Questions tagged [oa.operator-algebras]
Algebras of operators on Hilbert space, $C^*$-algebras, von Neumann algebras, non-commutative geometry
2,270 questions
4
votes
1
answer
208
views
Non-increasing property of a norm-like function over matrices
Let $P,Q$ be two real orthogonal projections on $\mathbb R^n$, and assume that they are permutation similar. More specifically, assume that each of them is permutation similar to a block diagonal ...
- 857
1
vote
1
answer
227
views
Question on monotonicity of a norm-like function for matrices
Let $P_1,P_2$ be two real orthogonal projections on $\mathbb R^n$, and assume that they are permutation similar. More specifically, assume that each of them is permutation similar to a block diagonal ...
- 857
4
votes
1
answer
173
views
Question on power-nonnegative matrix
Let $P_1,P_2\in M_n(\mathbb R)$ be two orthogonal projections, i.e., $P_1^2=P_1=P_1', P_2^2=P_2=P_2'$ and assume that they are unitarily similar.
Let
$$
A=(P_1P_2)\circ(P_2P_1),
$$
where $\circ$ ...
- 857
3
votes
0
answers
140
views
Operator-valued Schwartz function
Let $\mathcal{H}$ be a Hilbert Space and let $\mathcal{B}$ be the Hausdorff LCS of bounded operators in $\mathcal{H}$ equipped with the WOT-topology.
Let $A(t), t \in \mathbb{R}$ be $\mathcal{B}$-...
- 161
4
votes
1
answer
195
views
Density of commutators in traceless matrices with norm bounds
Consider the operator norm $\|\cdot\|$ in $\mathcal{M}_n(\mathbb{C})$ and let $[b,a]:=ba-ab$ stand for a commutator. Is it true that the set
$$\{[b,a]\colon\,\,\|a\|\leq1,\|b\|\leq1\}$$
is dense in
$$\...
- 437
6
votes
0
answers
111
views
Preservation of measurability of operator-valued functions under functional calculus
I wonder if the statement below is known.
Let $H$ be a separable Hilbert space, and let $F$ be a measurable function whose values are bounded self-adjoint operators in $H$. By measurable here I mean ...
- 149
1
vote
0
answers
209
views
A $C^*$ property for idempotent less algebras which is weaker than simplicity
In the literature is there a name or terminology for the following property P of $C^*$ algebras:
A unital $C^*$ algebra $A$ satisfies P if it does not have any non trivial idempotent ...
- 276
3
votes
1
answer
241
views
Weak operator measurability and $L^\infty(X, \mathcal{B}(H))$ for non-separable $H$
Let $H$ be a Hilbert space, $\mathcal{B}(H)$ the space of bounded operators $H\to H$, and $X$ a measure space. We would like to define $L^\infty(X, \mathcal{B}(H))$.
If $H$ is separable we can ...
- 3,083
5
votes
0
answers
144
views
Computation of Connes–Størmer relative entropy for the simplest noncommutative example
Let $u \in M_n(\mathbb{C})$ be a unitary matrix, and let
$\Delta_n \subset M_n(\mathbb{C})$ denote the diagonal subalgebra (a MASA).
Consider its conjugate subalgebra $u \Delta_n u^*$.
Is there an ...
- 463
0
votes
0
answers
81
views
F. & M. Riesz property for $q$-commuting operator algebras
Let $\mathfrak{T}_q$ be the universal $C^*$-algebra generated by unitaries $U_1,\ldots,U_n$ with commutation relations $U_i U_j = q_{ij} U_j U_i$ for $i \neq j$, where $q_{ij}$ are complex scalars ...
- 99
1
vote
0
answers
48
views
Peak sets and peaking functions for $q$-commuting operator algebras
Let $\mathfrak{T}_q$ be the universal $C^*$-algebra generated by unitaries $U_1,\ldots,U_n$ with commutation relations $U_i U_j = q_{ij} U_j U_i$ ($i \neq j$), where $q_{ij}$ are complex scalars with $...
- 99
1
vote
0
answers
48
views
Density of the set of non-cyclic vectors for adjoint elements in the GNS space generated by multiparameter $q$-commuting unitaries
Let $q = \{q_{ij} \in \mathbb{C} : |q_{ij}| = 1, q_{ij} = q_{ji}^{-1} \text{ for } i \neq j\}$ be a family of scalars satisfying the multi-parameter q-commutation relations. We assume the q-parameters ...
- 99
1
vote
0
answers
70
views
Split-signature wedge reflection positivity and Osterwalder-Schrader reconstruction
The Osterwalder-Schrader (OS) reconstruction theorem [1,2] establishes that Euclidean correlation functions satisfying reflection positivity (RP) in a codimension-1 hyperplane determine a Wightman QFT ...
5
votes
1
answer
189
views
closure of product of unbounded selfadjoint operators
Let $A, B$ be injective self-adjoint operators, both unbounded, acting on a common Hilbert space, with $D$ core for $B$ and $B(D)$ core for $A$. Is $AB\,\big|_D$ closable?
- 91
6
votes
2
answers
247
views
Question on nontrivial von Neumann algebras
Do we know nontrivial von Neumann algebras $A$ whose $A^2=\{x\oplus x: x\in A\}$ has no cyclic vector? What about if $A$ is a factor of type $\text{I}\,\text{I}_\infty$?
