Questions tagged [oa.operator-algebras]

Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry

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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, ...
Super-Measurable Analyst's user avatar
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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" ...
Clark Chong's user avatar
3 votes
1 answer
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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 ...
J. E. Pascoe's user avatar
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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 ...
Carlos De la Mora's user avatar
10 votes
1 answer
784 views

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}_{\...
Tobias Fritz's user avatar
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25 votes
2 answers
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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*-...
Lech Roch's user avatar
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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 ...
Ollie's user avatar
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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 ...
Chao You's user avatar
20 votes
2 answers
927 views

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....
Alain Valette's user avatar
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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-...
LingCheng's user avatar
21 votes
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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 ...
Suvrit's user avatar
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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 ...
Jiang's user avatar
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4 votes
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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${}^\ast$-...
Lech Roch's user avatar
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11 votes
2 answers
618 views

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 (...
Yemon Choi's user avatar
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2 votes
1 answer
219 views

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 ...
RSG's user avatar
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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 ...
user avatar
6 votes
1 answer
240 views

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 ...
Panchugopal 's user avatar
2 votes
2 answers
306 views

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\ \...
Alex Monras's user avatar
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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 ...
al-Hwarizmi's user avatar
1 vote
1 answer
2k views

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 ...
Garry's user avatar
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10 votes
2 answers
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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 ...
Ali's user avatar
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3 votes
1 answer
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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 ...
Lech Roch's user avatar
  • 505
7 votes
1 answer
460 views

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 ...
Hans's user avatar
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4 votes
0 answers
172 views

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(\...
Yemon Choi's user avatar
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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 ...
Nicolas Boerger's user avatar
8 votes
2 answers
5k views

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 ...
AatG's user avatar
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1 vote
1 answer
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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 ...
Cleft's user avatar
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8 votes
1 answer
294 views

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 ...
Cleft's user avatar
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5 votes
1 answer
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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\...
Paul McKenney's user avatar
1 vote
0 answers
456 views

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 ...
Nirakar Neo's user avatar
10 votes
3 answers
1k views

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?
Nicolas Börger's user avatar
3 votes
1 answer
198 views

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 ...
Valerie's user avatar
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4 votes
2 answers
363 views

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 ...
Qingyun's user avatar
  • 411
10 votes
2 answers
762 views

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 ...
SiOn's user avatar
  • 493
20 votes
3 answers
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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(\...
SiOn's user avatar
  • 493
7 votes
2 answers
786 views

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 ...
Asghar Ghorbanpour's user avatar
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?
Nemo's user avatar
  • 51
2 votes
1 answer
177 views

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 ...
Sayan's user avatar
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3 votes
0 answers
183 views

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? ...
Sayan's user avatar
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0 answers
194 views

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 ...
Kevin's user avatar
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3 votes
2 answers
407 views

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 ...
user avatar
4 votes
1 answer
461 views

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 $B(\ell_2)$? ...
Bojan Kwitek's user avatar
13 votes
2 answers
1k views

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 ...
truebaran's user avatar
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2 votes
1 answer
427 views

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 ...
saman's user avatar
  • 23
11 votes
2 answers
564 views

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. (...
Alexander Chervov's user avatar
5 votes
1 answer
2k views

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 ...
Yul Otani's user avatar
  • 342
11 votes
2 answers
1k views

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 ...
Koushik's user avatar
  • 2,076
8 votes
1 answer
476 views

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 ...
Richard's user avatar
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16 votes
1 answer
887 views

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) ...
Alain Valette's user avatar
1 vote
0 answers
265 views

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 ...
Koushik's user avatar
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