All Questions
Tagged with operator-algebras or oa.operator-algebras
2,153 questions
9
votes
1
answer
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Ping Pong and Free Group Factors
This question concerns alternative characterizations of free group factors. The ping pong lemma is a well-known criteria for the freeness of a group. I've often wondered if there is a ping pong like ...
- 7,057
1
vote
1
answer
331
views
An unconventional definition of the $ C^{*} $-algebraic reduced crossed product
Let $ (A,G,\alpha) $ be a $ C^{*} $-dynamical system, i.e., $ A $ is a $ C^{*} $-algebra, $ G $ is a locally compact Hausdorff group and $ \alpha $ is a strongly continuous action of $ G $ on $ A $ by ...
- 307
8
votes
3
answers
1k
views
Gelfand duality in NCG
In non-commutative geometry, Gelfand duality is the construction of multiplicative linear functionals of a commutative C*-algebra, which can be viewed as the space of all its irreducible complex ...
- 83
1
vote
0
answers
81
views
A consecutive resolution of continum algebras to a simple continum algebra
Motivated by classical Gelfand Naimark duality, the correspondence between the category of commutative $C^{*}$ algebras and the category of locally compact Hausdorff spaces, we ...
- 356
7
votes
1
answer
475
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 ...
- 261
5
votes
1
answer
342
views
NonCommutative Baire theorem
The classical Baire theorem says that the intersection of a sequence of open dense subsets of $X$, is dense, if the space is compact Hausdorff. In the language of $C^{*}$ algebras this is ...
- 356
2
votes
1
answer
122
views
Choi type matrix condition for completely positivity on a certain operator system spanned by some unitaries
Let $B_1$ and $B_2$ be $C^*$-algebras. Let $U_1, \ldots, U_n$ be some unitaries in $B_1.$ We consider the operator system $S$ spanned by $U_iU_j^*.$
Let $\phi: S \rightarrow B_2.$
Given that the ...
- 1,429
1
vote
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 ...
- 11
4
votes
2
answers
572
views
Matrices whose entries are essentially traces of products of unitary matrices.
A $n\times n$ matrix $A=[a_{ij}]$ is called "good", if there exists some $k$ and a set of $k\times k$ complex unitaries $U_i$, $1\leq i\leq n$ , such that $tr(U_i^{+}U_j)=ka_{ij}$, where $U^{+}$ ...
- 1,503
7
votes
1
answer
301
views
allowing `discontinuous functions' into a C* algebra
There follows a possible construction, and I would like to know if it or a similar construction has been done before (as I suspect), so that I can reference it, or if it obviously does not work! Any ...
- 1,143
3
votes
1
answer
187
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Conjugacy of circle actions on UHF C*-algebras
Consider pointwise continuous actions of the unit circle on the $2^{\infty}$-UHF C*-algebra A by *-automorphisms. Assume that two such actions have the same fixed point algebra, i.e., elements that ...
- 31
7
votes
2
answers
772
views
Can anyone calculate KK(A,B) when neither A or B are the complex numbers?
Here I am referring to Kasparov's KK-theory, a bivariant functor on the category of separable C* algebras. It is well known that $KK(A, \mathbb{C})$ is K-homology and $KK(\mathbb{C}, B)$ is K-theory, ...
- 29.2k
8
votes
4
answers
454
views
Possible values of the index for subfactor inclusions coming from conformal nets
This question is related to Can the minimal index of a subfactor take all values in {4cos^2(pi/n);n=3,4,5,...} u [4,infinity]?
I was wondering what one knows for the special case of conformal nets ...
- 2,121
8
votes
1
answer
296
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 ...
- 203
5
votes
1
answer
510
views
Reference for embedding an infinite direct product of matrix algebras into the hyperfinite $II_1$ factor
In some calculations I am writing up,
$\newcommand{\cR}{{\mathcal R}}$
I want to add - as a fairly throwaway remark - that any countable product (= $\ell^\infty$-direct sum) of matrix algebras can be ...
- 25.8k
4
votes
0
answers
114
views
Coming up with a represenation for sum of functions in the Fourier algebra
This is my first overflow question, so let me apologize in advance if this belongs on http://math.stackexchange.com.
Let $G$ be a discrete group.
Let $\lambda:G\to B(\ell^{2}(G))$ be the left ...
- 161
5
votes
1
answer
311
views
How well do we know relative commutants in $L(\mathbb{F}_\infty)$?
Let $H=K_1\oplus K_2$ be infinite dimensional Hilbert spaces. Voiculescu's free Gaussian functor gives us free group factors $L(H)$, $L(K_1)$, $L(K_2)$ acting on the full Fock space $\Gamma(H)$ and, ...
- 1,411
4
votes
1
answer
229
views
Does the group of compact perturbations of the identity act transitively on the compact operators?
Let $H$ be an infinite dimensional (separable if necessary) complex Hilbert space, and denote by $K(H)$ the ideal (in $B(H)$) of compact operators on $H$. Let $G_c=\{I+K\in B(H): I+K \text{ is ...
- 3,115
4
votes
1
answer
199
views
Does noncommutative Lp-convergence respect orderings?
Let $M$ be a von Neumann algebra and $\tau$ a faithful (semi-finite?) normal trace on $M$; as is standard, the $L^p$-norm is defined as $||u||_p=\tau(|u|^p)^{1/p}$. Let $\{u_i\}_{i=1}^\infty$ be a ...
- 777
4
votes
1
answer
407
views
Are all the R-R-bimodules completely reducible?
Let $R$ be the hyperfinite $II_1$ factor and let $X$ be any $R$-$R$-bimodule.
Question: Is $X$ completely reducible (i.e. a direct integral of irreducible $R$-$R$-bimodules)?
Example: If $(N \...
1
vote
0
answers
85
views
Characterization of the Subspace of Quasifree States of the CAR Algebra
Consider $\mathfrak U(\mathfrak H)$, the CAR algebra over a separable Hilbert space $\mathfrak H$. The states $E_{\mathfrak U}$ over this algebra are defined to be positive linear functionals of norm ...
- 605
3
votes
1
answer
427
views
Inner automorphisms and $K$-theory
It is known that any inner automorphism of a unital $C^{\ast}$-algebra $A$ induces the identity map on $K_{0}(A)$ because unitary equivalence implies Murray-von Neumann equivalence. What is known ...
- 169
4
votes
0
answers
202
views
Connectivity of the group of invertible elements of $C(S^{2})\otimes A$
For what type of $C^{*}$ algebras $A$, the group of invertible elements of $C(S^{2}) \otimes A$ is a connected group?
All finite dimensional $A$ satisfy this property.
Is it true to say ...
- 356
4
votes
0
answers
207
views
Extending Akemann's Non-Commutative Urysohn Lemma
Assume $A$ is a C*-algebra and $p,q\in A^{**}$ are compact projections.
Can we always find $a,b\in A^1_+$ with $p\leq a$, $q\leq b$ and $||pq||=||ab||$?
Note if $||pq||=1$ this is immediate, while ...
- 1,307
6
votes
1
answer
525
views
Strong convergence of projections in $B(H)$
(I asked this question at math stackexchange 4 months ago, but received no answers)
Let $\{e_{kj}\}$ be the canonical matrix units in $B(H)$, with $H$ separable. Define projections $q_k$ by
$$
q_k=\...
- 2,793
7
votes
2
answers
529
views
Telling group algebras apart
It's a big, famous, hard problem in operator algebras to determine if the von Neumann algebras $L(F_2)$ and $L(F_3)$ are isomorphic, or not. Here $F_n$ is the free group on n generators and $L(F_n)$ ...
- 18.7k
1
vote
0
answers
356
views
Determining the primitive ideal space of C-star algebras
Is there a general way of finding a primitive ideal space of $C^*$-algebra?
For example, if $C^*$-algebra is given by the universal $C^*$-algebra generated by two self-adjoint unitary elements, how ...
- 11
3
votes
1
answer
565
views
When does a $W^*$-algebra have a standard Borel spectrum?
EDIT: André Henriques has commented below that the correct separability condition is not weak-* separability as I have written below, but separability of the predual.
This post came out a bit long, ...
2
votes
0
answers
157
views
Support vectors and relative modular operator
I'm studying the relative modular operator and I'm looking for o good text to do it. Until now I'm using Araki's papers but I don't know how to deal with the support of a vector, $s^M(\xi)$, which is ...
4
votes
1
answer
183
views
A question about correlations between $ C^{*} $-algebras
I was studying J. M. G. Fell’s paper The Structure of Algebras of Operator Fields when I encountered the concept of a correlation between two $ C^{*} $-algebras.
Definition. Let $ A $ and $ B $ be $...
- 1,396
2
votes
1
answer
299
views
Is this left ideal of C*-algebra principal?
This is a follow up of this question. Let $I$ be closed left ideal of $C^*$-algebra $A$.
Assume we are given a sequence of left $A$-module morphisms $R_n:I\to A$ with $\sum_n \Vert R_n\Vert<\infty$...
- 1,697
2
votes
1
answer
327
views
Integration in C^* algebra
Let $\mathfrak{A}$ be a C${}^*$ algebra and $\mathbb{R}\ni s \mapsto \alpha_s$ a continuous family of its automorphisms. Is it true that
$$
\int d s \, f(s)\, \alpha_s(A)
$$
is well defined as a ...
- 552
8
votes
1
answer
904
views
The Haar state on compact quantum groups $A_u(Q)$ and $A_o(Q)$
Let $Q\in GL_n(\mathbb{C})$. The free unitary quantum group is the universal $C^*$-algebra $A_u(Q)$ with generators $u_{ij},1\leq i,j\leq n$ and relations making $u=(u_{ij})$ as well as $Q\bar{u}Q^{-1}...
- 1,702
6
votes
0
answers
502
views
Faithful and weakly-mixing representations of Property (T) groups in relation to left regular rep
Is it known that: Any countable Property (T) group (or more generally, a non-amenable group) has a faithful, weakly-mixing representation which is NOT weakly included in its left regular ...
- 647
1
vote
1
answer
273
views
examples of completely positive order zero maps to demonstrate a theorem
I'm interested explicit examples which can be used to demonstate the theorem:
Theorem: Let $A$ and $B$ $C^*-$algebras and $\phi:A\to B$ be a completely postive map of order zero. Set $C:=C^*(\Phi(A))\...
user62639
2
votes
0
answers
88
views
When is the multiplication map bounded? [duplicate]
Forgive me if my question is too elementary however I haven't found this issue discussed anywhere. Suppose that $A$ is arbitrary $C^*$-algebra and consider the multiplication map $A \overline{\otimes} ...
- 243
7
votes
1
answer
540
views
Proper morphisms of C*-algebras / Nondegenerate representations
Let $A \to B$ be a proper morphism of $C^*$-algebras. A nondegenerate representation of $B$ induces a nondegenerate representation of $A$. Does the converse hold?
I.e.: let $A \to B$ be a morphism of ...
10
votes
1
answer
786
views
Examples of groups without the n-positive approximation property
Let $G$ be a locally compact group and let $A(G)$ be the http://eom.springer.de/f/f120080.htm>Fourier algebra of $G$, which we view as the predual of the group von Neumann algebra $\mathcal M(G)$. ...
- 4,624
3
votes
1
answer
250
views
Hilbert $C^*$ Modules, dense submodules
I read a paper in which the author uses the propetry that if $A$ is a submodule of a
Hilbert $C^\ast$ module ($C$ is a $C^*$ algebra) such that
$A^\bot=0$ then $A$ is dense. I don't know how to ...
- 221
11
votes
1
answer
397
views
Is there a Dedekind-Frobenius group determinant for infinite groups?
If $G$ is a finite group and $\lbrace x_{g} \rbrace_{g\in G}$ are commuting formal variables, then one can form a matrix whose $(g,h)$ entry is $x_{gh^{-1}}$. The determinant of this matrix is a ...
- 7,057
2
votes
0
answers
117
views
Group actions on principal groupoids
Suppose that $\mathcal{G}$ is etale principal groupoid and that $G$ is a discrete (or finite) group acting freely on the locally compact unit space $\mathcal{G}^0$ (or assuming compactness, if ...
1
vote
1
answer
123
views
A relation among projections of a von Neumann algebra
This is a follow-up question on this. Let $A$ be a von Neumann algebra and $P$ be its projection lattice.
For $p,s,q \in P$, let us define $ p \perp q \mid s \iff ps^\perp q = 0$ where $s^\perp = 1-...
- 1,731
3
votes
2
answers
416
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 ...
user23860
3
votes
0
answers
168
views
The left regular representation of the Jacobi groups over local fields of characteristic >2 is type I?
Let $K$ be a non-archimedean local field of characteristic $>2$. Consider the Jacobi group $G=H_{2n+1}(K)\rtimes Sp_{2n}(K)$, which is the semidirect product of the Heisenberg group $H_{2n+1}(K)$ ...
- 1,702
1
vote
2
answers
407
views
Questions about special $C^*$-subalgebras and ideals.
Let $A$ be a $C^*$-algebra and $I$ be a two side closed (essential) ideal of $A$. Suppose that $p \in A\backslash I$ is a non trivial projection. Let $B=pIp$. My questions are:
(1) Is $B$ a $C^*$-...
- 147
15
votes
1
answer
1k
views
Convolution algebras for double groupoids?
There is a lot of work of course on convolution algebras of measured groupoids, and this gives "Noncommutative geometry". However there is a lot of interest in algebraically structured groupoids, for ...
- 12.3k
2
votes
2
answers
315
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\ \...
- 315
3
votes
2
answers
339
views
Commutator formula in infinite dimensions
The commutator formula states that for A,B elements of a Lie algebra,
$\lim_{n\to \infty}\left\{ \exp\left(-A\tfrac{t}{n}\right)\exp\left(-B\tfrac{t}{n}\right)\exp\left(A\tfrac{t}{n}\right)\exp\left(...
- 31
2
votes
1
answer
182
views
A Possible characterization of F.D or AF commutative $C^{*}$ algebras
By F.D or AF $C^{*}$ algebra,we mean finite dimensional or approximately finite dimensional $C^{*}$ algebra.
Let $A$ be a unital commutative $C^{*}$ algebra with the property that for every ...
- 356
10
votes
1
answer
783
views
When do tensor products of C*-algebras commute with colimits?
Let $I$ be a filtered poset, which you should think of as being huge. Let $A_i$ be an $I$-diagram of $C^{\star}$-algebras and let $A$ be the colimit of this diagram; if necessary, we can also assume ...
- 976