All Questions
Tagged with hilbert-spaces oa.operator-algebras
20 questions with no upvoted or accepted answers
11
votes
0
answers
388
views
Von Neumann Inequality in Banach spaces
It is known that the only Banach space that satisfies the von-Neumann inequality is the Hilbert space:
Theorem (see e.g. Pisier, "Similarity Problems and Completely Bounded Maps", p 27) For a Banach ...
- 5,529
6
votes
0
answers
175
views
Functional calculus for the Dolbeault operator over Hilbert C*-modules
$\newcommand{\odd}{\mathrm{odd}}\newcommand{\even}{\mathrm{even}}$Let $X$ be a complex manifold, you can assume it's compact, if necessary. We have the Dolbeault complex $$0 \rightarrow \mathcal{A}^{0,...
- 146
4
votes
0
answers
152
views
Maximally fine topologies on $B(H)$ making the unit ball compact
Let $H$ be a Hilbert space, and $B(H)$ its algebra of bounded operators. One of the reasons the Ultraweak topology is (in a way) more useful than the weak operator topology is that the Ultraweak ...
- 190
4
votes
0
answers
160
views
Solution without using any k-theory tools
Let $A$ be the UHF-algebra of type $2^{\infty}$. Suppose that $p$ and $q$ are two projections in $A$ and $\tau(p) = \tau(q)$, where $\tau$ is the unique normalized trace. Then there is a partail ...
- 581
4
votes
0
answers
185
views
A strongly open set which is not measurable in the weak operator topology
Let $H$ be a non-separable Hilbert space and $\{e_i\}_{i\in I}$ be an orthonormal basis for $H$. Let $J$ be a uncountable proper subset in $I$.
Let us put $$E=\{x\in B(H): \lVert xe_j\rVert <1: \...
- 4,058
3
votes
0
answers
152
views
Whether a projection can "overlap" certain projections yet not commute with them
Question here about how the projections of a von Neumann algebra $\mathcal{R}$ might be arranged, relative to a projection that is not in $\mathcal{R}$.
Stipulate the following:
$H$ is ...
- 271
3
votes
0
answers
144
views
Deformation and Representations
Let $\widetilde{U_q(sl_n)}$ denote a deformation of the algebra $U_q(sl_n)$. In particular, $\widetilde{U_q(sl_n)}$ is defined by the same generators and relations and $*$-operations as $U_q(sl_n)$ ...
2
votes
0
answers
316
views
What are alternative or equivalent definitions of a positive-definite function on a group?
The standard definition of a positive-definite function on a group goes as follows:
Let $\varphi : G \rightarrow L(H)$, where $G$ is a group (with an involution) and $H$ a Hilbert space. $L(H)$ is the ...
- 63
2
votes
0
answers
158
views
Question about the ergodic mean
This is a repost from this MathStackExchange question, where unfortunately I was not able to resolve this question.
I've read a thesis where there is an example on ergodic mean, where however there is ...
- 33
2
votes
0
answers
56
views
Existence of a suitable smooth kernel
Denote by $H=L^2([0,1])$ the Hilbert space of square integrable real valued functions on the interval $[0,1]$. Does there exist a nontrivial smooth real valued function $k$ on the unit interval such ...
- 4,135
2
votes
0
answers
116
views
Closable operators on Hilbert modules
For $T:{\frak{Dom}}(T) \to H$ a densely defined operator, with $H$ a (separable) Hilbert space, we know that $T$ is closable if its adjoint $T^*$ has dense domain in $H$.
Does this extend to the (...
- 993
1
vote
0
answers
89
views
Let $T\in B(H)$. Then prove that $T$ is a contraction if and only if the closed unit disc is spectral set for $T$
Let's first recall the definition of Spectral set for a bounded operator $T$ on a hilbert space $H$. Let $X\subseteq \Bbb{C}$ be compact such that $\sigma(T)\subseteq X$. Define $$\mathcal{R}(X):=\...
- 111
1
vote
0
answers
57
views
Continuity of a linear functional for sequences of projections
Let $T_+$ be the set of a positive trace-class operators over some separable Hilbert space and
$A: T_+ \to \mathbb{R}\cup \{\infty\}$ some linear functional.
In general, $A$ will not be continuous. ...
- 695
1
vote
0
answers
70
views
Is $F: B(H)\to B(H\otimes K)$, $X\mapsto X\otimes \mathbb{1}_K$ completely contractive?
Take Hilbert spaces $H$ and $K$. Consider a linear map $F: B(H)\to B(H\otimes K)$, $X\mapsto X\otimes \mathbb{1}_K$.
Is it true that $F$ is completely contractive? If it is, I would be very grateful ...
1
vote
0
answers
277
views
Adjoint for a non-densely defined unbounded operator on a Hilbert space
Let $\mathbf{H}$ be a Hilbert space, and $D$ an unbounded densely-defined operator on $\mathbf{H}$. As is well-known, every such operator admits an adjoint, with domain possibly different from that ...
- 993
1
vote
0
answers
120
views
Injective differential of linear operators on a Hilbertspace
Given a complex Hilbertspace $\mathcal{H}$ of dimension $\dim(\mathcal{H}) = d$ and the set $$\mathcal{F} := \{q\in L(\mathcal{H})\vert\quad \text{rank}(q) = 4 \quad \wedge \lambda^q_{1,2} < 0\ \ \...
- 143
1
vote
0
answers
74
views
If $f$ takes values in $L(H,L(H,\Bbb R))$ and $μ$ is a $H\hat ⊗_πH$-valued measure, how are $\int f\:dμ$ and $\int f⊗_π\text{id}_Hdμ$ related?
Let
$H$ be a separable $\mathbb R$-Hilbert space
$H\:\hat\otimes_\pi\:H$ denote the projective tensor product of $H$ and $H$
$(\Omega,\mathcal A)$ be a measurable space
$\mu$ be a $H\:\hat\otimes_\pi\...
- 167
1
vote
0
answers
233
views
Bochner integrals with values in a Hilbert $A$-module
I'm wondering whether there exists a generalisation of Bochner integration with values in a Hilbert $A$-module $M$, where $A$ is a general $C^*$-algebra rather than $\mathbb{C}$ (and whether there are ...
- 1,903
0
votes
0
answers
35
views
Operator-form correspondence without lower semiboundedness
When dealing with a normal unbounded operator $A$, it is often useful to frame questions about the operator in terms of questions about the associated form $\omega,$ which has domain $D(|A|^{1/2})$ ...
- 109
0
votes
0
answers
255
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Bounded operators with infinite matrix representations
I asked this question on StackExchange originally, but I'm giving it a go here as well.
Suppose that $A$ is a unital $C^*$-algebra, $\varphi\colon A\to B(H)$ is a unital, completely positive map and ...
- 123