All Questions
Tagged with operator-algebras or oa.operator-algebras
2,152 questions
3
votes
0
answers
77
views
+50
Tight upper bound for $m[Q^k - Q^{k+1}]$ for completely positive linear maps
Let $m: \mathcal{L}(\mathbb{R}^{d \times d}) \to \mathbb{R}$ be the function
$$
m[H] = \frac{\lambda_{\max}(H[\mathbf{I}])}{\lambda_{\max}(H)},
$$
where $\lambda_{\max}$ denotes the largest eigenvalue....
- 73
0
votes
1
answer
119
views
Inequality for commuting hermitian operators
Let $p_1$ and $p_2$ be a complete system of orthogonal projections on $R^n$, $n\geq 2$ (i.e., $p^2_i=p_i=p^*_i$ and $p_1+p_2=\bf{1}$) and $S_1,S_2$ be two commuting hermitian operators on $R^n$ (i.e., ...
- 73
2
votes
1
answer
143
views
Inequality for hermitian matrices
Let $p_1$ and $p_2$ be a complete system of orthogonal projections on $\mathbf R^n, n \geq 2$ (i.e., $p_i^2=p_i=p_i^*$ and $p_1+p_2=\bf{1}$) and $S_1, S_2$ be two hermitian operators such that $S_i \...
- 73
1
vote
0
answers
98
views
Unitary representations of Fuchsian and Kleinian groups
Let $\Gamma$ be a discrete group that is either Fuchsian ($\Gamma \subseteq \text{PSL}(2,\mathbb R)$) or Kleinian ($\Gamma \subseteq \text{PSL}(2,\mathbb{C})$).
I have a unitary representationL
$$
\...
- 357
19
votes
0
answers
472
views
On C*-rigidity problem for torsion-free groups
I'd like to address the $\mathrm{C}^\ast$-rigidity problem for
torsion-free groups (see
this paper),
which asks for non-isomorphic torsion-free groups with isomorphic
(reduced) group $\mathrm{C}^\ast$-...
- 10.1k
4
votes
0
answers
120
views
Projection behavior under the automorphism $\Phi$ on a diffuse semi-finite von Neumann algebra
Let $\mathcal{M}$ be a diffuse semi-finite von Neumann algebra acting on a Hilbert space $\mathcal{H}$, equipped with a faithful normal semi-finite trace $\varphi$. Let $\Phi : \mathcal{M} \to \...
- 113
1
vote
1
answer
117
views
Lower bound for a commutator trace
I have this Hilbert space of square-integrable complex-valued functions on a square, $\mathbb{L}^2([0,1]^2)$. And let $M_x$, $M_y$, and $M_{x+y} = M_x+M_y$ be the operators of multiplication by the ...
- 121
0
votes
0
answers
91
views
Studying flows on $L^2(\mathbb{R}^2)$ given by vector fields using unitary operators
Background: In the $xy$-plane (or a 2-sphere if you are concerned about compactness), we can think about three different types of flow given by vector fields.
(1) pushes everything towards the origin,
...
- 1,143
13
votes
0
answers
331
views
Lie theory for quantum groups?
$\DeclareMathOperator\SU{SU}$I know about quantum groups from two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in ...
- 357
1
vote
0
answers
86
views
Proof mistake of: $M_0A(G) = B(G)$ for a locally compact group
I am posting my question of mathstack exchange here. (see: My post on MSE)
Let $G$ be a locally compact group with Haar measure $\mu$, and $B(G),A(G),C_r^*(G),L(G)$ be its Fourier-Stieltjes algebra, ...
- 261
1
vote
0
answers
69
views
Compressions and (ir)rational trace
Let $\mathcal{M}$ be a type $II_1$ factor with tracial state $\tau$ and $P$ be a projection in $\mathcal{M}$ such that $\tau(P)=1/n$ for some natural number $n.$ It is known (Ananatharaman-Popa "...
3
votes
0
answers
267
views
Cohomology for quantum groups
I'm interested in quantum groups for two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in the sense of Drinfeld &...
- 357
9
votes
0
answers
163
views
Moore-Penrose partial isometries and hermitian elements
Let $A$ be a unital Banach algebra. An element $a \in A$ is hermitian if $\|\mathrm{exp}(ita)\|=1$ for every $t \in \mathbb{R}$. An element $a \in A$ is Moore-Penrose invertible if there exists $b \in ...
- 3,497
-6
votes
1
answer
180
views
An analog of Anderson's result in C* algebra setting [closed]
Let $\mathcal{A}$ be a unital $C^{*}$-algebra and $S(\mathcal{A})$ denote the states space of $\mathcal{A}$.
For $a\in \mathcal{A}$ , define $W(a) =\{\phi(a):\phi\in S(\mathcal{A})\}$
It's known that $...
- 307
-3
votes
1
answer
162
views
Amenable non-Hausdorff groupoids
Is there any clear definition of amenable non Hausdorff groupoids? It should be possibly non-separable nuclear C*-algebras? Please let me know if there is any existing literature talking about this.
- 15
1
vote
1
answer
202
views
Scattered C*-algebras
A $C^*$-algebra $\mathcal{A}$ is called scattered if $\mathcal{A}^{**}$ is atomic, i.e. a direct sum of type I factors.
Obviously, $\mathcal{K}(H)$ is scattered. Other examples?
Observe also that ...
2
votes
0
answers
232
views
$\mathcal{B}(\mathcal{H})$ as the reduced $C^*$-algebra of a groupoid
Given an infinite dimensional Hilbert Space $\mathcal{H}$ what is the underlying locally compact Hausdorff etale groupoid $G$ such that $C_r^{\ast}(G)$ is $\ast$-isomorphic to $\mathcal{B}(\mathcal{H})...
- 15
1
vote
1
answer
89
views
Continuous functions on HLS groupoids
I am reading a paper about property (T) for groupoids: Topological property (T) for groupoids. In section 4.4 they discuss the HLS groupoids which I describe define here.
Let $\Gamma$ be a discrete ...
- 261
1
vote
2
answers
86
views
Entrywise $\infty$-norm of squared difference of square roots of matrices
For a positive $n \times n$ definite real matrix $M$ we denote by $\sqrt{M}$ the positive square root of $M$. For an $n \times n$ matrix $A$ denote its entrywise infinity norm by
$$\|A\|_{\infty,\...
- 177
0
votes
1
answer
119
views
Reference request: hyperfinite cross product
Given a countable essentially free ergodic non-singular group action $G \curvearrowright (X, \mu)$ on a standard measure space, suppose $\mu$ is a non-atomic probability measure and $\alpha: G \...
6
votes
1
answer
170
views
Do projections in an $AW^\ast$-algebra form an orthomodular lattice?
I’m currently studying orthomodular lattices arising out of operator algebras. One of the most standard examples is the projection lattice of a von Neumann algebra - if $M$ acts on a Hilbert space $H$,...
- 2,830
3
votes
1
answer
116
views
Does a bounded positive modular sesquilinear form on a $C^\ast$-algebra induces an element of its multiplier algebra?
This is a question that originates from my attempt at this question. Specifically, for a $C^\ast$-algebra $A$, I am attempting to construct a map $\phi: A \times A \to A$ s.t.,
$\phi$ is sesquilinear,...
- 2,830
3
votes
1
answer
191
views
A possible spectral characterization of commutative $C^*$ algebras
Let $A$ be a $C^*$ algebra. Assume that
the spectrum $Sp(a_1a_2\ldots a_{n-1}a_n)$ is unchanged as a set after a permutation of $a_i$'s. (unless possible emerge or removing 0 from the spectrum)
Does ...
- 356
3
votes
0
answers
73
views
What are the Cuntz semigroups of the Cuntz algebras?
Are the Cuntz semigroups known for the Cuntz algebras $\mathcal{O}_n$ ($1<n<\infty$)? I searched the literature and couldn't find it anywhere. I'm especially looking for $W(\mathcal{O}_n)$ but ...
- 191
1
vote
0
answers
95
views
Reference for structure of subhomogeneous C*-algebras?
Recall that a C*-algebra is subhomogeneous if there is some $n$ such that the dimension of $H$ is at most $n$ for every irreducible representation $\pi \colon A \to \mathcal{B}(H)$ on some Hilbert ...
- 733
0
votes
0
answers
123
views
crossed product by compact groups
Do we need the ambient measure on G to be a Haar measure in order to form the crossed product by a compact group of a von Neumann algebra M? If the measure is indeed Haar, then we can obtain the ...
- 15
1
vote
1
answer
122
views
distance in the matrix algebra w.r.t. the nuclear norm
Let $\varphi\in\mathcal{M}_n(\mathbb{C})$ and let $Z:=\mathbb{C}\cdot I=\{zI\colon\,z\in\mathbb{C}\}$ be the one-dimensional subspace spanned by the identity matrix $I$. Let moreover $\|\cdot\|_{\...
- 375
7
votes
1
answer
280
views
Norm in the minimal tensor product of C*-algebras
Let $A$ and $B$ be two $C^*$-algebras, and let $A \otimes B$ denote their minimal tensor product. Given positive, linear functionals $\varphi$ on $A$ and $\psi$ on $B$, we obtain a positive, linear ...
- 3,497
6
votes
0
answers
110
views
Standard form of fiber product of von Neumann algebras
Let $Z$ be an abelian von Neumann algebra, and let $A$ and $B$ be two von Neumann algebras that receive central maps $Z \to Z(A)$ and $Z \to Z(B)$.
We may then construct the fiber product of $A$ and $...
- 43.2k
4
votes
1
answer
251
views
Show that $\Lambda_\varphi(x_n)\to \Lambda_\varphi(x)$ for an nsf weight $\varphi$ on a von Neumann algebra
Let $\varphi$ be an nsf weight on a von Neumann algebra $M$. Fix a square-integrable element $x\in \mathscr{N}_\varphi$. Put
$$x_n := \sqrt{\frac{n}{\pi}}\int_{-\infty}^{+\infty} \exp(-nt^2) \sigma_t^\...
- 175
1
vote
0
answers
76
views
A representation of positive matrix
Let $\mathcal H$ be a Hilbert space. Let $-\frac{1}{2}<r<0.$ Denote $c_p:=\int_{0}^\infty\frac{t^r}{1+t}dt.$ Suppose $A$ be a positive invertible operator in $B(\mathcal H).$ Is it true that $A^...
- 1,879
3
votes
0
answers
141
views
Location of nontrivial Gleason parts on the topological boundary of the polydisc - Was it described anywhere else besides in Bekken's PhD Thesis?
My PhD advisor and me need the exact description of the location of the non-trivial Gleason parts on the topological boundary of the polydisc $\mathbb{D}^n$.
It was described in Otto B. Bekken's PhD ...
- 63
5
votes
1
answer
183
views
Question about modular group (Modular theory in operator algebras, section 2.14)
Consider the following fragment from Stratila's book "Modular theory in operator algebras", section 2.14, p20:
I'm trying to understand the claim $(3)$ (see the red box). The main strategy ...
- 175
4
votes
1
answer
275
views
What are the norms of the generators of the standard Podleś sphere?
Fix a real number $0<q<1$. We consider the standard Podles sphere $A_q$ as the universal unit $C^*$-algebra generated by $a$ and $b$ with relations
\begin{equation*}
\begin{split}
&a=a^*,~ ...
- 9,009
7
votes
2
answers
349
views
Can the Banach algebra structure on $B(E)$ be (almost) retrieved from its Banach space structure?
This is basically just out of curiosity. Also, since my research area is in von Neumann algebras and my knowledge of general Banach algebras as well as general Banach spaces is somewhat limited, I ...
- 2,830
1
vote
0
answers
55
views
Characterizing one-sided M-projections on real C*-algebras
Let $A$ be a real C*-algebra, and let $P: A \to A$ be a bounded linear projection. We say that $P$ is a left M-projection if the map
$$
v_P: A \to C_2(A), \quad x \mapsto \begin{pmatrix} P(x) \\ x - P(...
- 11
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 $\mathcal{A}$ ...
- 173
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 ...
- 18.7k
4
votes
0
answers
147
views
Isomorphism between the reduced C*-algebra of a groupoid and the crossed product of inverse semigroups
In Paterson's book "Groupoids, Inverse Semigroups and their Operator Algebras" he proves that for any r-discrete groupoid $G$ with unit space $G^0$, its full $C^* $-algebra $C^* (G)$ is ...
- 261
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 ...
- 356
3
votes
0
answers
97
views
Is a localised "restricted symmetry" automorphism implementable as a unitary operator on the GNS Hilbert space?
I have a pure state $\omega$ on a quasilocal algebra $\mathcal{A}$ on a 2d lattice $\Gamma = \mathbb{Z}^2$ with a $\mathbb{C}^d$ vector space on each site. Let there be a unitary symmetry action $U_g(...
- 569
3
votes
0
answers
109
views
Faithful traces on reduced $C^*$-algebra of a measured groupoid
Let $G$ be a measured étale groupoid with quasi-invariant measure $\mu$ (that induces the reduced $C^* $-algebra, meaning it has full support) with associated equivalent measures $\nu,\nu^{-1}$.
Is ...
- 261
2
votes
0
answers
142
views
A $C^*$-algebra with the bidual $B(H).$
Let $H$ be a separable Hilbert space and let $A$ be a non-unital $C^*$-subalgebra in $B(H)$ such that the second dual $A^{**} \equiv B(H).$ Does $A$ coincide with the ideal of compact operators $K(H)?$...
6
votes
0
answers
126
views
How obtain the right definition of smooth elements in a $C^*$-algebra?
In Alain Connes' $C^*$-algèbres et géométrie différentielle (an English translation is here,), for a $C^*$-algebra $A$, we consider a $C^*$-dynamic system $(A,G,\alpha)$, where $G$ is a Lie group and $...
- 9,009
1
vote
0
answers
84
views
Can a limit of degenerate two-cocycles be non-degenerate?
Let $G$ be a discrete abelian group and $\omega\colon G\times G\to\mathbb{T}$ a two-cocycle on $G$. We say that $\omega$ is non-degenerate if for every $e\neq g\in G$ there exists $h\in G$ such that $\...
- 29
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 ...
- 143
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
$...
- 175
5
votes
2
answers
1k
views
Are umbral moonshine and umbral calculus connected?
In a 2013 article, Cheng, Duncan and Harvey introduce the concept of umbral moonshine as a generalization of monstrous moonshine. The terminology they use, starting with the title, is common in umbral ...
- 491
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
0
votes
0
answers
57
views
Decomposition of a contractive representation into an orthogonal sum for the $n$-dimensional case. Has this been done yet?
I know that it has been done for the two-dimensional case. Marek Kosiek showed it in his work "Decomposition of operator representations of the algebra $R(K_1 \times K_2)$" and "...
- 63