All Questions
Tagged with operator-algebras or oa.operator-algebras
2,153 questions
2
votes
0
answers
166
views
How simplify the pentagonal equation from two fusion rings?
A semi-simple finite dimensional Hopf algebra $\mathbb{A}$, and its dual $\mathbb{A}^{*}$ produce two fusion categories $\mathcal{C}$ and $\mathcal{C}^{*}$ and then two fusion rings $\mathcal{R}_{1}$ ...
7
votes
0
answers
573
views
References for "folklore" on strong amenability of (group) C*-algebras?
[Apologies in advance for the prolixity - but I was unsure how much of the story is familiar.]
$\newcommand{\ptp}{\widehat{\otimes}}
\newcommand{\co}{\operatorname{co}}
\newcommand{\Cst}{\operatorname{...
- 25.8k
2
votes
1
answer
535
views
about decomposition of a non-negative definite operators
Hello,
Many years before, I had the following problem.
We first give a definition. Given a non-negative definite real-valued definite matrix $n^2\times n^2$ matrix $M$, it is called separable if it ...
- 1,649
0
votes
0
answers
56
views
When is a cycle in $KK^G(A,A)$ with zero operator the identity cycle?
Given a cycle of the form $(\pi,H,0)$ in $KK^G(A,A)$, when is it equivalent to the identity cycle $1_A=(i_A,A,0)$?
The operator $T=0$, and $\pi:A \rightarrow L(H)$ may be injective.
Any criterions ...
- 58
3
votes
1
answer
504
views
In quantum dynamical simulations, what is the symmetric (Riemannian) analog of a Poisson bracket?
The question narrowly posed is:
What is the accepted name of the bracket operation that is obtained by replacing the (antisymmetric) symplectic structure of the Poisson bracket with a (symmetric) ...
- 1,389
2
votes
1
answer
554
views
About properties of groupoid C*-algebras
I'm interested in the following kind of questions about groupoid $C^*$-algebras.
1) If $G_1 \times_{H} \ G_2$ is a fibre product of (nice) groupoids do we have something like $$C^\star(G_1 \times_{H} ...
- 2,029
10
votes
0
answers
509
views
Lacunary hyperbolic groups and weak amenability
In the paper called Lacunary Hyperbolic group, Y. Ol'shanskii, D. Osin and M. Sapir define and characterize the lacunary hyperbolic groups, which contains the hyperbolic groups but also Tarski's ...
- 195
1
vote
0
answers
102
views
Differentiable Path of Operators and their Inverses
Let $\mathcal{H}$ be a separable Hilbert space. Consider a differentiable map $\mathbb{R} \rightarrow \mathcal{B}(\mathcal{H}), t \mapsto A(t)$, where $\mathcal{B}(\mathcal{H})$ is the space of ...
- 11
0
votes
1
answer
202
views
injective *morphisms of C(D), the C^* algebra of continuous functions on the closed unit disk D
This question relates to one on topology and C^*-algebras that was asked two days ago, namely at the link: C*-algebras with no nontrivial endomorphisms link text
Let D be the closed unit disk in ...
0
votes
2
answers
377
views
"Frobenius-finite" linear operators on a Hilbert Space
Let $H = L_2(S)$ be the complex Hilbert space over $S$ with the counting measure. (There might be another term for this concept, but) I define a continuous linear operator $L$ on $H$ with matrix ...
user5810
33
votes
0
answers
1k
views
Subalgebras of von Neumann algebras
In the late 70s, Cuntz and Behncke had a paper
H. Behncke and J. Cuntz, Local Completeness of Operator Algebras, Proceedings of the American Mathematical Society, Vol. 62, No. 1 (Jan., 1977), pp. 95-...
- 25.5k
0
votes
0
answers
200
views
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 ...
6
votes
0
answers
178
views
Is a circle action on M_n necessarily inner?
An action $\alpha$ of a locally compact topological group G on a unital $C^*$-algebra $A$ is called inner if there exists a continuous group homomorphism $u\colon G\to U(A)$ such that $\alpha_g(a)=...
- 717
2
votes
1
answer
653
views
Strict positivity in dense subalgebras of $C^{*}$-algebras
Let $A$ be a $C^{*}$-algebra, represented on a Hilbert space $H$, and $D$ a selfadjoint unbounded operator on $H$ (note that we do not impose that $D$ have compact resolvent). Let
$\mathcal{A}:=${$a\...
- 213
1
vote
1
answer
97
views
modul projection map
Let H be a Hilbert space and B(H)be space bounded linear operators on H. Let A be a commutative maximal sub-algebra of B(H). Is there a conditional expectation ( a norm one projection) from B(H) to A?
- 43
1
vote
1
answer
293
views
Second adjoint operators on non quasi-reflexive Banach spaces
I am interested in 'algebraic-density'-type properties of second adjoint operators in the algebra of bounded operator on a second dual of a Banach space. Incidentally, I have a problem with ...
- 11
1
vote
1
answer
518
views
Completely equivalent operator norms on $*$-Banach algebras.
Let $A$ be an algebra over $\mathbf{C}$ with an involution operator, and let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two $equivalent$ operator norms, making $A$ into a $*$-Banach algebra (we denote them as ...
- 613
1
vote
1
answer
162
views
Inductive limit of mapping tori
I have two mapping tori $A_{n}$={$f\in C([0,1], M_{n}(\mathbb{R}))\mid f(1)=\alpha_{1}(f(0))$}, $B_{n}$={$f\in C([0,1], M_{n}(\mathbb{R}))\mid f(1)=\alpha_{2}(f(0))$} where $\alpha_{1}, \alpha_{2}$ ...
- 169
0
votes
1
answer
232
views
simultaneously Approximated by self-adjoint elements.
We know that a $C^*$-algebra $A$ has real rank zero iff every self-adjoint element of $A$ can be approximated in norm by self-adjoint element with finite spectra. My question is:
If we have two self-...
- 147
10
votes
0
answers
320
views
Can we find arbitrarily many elements of SU(2) generating a good copy of MAX($\ell_1^n$) inside VN(SU(2))?
In trying to prove that the answer to the title is "no", I was led to the following problem (which I think is equivalent to the question asked in the title, but can be stated independently). If ...
- 25.8k
4
votes
1
answer
311
views
Continuous functions on the states of a C*-algebra and its elements
Let $\mathcal A$ be a C*-algebra and $s(\mathcal A)$ the set of states on $\mathcal A$, with the weak* topology, as a subspace of the dual space. Suppose $f: s(\mathcal A) \to \mathbb C$ is a ...
- 41
13
votes
1
answer
404
views
Self map of unitary group
Let $H$ be a Hilbert space and let $u_1 \in U(H)$ be a unitary operator on $H$. Consider the self-map $w: U(H) \to U(H)$ which is given by
$$w(v) := v^2 u_1 v^{-1}.$$
Since $U(H)$ is connected, there ...
- 25.5k
9
votes
1
answer
509
views
Maximal localizations of von Neumann algebras
Suppose M is a von Neumann algebra.
Denote by L its maximal noncommutative localization,
i.e., the Ore localization with respect to the set of all left and right regular elements,
i.e., elements whose ...
- 37.8k
8
votes
1
answer
460
views
Finite dimensionality of certain $C^{\star}$-algebras
In the discussion about the question Finite-dimensional subalgebras of $C^{\star}$-algebras the following separate question came up:
Let $H$ be a Hilbert space and $a_1, \dots, a_n \in B(H)$ be self-...
- 25.5k
1
vote
0
answers
198
views
Reduced C*-algebras of locally compact etale Hausdorff groupoids
Let $G$ be an étale locally compact Hausdorff groupoid (possibly second-countable) and let $a\in C_{\textrm{red}}^*(G)$. Is it true that for all $\varepsilon>0$ there is $s\in C_c(G)$ satisfying ...
1
vote
1
answer
175
views
Nonlinear Operators(with the group property?)
Let V be a finitely generated vector space with dimension(V) = $n \in \mathbb{N}>1$. Now let T: $ V \to V$ be a map such that $\forall \hat{v},\hat{w} \in V$, $\; T(\hat{v}+\hat{w}) \neq T(\hat{v})+...
- 123
7
votes
1
answer
385
views
A coalgebraic description of the hyperfinite II_1 revisited
Back here I was asking for a coalgebraic characterisation of the hyperfinite $II_1$ factor. Recall the latter's construction by forming the inductive limit of a chain of matrix algebras $R \to M_2(R) \...
- 5,094
5
votes
2
answers
862
views
Hilbert $C^*$-modules and approximate units
Hi,
Given a $\sigma$-unital $C^*$-algebra $A$ and a full Hilbert $A$-module $E$, is it possible to find an approximate unit $ \{\epsilon_i\}, i\in I$ in $A$ such that each $\epsilon_i$ is of the ...
- 93
2
votes
1
answer
259
views
Nuclearity of certain semigroup crossed product C*-algebras
This question is related to this question link.
Suppose we have an (abelian) semigroup $S$ acting by endomorphisms on a $C^*$-algebra A giving rise to a semigroup crossed product $B = A\rtimes S$. ...
- 2,029
0
votes
1
answer
1k
views
Linear Mapping and integration
I have been reading the paper - "Introduction to Quantum Fisher Information".
In section 1.2 the author talks about the linear map $\mathbb{J}_D$, which he defines as follows:
Let $D \in M_n$ be a ...
- 145
4
votes
1
answer
389
views
When can the group von Neumann algebra of a one-relator group be isomorphic to a free group factor?
Let $G=\langle a,b | R \rangle$ be a one-relator group. When can the left group von Neumann algebra $LG$ be isomorphic to a free group factor? Jesse and Andreas have "trapped the lion" pretty well ...
- 7,057
4
votes
0
answers
282
views
Extensions of completely positive mappings
I would like to ask the following two questions.
Let $1_{\mathcal{H}}\in \mathcal{A}\subset\mathcal{B}\subset\overline{\mathcal{A}}^{SOT}\subset\mathbb{B}(\mathcal{H})$ be a sequence of $C^{\ast}$-...
- 619
12
votes
0
answers
310
views
Subfactors of $L(F_{\infty})$
It is a well known result that any subfactor of the hyperfinite $II_{1}$ factor is hyperfinite. I wonder if there is any finite index version of this for free group factors. In particular is it true ...
- 629
4
votes
0
answers
96
views
States and extremal states of quantum SU(2) and the Podleś sphere
Is there any description (preferably somehow related to the original generators) for the state space (as in C*-algebras) of quantum SU(2) and the Podleś sphere? If so (this is pushing my luck) are the ...
- 1,143
2
votes
0
answers
158
views
About the classification of infinite depth irreducible finite index maximal subfactors
The Temperley Lieb subfactors $A_{\infty}$ are the first examples of infinite depth irreducible finite index maximal subfactors. We can see these subfactors as coming from the simple Lie group $SU(2)$....
4
votes
0
answers
227
views
Recursive formula for joint moments in free probability
Suppose $\mathfrak{A}$ is an algebra (over $\mathbb{C}$, let's say), $\phi$ a linear functional on $\mathfrak{A}$, and $A_1, A_2 \subset \mathfrak{A}$ subalgebras which are $\phi$-freely independent. ...
- 275
4
votes
1
answer
734
views
Approximate unit for the algebra C*(h) consisting of projectors
Let E be a Hilbert C*-module over some C*-algebra and let $h \in K(E)$. Due to B. Blackadar's, "K-Theory for Operator algebras" Thm. 17.11.4 for a separable C*-algebra $A$, represented by ...
- 613
0
votes
0
answers
218
views
Series of linear maps: on a paper by Evans and Hanche-Olsen
I was reading this paper by Evans and Hanche-Olsen. In theorem 2, there are six equivalent statements given. I write just two of them, which I want to use.
Let $L$ be a bounded self-adjoint
...
- 421
0
votes
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 ...
- 1
4
votes
0
answers
151
views
Is there a maximal finite depth infinite index irreducible subfactor?
A subfactor $N \subset M $ is irreducible if $N' \cap M = \mathbb{C} $.
It's maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $.
It's cyclic if its lattice of ...
3
votes
0
answers
151
views
Is there a Fourier transform for principal r-discrete groupoids?
I have been looking for an analog of the Fourier transform for groupoids coming from tilings (which are generally principal and r-discrete), however all the generalizations I have found assume that ...
- 1,010
1
vote
0
answers
243
views
Norm bound of the entrywise logarithm of a stochastic matrix stationary matrix
Hello,
Denote $\log_\star$ as the entrywise logarithm operation, and let $A$ be some row-stochastic matrix such that $\lim_{p\rightarrow\infty}A^p$ exists and all its entries are non-zero.
As a part ...
- 225
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? ...
- 95
0
votes
0
answers
152
views
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" ...
- 365
0
votes
0
answers
150
views
A question about multiplier algebra of $C_0(G)\otimes C_b(G)$ for a locally compact group $G$
Let $G$ be locally compact group. How we can show that
$$
M(C_0(G)\otimes C_b(G))=C_b(G,C_b(G)).
$$
($M(C_0(G)\otimes C_b(G))$ is the multiplier algebra of $C_0(G)\otimes C_b(G)$)
- 1
3
votes
0
answers
132
views
Can you tell if a subfactor is finite depth by the growth rate of the standard invariant?
Let $N\subset M$ be a finite index inclusion of $II_1$ factors. To the inclusion we associate the tower of higher relative commutants
$\begin{array}{ccccccc}
\mathbb{C} = N'\cap N & \subset &...
- 2,441
9
votes
0
answers
443
views
Residual finite dimensionality of surface groups
Alex Lubotzky and Yehuda Shalom have shown in Finite representations in the unitary dual and Ramanujan groups., (Discrete geometric analysis, 173–189, Contemp. Math., 347, Amer. Math. Soc., Providence,...
- 25.5k
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(\...
- 25.8k
0
votes
0
answers
144
views
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-...
8
votes
0
answers
345
views
Is there a finite-index finite-depth II$_1$ subfactor which is more than $7$-super-transitive?
Background: See Noah and Emily's posts on subfactors and planar algebras on the Secret Blogging Seminar.
There are plenty of examples of $3$-super-transitive (3-ST) subfactors; Haagerup, $S_4 < ...
- 7,800