Questions tagged [oa.operator-algebras]
Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry
2,097
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When does a $W^*$-algebra have a standard Borel spectrum?
EDIT: André Henriques has commented below that the correct separability condition is not weak-* separability as I have written below, but separability of the predual.
This post came out a bit long, ...
0
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0
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152
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Need help determining whether a certain map is a $C^\ast$ homomorphism
Hello, I need help determining whether the map I defined between two algebras is a well-defined homomorphism of $C^\ast$-algebras. I ran into this problem while trying to define a "rotation map" ...
3
votes
1
answer
298
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Metrics and completions on the direct limit of matrices of all sizes over arbitrary fields
Let $M_n$ denote the $n$ by $n$ matrices.
Consider the homomorphisms
$$\phi_{n,kn}: M_n \rightarrow M_{kn}$$
which takes a matrix $A \in M_n$ to $A \otimes I_k \in M_{kn}.$
This gives a sensible way ...
0
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0
answers
199
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Weights on Von Neuman factors
Let A be a type $I$ factor on a Hilbert space H. Let $\varphi$ be a semi-finite normal weight on $A^{+}$ is it possible to say that then there exist Hilbert spaces $H_2 \subset H_1$ and an isomorphism ...
10
votes
1
answer
784
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Tannaka duality for C*-algebras?
Tannaka-Krein
duality shows
how to recover a group $G$ from its category $\mathbf{Rep}(G)$ of finite-dimensional
complex representations and the forgetful functor $F:\mathbf{Rep}(G)\to
\mathbf{Vect}_{\...
25
votes
2
answers
1k
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Can nuclearity be determined by tensoring with a single C*-algebra?
A C*-algebra is nuclear if the algebraic tensor product $A\odot B$ ($B$ is any other C*-algebra) admits a unique C*-norm. This definition requires testing the condition for nuclearity with `all' C*-...
4
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2
answers
550
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Are the groups $C( \mathbb{R} ; U(n) )$ isomorphic?
Let $C(\mathbb{R};{U}(n))$ denote the topological group of continuous functions $\mathbb{R}\to {U}(n)$ with pointwise multiplication and compact-open topology. My question is:
Are these groups ...
1
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0
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230
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From positive definite function to Følner sequence ----- a question on amenability and nuclearity
We know that amenability of countable discrete group $\Gamma$ has many equivalent characterizations. In particular, there are two: a) there is a sequence of finitely supported positive definite ...
20
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2
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927
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Which p-adic algebraic groups are type I?
It was proved by Jacques Dixmier (Sur les représentations unitaires des groupes de Lie algébriques, Annales de l'institut Fourier, 7 (1957), p. 315-328, doi: 10.5802/aif.73, MR 20 #5820, Zbl 0080....
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144
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isomophism, commutator
Let X be a Banach space. $B(X)$ is the algebra of all bounded linear operators on X.
$\phi: B(X)\rightarrow B(X)$ is a isomoprphisn, $\varphi: B(X)\rightarrow B(X)$ is a isomorphism or negative anti-...
21
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855
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Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials
While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
5
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236
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Find a lower bound for a pre-invariant $Fol(L(F_m), X_m)$
In the paper of Bannon and Ravichandran, A Folner invariant for type $\rm{II}_1$ factors, they defined an invariant $Fol(M)$ for a separable type $\rm{II}_1$ factor $M$, especially for the free group ...
4
votes
0
answers
372
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Hans Saar's thesis
I would love to have a look on some results which are claimed by some people to be in Saar's thesis:
H. Saar, Kompakte, vollständig beschränkte Abbildungen mit Werten in einer nuklearen C${}^\ast$-...
11
votes
2
answers
618
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Quasinilpotent elements of group C-star algebras
If $G$ is a discrete torsion-free group, can its (reduced or full) group C-star algebra contain non-zero quasinilpotent elements? I've seen various examples in the group von Neumann algebra setting (...
2
votes
1
answer
219
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Arveson index and curvature
Can someone explain me what is the intuitive idea behind Arveson Index and curvature of $E_0$ semigroups. I was reading the standard paper of Arveson, but is lost and yet to get intuition about it. An ...
3
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0
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292
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What information about a locally compact group $G$ is encoded in $C_r^\ast(G)$ which is not in $L^1(G)$?
Let $G$ be a locally compact group and let $ C_r^\ast(G) $ denote its reduced group $C^\ast$-algebra. Many features of a $G$ can be realized from $L^1(G)$ or $C_r^\ast(G)$. For example, $G$ is ...
6
votes
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240
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Hyperfiniteness of CCR algebra
Hi, It is known that the double commutant of the CCR algebra in it's GNS space
with respect to some quasi-free states are always type III factors.
My question is; Will some of them be hyperfinite ...
2
votes
2
answers
306
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What is the dual of an semidefinitely representable (SDR) cone?
The Question
Let $V\simeq \mathbb{R}^r$ be an $r$-dimensional vector space with the usual Euclidean inner product.
Let $\mathcal K\subset V$ be a cone defined as
$$
\mathcal K=\Big\{x\in V\ \...
0
votes
0
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340
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abstract algebra for component wise operations on "vectors" or what it might be called
I have a quite tough problem to solve and need an algebra that allows to "vectors" following operations:
- multiplication between two vectors are componentwise that means v=(v1, v2, v3,...) multiplied ...
1
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1
answer
2k
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PhD in operator algebras and non-commutative geometry [closed]
I do not know whether it is a good place to ask this question or not.
I want to PhD in operator algebras and non-commutative geometry. What are the best places in the world for that? I want a good ...
10
votes
2
answers
1k
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B(H) as a direct sum of a closed two sided ideal and a subalgebra
Let $B(H)$ is the C*-algebra of all bounded linear operators on
Hilbert space $H$. Are there a closed two-sided ideal $I$ and a
subalgebra $A$ of $B(H)$ such that $B(H)=I \oplus A$ (direct sum I and ...
3
votes
1
answer
256
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Is the image of a vNA under a SOT-continuous morphism still a vNA?
It looks very easy but I must admit I am struggling with this problem.
Okay, let $M$ be a von Neumann algebra acting on a Hilbert space $H$ and let $K$ be another Hilbert space. Suppose $h\colon M\to ...
7
votes
1
answer
460
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Taking direct sums in $K$-theory in Kirchberg-Phillips classification
A theorem by Kirchberg and Phillips states that two unital separable nuclear simple purely infinite $C^*$-algebras (so called Kirchberg algebras) satisfying the Universal Coefficient Theorem are ...
4
votes
0
answers
172
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reference for direct finiteness of the ring of affiliated operators
Let $\Gamma$ be a group, $N(\Gamma)$ its group von Neumann algebra,
$\newcommand{\cUG}{{\mathcal U}(\Gamma)}$
and $\cUG$ the ring of all densely-defined, closed operators $\ell^2(\Gamma)\to\ell^2(\...
3
votes
1
answer
186
views
Local cross sections for Unitary group in a hilbert space
Let $U(\mathcal{H})$ be the group of unitary operators on a Hilbert space with the norm topology. Let $H\subset U(\mathcal{H})$ be a closed subgroup.
Under which condidions (on the ...
8
votes
2
answers
5k
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When is spectral norm of AB equal to that of BA?
I have $A^{1/2} B A^{1/2} \preceq I$ for two PSD matrices $A$ and $B$, and I'd like to know if that implies $\|AB\|_2 \leq 1.$
The argument I was using to show this is that for any two square ...
1
vote
1
answer
489
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Is this result of Spain correct?
Let us have a look on the proof of Theorem 2 in [P. G. Spain, Boolean algebras of projections, Proceedings of the Edinburgh Mathematical Society (Series 2) 19, 03, March 1975, 287-289]
The author ...
8
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1
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294
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Masas in second duals of Banach algebras
Lel $B$ be a Banach algebra and give $B^{**}$ one of the Arens products in order to make it a Banach algebra. Then the canonical embedding $\kappa\colon B\to B^{\ast\ast}$ is a homomorphic embedding ...
5
votes
1
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482
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When is a $*$-homomorphism between multiplier algebras strictly continuous?
(This question was posted on MSE here but didn't get any answers.)
The strict topology on the multiplier algebra M(A) of a C*-algebra A is that generated by the seminorms
$$ x\mapsto \|ax\|\quad x\...
1
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0
answers
456
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Factorization through Hilbert Space
I am working on my Masters thesis which is on the Grothendieck's inequality. My aim is to study its formulation in different contexts starting from commutative case and then covering non-commutative ...
10
votes
3
answers
1k
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Compact subgroups of the unitary group of operators in a hilbert space
Is there a characterization for the compact subgroups of the unitary operators in a Hilbert space, where the unitaries are furnished with the norm topology? What about other topologies?
3
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1
answer
198
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Dense subspaces in primitive ideals of C-star algebras
Let $G$ be a unimodular locally compact group (my main examples are algebraic groups over local fields. Thefore we can assume $G$ is Type I, if necessary). Then there are at least three group algebras ...
4
votes
2
answers
363
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system of homogeneous matrix equations
Edit: Maybe I should make it clear that by a solution I mean a pair of matrices $(A,B)$ such that the identity below holds for any complex numbers $x,y$.
One of my friend asked me the following ...
10
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2
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762
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General recipe for building C*-algebras out of combinatorial object
I want to ask what should be a nice way to build C*-algebras out of objects like groups, inverse-semigroups, semigroups, ringgs or graphs. I know there are well known construction of C*-algebras out ...
20
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3
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3k
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Realizing universal $C^*$-algebras as concrete $C^*$-algebras
How do I in general realize a universal C*-algebra generated by some generators and relation as concrete C*-algebras? For example, I know that universal C*-algebra generated by a single unitary is $C(\...
7
votes
2
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786
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When is a Pseudo-differential operator trace class or in Dixmier ideal?
Let's denote the set of all Pseudo-differential operators with symbol of “order” $d$ by $\Psi_d(M)$ and Sobolev space on $M$ by $H_s(M)$. It is known that
If $P\in\Psi_d(M)$ Then $P$ extends to a ...
2
votes
1
answer
330
views
Cyclic vectors for C* algebras
Let A be a C* algebra of operators on a Hilbert space H. Can it happen that for some x in H the set Ax is dense in H but it is not the whole H?
2
votes
1
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177
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Second quantization of partial isometry
If we have a unitary map from Hilbert space $H$ to $H$, we get a unitary map from $e^{H}$ to
$e^{H}$, where $e^{H}$ is the symmetric Fock space of $H$. But if we replace the unitary with partial ...
3
votes
0
answers
183
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Is the construction of ring C*-algebra functorial?
Cunz and Li defined defined C*-algebras for arbitrary rings with of course some condition. One can look at their article (http://arxiv.org/abs/0905.4861). My question: is the construction functorial? ...
0
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194
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What methods have been used to study AW*-algebras up to now?
I am interested mainly in ring theory and homological algebra. Now I want to know about the research methods of AW*-algebras. So I want to know the answer to the question:"what methods have been used ...
3
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2
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407
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Stabilization in Banach algebras
In $C^\ast$-algebras we use $K(H)$, the algebra of compact operators on a separable Hilbert space, for stabilization of a $C^\ast$-algebra, i.e. $S(A):=A\otimes K(H)$. Is there any similar ...
4
votes
1
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461
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Kadison-Singer problem in exotic Hilbert spaces
The Kadison-Singer problem is considered in relation to the separable Hilbert space:
KS: Does every pure state on the diagonal (atomic) masa of $B(\ell_2)$ has a unique extension to $B(\ell_2)$?
...
13
votes
2
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1k
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Calkin Algebra and the embedding
Let $H$ be a separable, infinite dimensional Hilbert Space and $Calk(H):=B(H)/K(H)$ denotes
the Calkin algebra. There is obvious surjection $\pi: B(H) \to Calk(H)$ but I'm interested
in somehow ...
2
votes
1
answer
427
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crossed product
on Williams Crossed product book,on page 198, it is mentioned that there is only one regular representation for C_c(G), and that is the left regular representation.
I know that this representation is ...
11
votes
2
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564
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Seeing topological (geom.) properties of the space via corresponding C^*-algebra
Compact Hausdorff spaces bijectively correspond to C^*-algebras with identity. One needs to consider the algebra of continuous functions C(X) to go in one direction and spectrum to go in the other. (...
5
votes
1
answer
2k
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definition of operator valued integral with spectral measure
I am trying to make sense of some operators that come up on Buchholz and Summers' work on warped convolutions (two works on arxiv: 2008 and 2011).
There, they work on a Hilbert space $H$ and on the ...
11
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2
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1k
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Intuitive meaning of Double Commutant Theorem
Is there any intuitive explanation of the Double Commutant Theorem for Von Neumann Algebras? By intuitive I mean in terms of Quantum Mechanics. For example, duality of states and observables in the ...
8
votes
1
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476
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Eigenvalues of the free sphere
Consider the usual sphere $S^{n-1}\subset\mathbb R^n$. By Stone-Weierstrass $C(S^{n-1})$ is generated by the standard coordinates $x_1,\ldots,x_n:\mathbb R^n\to\mathbb R$, and in fact we have the ...
16
votes
1
answer
887
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Explicit path in the unitary group of a $C^*$-algebra
For $G$ a discrete group, there is a canonical inclusion $g\mapsto u_g$ of $G$ into the unitary group of the reduced $C^*$-algebra $C^*_r(G)$. Denote by $[u_g]$ the class of $u_g$ in the (topological) ...
1
vote
0
answers
265
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Nuclear Space problem
I need to show that if X is compact,then C(X) is nuclear.Also is the condition X is metrisable
necessary. I am at present attending a conference "Recent Aadvances in Operator Theory". This
problem ...