115
2
votes
1
answer
189
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Need a reference of a fact given in B. Blackadar's Operator Algebras
I am reading Blackadar's book on Operator algebras. In $\Pi 9.6.5$ Blackader says that
Maximal Tensor products commute with arbitrary limits.
In the same book the proof of this fact is not given....
- 1,115
7
votes
1
answer
311
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Homomorphism to multiplier algebra of groupoid $C^\ast$-algebra
If I have a functor $X\to Y$ between topological groupoids with appropriate Haar measures, such that $X_0 \to Y_0$ is injective and a homeomorphism onto its image, then I should have (or rather, I ...
- 35.5k
6
votes
4
answers
1k
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Resource recommendation: Spectral theory and $C^*$ algebras
I have formally studied functional analysis, both as university courses, and by myself, but this is one area of mathematics I find so huge and complicated, I have a hard time properly getting into it.
...
- 2,792
4
votes
1
answer
201
views
closure of a separating set of pure states
Let $A$ be a unital C*-algebra, and let $\mathcal R$ be a separating family of irreducible representations of $A$. Each vector state of a representation in $\mathcal R$ is a pure state, and the span ...
- 974
3
votes
1
answer
261
views
CBAP for the full group $C^*$-algebra
Let $G$ be a weakly amenable group, in the sense that it has a net of finitely supported functions $\varphi:G\to \mathbb{C}$ which converge point wise to 1 and their cb norm is bounded uniformly by ...
- 99
5
votes
1
answer
242
views
Spectral decomposition of a C$^*$algebra with respect to an action of a compact abelian group
Let $G$ be a compact abelian group (finite dimensional, but not finite) and $A$ be a $C^*$-algebra. Consider an action $\alpha: G\to Aut(A)$. In analogy with the case of finite abelian group, I ...
- 743
18
votes
3
answers
1k
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In which sense the GNS-construction is a functor?
I asked this at mathstackexchange a week ago, without success.
I think the Gelfand–Naimark–Segal construction must be a functor in some sense, but I can't find an explicit statement anywhere. Can ...
- 7,426
1
vote
0
answers
258
views
Is an exact operator, unitary equivalent to a banded operator?
Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Definition : Let $(e_{n})_{n \in \mathbb{N}}$ be an orthonormal basis.
$T \in B(H)$ is ...
6
votes
1
answer
1k
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Reference needed for: every idempotent in a C*-algebra is similar to a hermitian one
The result stated in the title is thoroughly standard - or that's the impression I got.
I seem to remember seeing it stated somewhere in a book I was reading in the library, and then reverse-...
- 25.8k