All Questions
Tagged with operator-algebras or oa.operator-algebras
2,152 questions
4
votes
1
answer
275
views
What are the norms of the generators of the standard Podleś sphere?
Fix a real number $0<q<1$. We consider the standard Podles sphere $A_q$ as the universal unit $C^*$-algebra generated by $a$ and $b$ with relations
\begin{equation*}
\begin{split}
&a=a^*,~ ...
- 2,830
0
votes
1
answer
144
views
Is a NC sphere a (one point) compactification of a NC plane?
Inspired by this question About noncommutative sphere and inspired by the fact that the classical sphere is the one point compactificatiin of $\mathbb{R}^2$ we ask the question below:
Is the non ...
- 2,830
14
votes
2
answers
871
views
Which finite dimensional Banach spaces can be represented isometrically as spaces of bounded operators on a finite dimensional Hilbert space?
Background:
It is known that every Banach space $X$ can be embedded isometrically as a subspace in the space $C(K)$ of continuous functions on a compact Hausdorff space $K$. Indeed, one can take $K$ ...
- 6,406
7
votes
2
answers
349
views
Can the Banach algebra structure on $B(E)$ be (almost) retrieved from its Banach space structure?
This is basically just out of curiosity. Also, since my research area is in von Neumann algebras and my knowledge of general Banach algebras as well as general Banach spaces is somewhat limited, I ...
- 2,605
6
votes
1
answer
255
views
Example/Existence of Positive Linear Functional which is NOT Hermitian
We know that if $\mathcal{A}$ is a unital $C^*$-algebra and if $f:\mathcal{A}\to\mathbb{C}$ is a positive linear functional then it is Hermitian. It simply follows from the fact that in $\mathcal{A}$ ...
- 2,830
3
votes
0
answers
74
views
Are all enveloping algebras $\mathcal{U}(\mathfrak{g})$ locally compact quantum groups?
Let us consider the enveloping algebra $\mathcal{U}(\mathfrak{g})$ of some Lie algebra $\mathfrak{g}$.
Under what assumptions about $\mathfrak{g}$, does the enveloping algebra generate a locally ...
- 143
8
votes
2
answers
302
views
Differing notions of Morita equivalence for operator algebras
Rieffel first studied Morita equivalence for $C^*$-algebras and von Neumann algebras in "Morita equivalence for C∗-algebras and W∗-algebras" Zbl 0295.46099 as a direct generalisation of the ...
- 525
22
votes
2
answers
2k
views
Non weakly-group-theoretical integral fusion category
Is there an integral fusion category of rank $7$, FPdim $210$ and type $(1,5,5,5,6,7,7)$ with the following fusion rules (or the little $\color{purple}{\text{variation}}$ below)?
$$\scriptsize{\begin{...
1
vote
0
answers
91
views
Subfactors with integer Jones index
Is there any integer (Jones) index subfactor which is not extremal?
- 393
1
vote
1
answer
186
views
Takesaki II "Bimodule" question
Consider the following fragment from Takesaki's book "Theory of operator algebras", chapter IX Non-commutative integration, Section 3 on p187-188:
I have trouble understanding the equality
$...
- 2,830
1
vote
0
answers
70
views
Affiliating the whole algebra of 'coordinates' with a locally compact quantum group
When constructing a quantum deformation of a classical (matrix) locally compact group, we usually start with the *-Hopf algebra $A$ of matrix entries/coordinates. We then deform this algebra ($A_q$) ...
- 143
13
votes
0
answers
174
views
Existence of more than two C*-norms on algebraic tensor product of C*-algebras
Let $A$ and $B$ be two C*-algebras. Then $(A,B)$ is called is a nuclear pair if there is a unique $C^*$-norm on the algebraic tensor product $A\odot B$.
If $A$ or $B$ is nuclear, then all pairs $(A,B)$...
- 536
1
vote
0
answers
176
views
Unitary operators in $\mathcal{U}(L^2(X,\mu))$ not coming from $\operatorname{Aut}(X,\mu)$
$\DeclareMathOperator\Aut{Aut}$Let $(X,\mu)$ be a measured space with probability measure $\mu$ and $\Aut(X,\mu)$ denote the group of measure preserving bijections of $(X,\mu)$. It is easy to see ...
- 12.9k
0
votes
0
answers
111
views
Integral decomposition
Let $\mathcal{A}$ be a separably acting von Neumann algebra and let $$\mathcal{A}\cong \int_{\Gamma}^{\oplus} \mathcal{A}_{\gamma}\,d\mu(\gamma)$$ be its direct integral decomposition into factors $\...
- 12.9k
0
votes
0
answers
55
views
Status of generalization of timelike tube theorem to algebras of causal completions
The timelike tube theorem states that the additive algebra $A_{\text{add}}(U)$ of operators in a spacetime region $U$ is equal to the additive algebra $A_{\text{add}}(E(U))$ of the timelike envelope $...
CommunityBot
- 1
6
votes
2
answers
198
views
Proof that every commutative locally compact quantum group arises from a locally compact group
It is well-known that there is a bijection (up to isomorphisms) between locally compact quantum groups whose algebra is commutative and classical locally compact groups. I seem to cannot find a proof ...
- 4,351
3
votes
0
answers
117
views
Is every strongly singly generated type $II_1$ factor generated by a pair of hyperfinite $II_1$ subfactors with a common MASA?
A type $II_1$ factor $M$ is strongly singly generated (SSG) if every amplification $M^{t}$ of $M$ is singly generated as a von Neumann algebra. Interesting characterizations of SSG type $II_1$ factors,...
- 7,047
2
votes
0
answers
93
views
Amenability and the unitary group of an operator algebra
Let $M$ be a von Neumann algebra and $U(M)=\{x\in M: x^*=x^{-1}\}$ be its unitary group. In this post, we equip $U(M)$ only with the relative weak$^*$ topology $\sigma(M,M_*)$. Then, $U(M)$ is a ...
- 2,605
1
vote
0
answers
154
views
Kasparov's descent homomorphism for higher KK groups
I am currently trying to understand equivariant $KK$-theory. I think I roughly get the idea of Kasparov's descent homomorphism
$$KK^G(A,B) \rightarrow KK(A \rtimes G,B \rtimes G).$$
but what still ...
- 1
2
votes
1
answer
231
views
Characterization of certain subalgebras of $M_2(\mathcal{A})$ where $\mathcal{A}$ is a $C^*$-algebra
Let $\mathcal{A}$ be a $C^*$-algebra generated by a single element $a \in \mathcal{A}$. Suppose that it is also generated by another element $b \neq a$. Consider a subalgebra $\tilde{\mathcal{A}}$ of ...
- 2,830
0
votes
0
answers
115
views
$C^*$ algebra generated by conjugation of an element
Assume $\mathcal{A}$ is a unital $C^*$ algebra and consider some positive-definite element $\Psi\in M_n(\mathcal{A})$. Can we say something about $C^*(\langle \Psi^{-\frac{1}{2}}E_{i,i}\Psi^{\frac{1}{...
- 63.9k
3
votes
1
answer
244
views
inclusion of von Neumann algebras implies reversing inequality of its modular operators
I'm working with von Neumann algebras and I stumbled with this statement in a work of Borchers (1999)
Since $\mathcal N \subseteq \mathcal M$, it follows by standard arguments that $\Delta_{\mathcal ...
CommunityBot
- 1
3
votes
2
answers
132
views
$w^*$-limit of projections in von Neumann algebra
Let $\mathcal M$ be a semi finite von Neumann algebra with a normal faithful semi finite trace $\tau$. Let $(e_i)_{I\in I}$ be a net of projections in the von Neumann algebra which converges to an ...
- 4,351
-1
votes
1
answer
102
views
Is an $A$-$B$—$C^*$-correspondence a representation of a $G$-$C^*$-algebra, $\rho \colon A \otimes_{ \alpha } B \to \mathcal{L} ( \mathcal{H} )$?
Let $R$ and $S$ be two rings.
It is known that an $R$-$S$-bimodule is actually the same thing as a left module over the ring $R \otimes_{\mathbb{Z}} S^{\mathrm{op}}$, where $S^{\mathrm{op}}$ is the ...
- 43.2k
1
vote
1
answer
146
views
Form of a hereditary subalgebra of $C^*$-algebra $C_0(X)$
I would like to show that:
"every hereditary subalgebra $U$ of a $C^*$-algebra $C_0(X)$ for a locally compact Hausdorff Space $X$ has the form $J_E := \{f \in C_0(X) : f|_E=0 \}$ for a closed ...
- 63.9k
3
votes
0
answers
96
views
Excising the trace of a $II_1$-factor
Recall that a state $\varphi$ on a $C^*$-algebra $A$ is said to be excised by projections if there exists a net of projections $e_i \in A$ such that $\| e_i a e_i - \varphi(a) e_i\| \to_{i} 0$ for all ...
- 297
4
votes
1
answer
137
views
Fixed point algebra of a non-amenable factor
Let $M$ be a non-amenable factor. Suppose $\alpha$ is an action of a group $\Bbb R$ on $M$.
Define $$M^{\alpha}:=\{x\in M: \alpha_t(x)=x \text{ for any} \; t\in \Bbb R\}.$$
If we know that there exits ...
- 4,351
0
votes
1
answer
84
views
Reference for the G-equivariant Stinespring dilation theorem
Is there a good reference for the G-equivariant Stinespring dilation theorem? I can't find the theorem stated anywhere.
Thanks in advance.
2
votes
0
answers
121
views
Invariant weights associated to algebraic quantum groups
Consider an algebraic quantum group $(A, \Delta)$ in the sense of Van Daele, i.e. a regular multiplier Hopf $^*$-algebra with a positive left integral $\varphi$ and a positive right integral $\psi$.
...
- 175
14
votes
1
answer
2k
views
Infinite tensor product of states
Tensor products of finite number of different objects are always well described in the literature. However, the situation of infinite tensor products seems to be much tougher.
Even in the simplest ...
- 3,738
4
votes
2
answers
452
views
Is every bounded representation of Z unitarisable when all sets are measurable?
For the purpose of this question, a group is amenable iff there exists a Følner sequence.
Dixmier unitarisability problem asks whether a (countable discrete) group G is amenable iff every bounded ...
- 63.9k
11
votes
2
answers
2k
views
What's the matrix of logarithm of derivative operator ($\ln D$)? What is the role of this operator in various math fields?
Babusci and Dattoli, On the logarithm of the derivative operator, arXiv:1105.5978, gives some great results:
\begin{align*}
(\ln D) 1 & {}= -\ln x -\gamma \\
(\ln D) x^n & {}= x^n (\psi (n+1)-\...
- 10.5k
6
votes
1
answer
322
views
Pairwise orthogonality for partitions of unity in a *-algebra
Let $\mathcal{A}$ be a $*$-algebra with unit $1_{\mathcal{A}}$. As in the $\mathrm{C}^*$-setting, a projection is an element $p\in\mathcal{A}$ such that $p=p^2=p^*$. A partition of unity is a finite ...
- 6,406
5
votes
0
answers
265
views
Failure of Tomiyama's property ($F$) for reduced group $C^*$-algebras
Are there known examples of discrete groups such that the minimal tensor product of their reduced group $C^\ast$-algebras does not have Tomiyama's property ($F$)?
Such groups must necessarily be non-...
- 51
0
votes
1
answer
118
views
Minimal norm problem whose unknown is an operator
Generally given an Hilbert space $X$ with and a bounded linear operator $H : X \to X$ given a vector $y \in X$ we seek an $x \in X$ such that
$$
f(x) = \frac{1}{2} \left\lVert Hx - y \right\rVert_2^2
$...
- 39k
0
votes
3
answers
752
views
center of the algebra of bounded operators [closed]
Suppose that $X$ is a Banach space. How to prove that the center of the algebra $B(X)$ (the algebra of bounded operators on $X$) consists only of operators of the form $aI$, where $a$ is scalar and $I$...
- 6,357
10
votes
0
answers
397
views
Is $\mathcal{B}(\mathcal{H})$ a groupoid $C^*$-algebra?
Let $\mathcal{H}$ be a complex Hilbert space, and $\mathcal{B}(\mathcal{H})$ be the $C^{\ast}$-algebra of bounded operators on $\mathcal{H}$. Is there an étale groupoid $\mathcal{G}$ such that its $C^{...
- 1,485
14
votes
1
answer
675
views
Quantum braid group
Recently I learned the definition of the quantum permutation group $A_s(n)$, which starts from the fundamental representation by permutation matrices and exchanges the entries by noncommuting ...
- 1,027
1
vote
1
answer
80
views
A subfactor of finite index
We say $N$ is a subfactor of finite index of the factor $M$ if there is a normal faithful conditional expectation of $M$ onto $N$ that has finite index.
If $N$ is a subfactor of finite index of the ...
- 5,425
20
votes
2
answers
1k
views
P-adic C* algebras
I understand that there is a definition of p-adic Banach algebras and that a significant amount of functional analysis can be developed in the non-archimedean setting. Is there a p-adic version of C*-...
- 1,485
13
votes
0
answers
573
views
Classical (i.e. commutative) spaces with quantum symmetry but no classical symmetry
In a recent preprint (arXiv:2311.04889), my coauthors and I constructed a sequence of graphs with no classical symmetry which nevertheless have quantum symmetry.
For graphs this had been an open ...
- 2,339
4
votes
0
answers
134
views
Automorphism-invariant positive linear functionals on $C*$-algebras
Let $A$ be a $C^*$-algebra. Does there exist a non-trivial positive linear functional $\nu\in A^*$ which is $\mathrm{Aut}(A)$-invariant? That is, $\nu\circ\alpha=\nu$ for all $\alpha\in\mathrm{Aut}(A)$...
- 1,959
6
votes
2
answers
477
views
Linear map between projective finitely generated Hilbert modules is adjointable
Let $A$ be a (unital) $C^*$-algebra and $X,Y$ right Hilbert $A$-modules which are finitely generated and projective. It seems to be well-known that if $T: X \to Y$ is an $A$-linear map, then $T$ is ...
- 63.9k
2
votes
0
answers
209
views
Literature on Lyndon words and the Lie commutator
Since I lost my paper notes in a domestic conflagration in Japan some ten years ago, I've occasionally tried to recall one particular author who wrote in the 1900s about Lyndon words / strings, or ...
- 10.5k
1
vote
0
answers
94
views
positive invertible maps which are not *-automorphisms
Let $A$ be a unital C*-algebra. Is there a unital positive self-map $F:A\to A$ which is invertible (i.e. injective and surjective) but not a $*$-automorphism? If yes, how does appear its Gelfand-...
- 41
3
votes
1
answer
162
views
Rescaling Fourier coefficients of a continuous function by a bounded sequence
This question stems out of: which sequences $(a_n)_{n\in\mathbb{Z}}$ of complex numbers have the property that if there exists a continuous function $f$ on the circle with Fourier coefficients $b_n$, ...
- 24.2k
7
votes
1
answer
161
views
type II$_1$ subalgebra of type III$_1$ factor
Let $M$ be a type III$_1$ factor and $N$ be the type II$_1$ subalgebra of $M$.
What is the type of $N'\cap M$? Can it be any type?
- 4,351
6
votes
4
answers
8k
views
Characterization of the non-negative definite functions $f(x,y)$
The common definition of the non-negative definite functions is as follows:
Definition 1: A continuous complex-valued function $f(x)$ is called non-negative definite, if for any real numbers $x_1,\...
- 4,713
1
vote
1
answer
407
views
The problems of global asymptotic freeness
Let $X_{N}\in\mathcal{M}_{N}\big(L^{\infty-}(\Omega,\mathbb{P})\big)$ be a $N\times N$ random complex matrix such its entries $(x_{ij}, 1\leq i, j\leq N)$ be $i.i.d.$, centred with variance $1$. $X_{...
CommunityBot
- 1
1
vote
0
answers
108
views
Infinite tensor product of Hilbert spaces [duplicate]
Recently while reading an article I came across the usage of infinite tensor product of Hilbert spaces. I have got a basic understanding of doing computations in infinite tensor product while reading ...
- 11