All Questions
Tagged with oa.operator-algebras fa.functional-analysis
778 questions
5
votes
1
answer
199
views
Is the unit ball of $B(H)$ a Baire space (with the SOT)?
Let $H$ be a Hilbert space, and let $B(H)$ be the set of bounded linear operators $t \colon H \to H$. Recall that we say $t_i \to t$ in the strong operator topology if $t_i \xi \to t \xi$ for every $\...
- 733
2
votes
1
answer
164
views
Cocompact lattices in $\mathrm{Sp}(n, 1)$
This is a continuation from my previous question. I am reading the following paper of Cowling-Haagerup, and I was wondering whether there are uniform lattices in $\mathrm{Sp}(n, 1)$. Is there some way ...
- 131
2
votes
0
answers
177
views
Banach isomorphisms between von Neumann algebras
It seems that most people are talking about $*$-isomorphisms of von Neumann algebras. However, I can not find any references for the Banach isomorphisms, i.e., let $A,B$ be two different von Neumann ...
- 617
4
votes
0
answers
115
views
Questions related to a paper of Cowling-Haagerup and uniform lattices of $\mathrm{Sp}(1,n)$
I am reading the following paper of Cowling and Haagerup for my master’s thesis. I am new to this area so I am not very conversant. So I do apologize if the questions are silly.
Question 1. In the ...
- 131
3
votes
1
answer
185
views
Is the weighted shift strong frequently hypercyclic?
One sided Shift
Let be $M$ separable metric space. Consider $X=M^{\mathbb{N}}$ the sequence space equipped with the product metric $d(x,y)=\sum_{i=1}^\infty |x_i-y_i|/2^i$ . Define the shift map $\...
- 757
0
votes
1
answer
93
views
Does ${\rm Tr}(l(a)+l(b)) ={\rm Tr}(l(\alpha_1 a +\alpha_2 b ))$ imply that $l(a)l(b)=r(a)r(b)=0$?
Let $H$ be a Hilbert space and let $a,b\in B(H)$ be such that
$${\rm Tr}(l(a))={\rm Tr}(r(a))<\infty , {\rm Tr}(l(b))={\rm Tr}(r(b))<\infty,$$
$${\rm Tr}(l(a)+l(b)) ={\rm Tr}(l(\alpha_1 a +\...
- 617
4
votes
0
answers
237
views
Does SO(n) have Lafforgue's Strong Property (T)?
On page 13 of the monograph of Bekka, de la Harpe, and Valette on Kazhdan's property (T), it is written "for $n \geq 3$, the compact group $\mathrm{SO}(n)$ has the strong property (T)," ...
3
votes
1
answer
244
views
Takesaki: question about lemma in section "Left Hilbert algebras and weights"
To make this question relatively self-contained, this post is quite long, but the question itself is rather short.
Consider the following fragments in Takesaki's second volume "Theory of operator ...
- 175
6
votes
1
answer
288
views
Sigma-weakly dense *-subalgebra of von Neumann algebra has increasing net of positive elements convergent to the identity
Let $M$ be a von Neumann algebra and $A\subseteq M$ a $\sigma$-weakly dense $*$-subalgebra of $M$. Does there exist an increasing net $\{a_i\}_{i\in I}\subseteq A\cap M^+$ such that $a_i\to 1$ in the $...
- 175
2
votes
1
answer
471
views
Takesaki II proposition 3.15 proof about modular automorphism groups: mistake in book?
Consider the following fragment from Takesaki's second volume "Theory of operator algebras" in chapter VIII "Modular automorphism groups" p122-123.
Why is it possible to choose an ...
- 175
1
vote
1
answer
410
views
Takesaki lemma: existence Gelfand-Pettis integral
Consider the following fragment from Takesaki's second volume of "Theory of operator algebras" (lemma 2.4, chapter VI "Left Hilbert algebras").
In another post, it was explained ...
- 175
1
vote
0
answers
89
views
Let $T\in B(H)$. Then prove that $T$ is a contraction if and only if the closed unit disc is spectral set for $T$
Let's first recall the definition of Spectral set for a bounded operator $T$ on a hilbert space $H$. Let $X\subseteq \Bbb{C}$ be compact such that $\sigma(T)\subseteq X$. Define $$\mathcal{R}(X):=\...
- 111
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
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
0
votes
1
answer
143
views
Differential form of the multidimensional "orthogonal dilation" operator
For a one-dimensional $f(x)$, the dilation operator $f(ax)$ can be expressed as $\exp(g(D))f(x)$, where $g$ is a closed-form function. This is easily checked by e.g. formal Taylor series expansion.
...
- 286
6
votes
1
answer
574
views
Integration in Banach algebra
Let $\mu$ be a Borel measure on the real line $\mathbb{R}$ taking values in a separable Banach algebra $A$. Assume that $\mu$ is such that the total variation measure $|\mu|$ is finite. Let $f$ be a ...
- 552
5
votes
1
answer
204
views
Continuity of the extension of a tracial state with respect to the strong operator topology
Problem: Let $M\subseteq B(H)$ be a finite von Neumann algebra with a faithful tracial state $\tau$. Let $\widetilde{M}$ be the $\tau$-measurable operators on $M$ (recalled below). Extend the trace $\...
- 85
3
votes
1
answer
226
views
$\tau$-measurable operator
Problem: Let $M$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $\tau$. Let $m$ be a positive element in $M$ and let $e_{(0,\infty)}(m)$ be the spectral projection of $m$ ...
- 85
0
votes
0
answers
92
views
Proof of the isomorphism of the Toeplitz algebra and the algebra generated by the element and the relation
Please tell me where can I see the proof of this well-known fact?
enter image description here
1
vote
0
answers
384
views
Densely defined and closed operator
Question: Let $N\subseteq B(H)$ be a finite von Neumann algebra with a faithful normal trace $\tau$ and let $n\in N$. Let $B$ be a von Neumann subalgebra of $N$. Let $\mathbb{E}_B: N\rightarrow B$ be ...
- 85
1
vote
0
answers
85
views
Spectral projection with height less than $\lambda$
Let $x\geq 0$ be a positive element in a von Neumann algebra $\mathcal M.$ Then b y functional calculus the projection $e_\lambda=1_{[0.\lambda)}(x)$ has the property that $e_\lambda$ commutes with $x$...
- 1,879
3
votes
1
answer
192
views
Characters of algebra of Schwartz functions
Consider the (non-unital) $\mathbb{C}$-algebra (point-wise multiplication) of $\mathcal{S}$ of Schwartz functions on $\mathbb{R}$.
Question: Does there exist some character (non-zero multiplicative ...
- 960
2
votes
0
answers
354
views
Weakly null sequences in projective tensor products
First, I'd like to record a question that may still be open. The snippet below is taken from DiestelPuglisi2009.
Second, let $E$ be a Banach space, $(u_n)$ be a weakly null sequence in the projective ...
- 2,605
3
votes
0
answers
295
views
Dunford-Pettis like properties for Banach spaces of operators
Let $E$ be a Banach space and $A\subseteq B(E)$ be a Banach subspace of operators on $E$.
Suppose $A$ satisfies the property (RCC) given below:
$$
\left.\begin{array}{l}
(x_n)\subseteq A \textrm{ ...
- 2,605
3
votes
0
answers
115
views
Recovering the matrix when the Schur decomposition of its blocks are known
Let E be a real symmetric matrix in $M_n(\mathbb{R})$ where $ n=2m$ and
$$E=\left(\begin{array}{cc}
G & X \\
X^t & H
\end{array}\right)$$
where $G,H,X$ are $m\times m$ matrices.
Suppose that $...
- 4,058
5
votes
1
answer
212
views
States "absorbed" by a Haar idempotent on a compact quantum group
Firstly, a small question of nomenclature. If $(S,\bullet)$ is a magma, is there good terminology to relate $a$ to $b$ when
$$a\bullet b=b=b\bullet a?$$
Can we say that $b$ absorbs $a$? Can we say ...
- 1,027
3
votes
0
answers
162
views
The essential norm where some Fourier coefficients are fixed
Let us denote $C_{2\pi}$ by the set of all $2\pi$-periodic continuous functions $f:\mathbb{R}\to \mathbb{R}$.
Q. Let $\phi\in C_{2\pi}$. Is the following statement valid?
$$\|\phi\|_2=\inf_{g\in C_{2\...
- 4,058
2
votes
1
answer
250
views
Norm continuity of the predual of a von Neumann algebra
Let $M$ be a von Neumann algebra and let $(p_i)$ be a net of projections in $M$ decreasing to $0$. Let $f\in M_{\ast} $, the predual of $M$.
It is well known that
$\| p_i f \|_{M_\ast}\to_{i} 0$
for ...
- 617
1
vote
1
answer
195
views
Concrete example of non-norm-attaining bounded linear operator on disc algebra
A bounded linear operator $T$ from a Banach space $X$ to a Banach space $Y$ is called norm-attaining, if there exists a vector $x\in X$ with $\|x\|=1$ such that
$$\|Tx\|=\|T\|.$$
Let $\mathbb{D}=\{z\...
- 149
4
votes
1
answer
334
views
Support projection vs closed support projection of a normal state in enveloping von Neumann algebra
I preface this by saying that I am fairly new to the enveloping von Neumann algebra scene, so there may be some gaps in my understanding.
Given a $C^*$-algebra $A$ and a state $\phi$ on $A$, one may ...
- 135
0
votes
0
answers
144
views
Type III von Neumann algebra
Let $\mathcal M$ be a type $\mathrm{III}$ von Neumann algebra. Is it true that for all $n\geq 1,$ $\mathcal M$ contains a copy of $M_n$ as a von Neumann subalgebra? By Theorem 9.24 of these lecture ...
- 1,879
1
vote
0
answers
123
views
On Riesz decomposition of Volterra operator
Let $T:L^2((0,1)) \to L^2((0,1))$ be the Volterra operator defined by
$$ Tf(x) = \int_0^x f(t)\,dt.$$
Given any $\lambda\neq 0$ it is well known that there exists some positive integer $n=n(\lambda)$ ...
- 4,135
9
votes
2
answers
516
views
Why operator systems?
A $\mathrm{C}^*$-algebra $\mathcal{A}\subset B(\mathsf{H})$ is a norm-closed, self-adjoint subalgebra of bounded operators on a Hilbert space. If we then take a unital self-adjoint (possibly closed) ...
- 1,027
1
vote
0
answers
76
views
Spectral measure for a finite set of mutually commuting normal operators
The following question is from Exercise $\S 11.11$ in A Course in Operator Theory written by John B. Conway:
Suppose $\{N_1, \cdots, N_p\}$ is a finite set of mutually commuting normal operators in $...
- 307
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
1
vote
0
answers
81
views
Tracial linear functionals on an amenable Banach algebra
This post is related to an earlier question about Kazhdan property (T). The purpose of the snippet below is to briefly summarize the background for the question in this post.
Question: Does there ...
- 2,605
8
votes
1
answer
286
views
Commutator ideal in nonunital C*-algebra
Let $A$ be a C*-algebra that has no one-dimensional irreducible representations, that is, there is no (closed) two-sided ideal $I\subseteq A$ such that $A/I\cong\mathbb{C}$.
Let $J$ denote the (not ...
- 3,497
4
votes
1
answer
141
views
"Open systems" version of Stone's Theorem for one-parameter groups of quantum operations
Let $H$ be a Hilbert space, which we interpret as a space of quantum states.
If $U(t):H\to H$ is a unitary norm-continuous one-parameter group with $U(0)=I$, (essentially) Cauchy's functional ...
- 854
4
votes
1
answer
133
views
A $C^*$ algebraic analogy of the concept of complemented subspace in the particular case of $\ell^\infty$
Let $A$ be a $C^*$ algebra. A $C^*$ subalgebra $C\subset A$ is said to be $C^*$ algebraic complemented of $A$ if there exist a $C^*$ subalgebra $D\subset A$ with $A=C\oplus D$ and the obvios mapping $...
- 356
2
votes
1
answer
226
views
Showing a 2-by-2 matrix is a contraction
Let $S\subseteq\mathbb{T}:=\{z\in\mathbb{C}:\vert z\vert=1\}$ be a compact set such that $\operatorname{conv}S\supseteq\{z\in\mathbb{C}:\vert z\vert\leq\frac{1}{\sqrt{2}}\}$ and $B\in M_2(\mathbb{C})$....
- 231
3
votes
1
answer
475
views
Extension of a bounded linear functional
Let $\mathcal{H}$ be a finite-dimensional Hilbert space and $\mathcal{A}\subseteq\mathcal{B(\mathcal{H})}$ be an operator system. Suppose $T_n$ is a collection of all $n$-by-$n$ matrices equipped with ...
- 231
2
votes
1
answer
179
views
Extension of the projective norm to a cross norm
Let $\mathcal{H}$ be a finite-dimensional Hilbert space and $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ be an operator system. Is it possible to extend the projective norm (the greatest cross norm) ...
- 231
1
vote
0
answers
106
views
A locally convex $C^*$ algebraic structure on the disk algebra
A locally convex $C^*$ algebra is a locally convex topological vector space $A$ whose topology is generated by a familly of complete $C^*$ semi norm and $A$ is a $*$ algebra. Moreover all algebra ...
- 356
0
votes
1
answer
101
views
"Project" an operator outside of a von Neumann Algebra into it
Suppose $W$ is a proper von Neumann Algebra contained in $B(H)$ and the identity in $W$ is the identity mapping of $H$ (namely, $W$ does not have non-trivial null space).
Given a self-adjoint $T\in W$...
- 307
1
vote
0
answers
135
views
Description of state space of $C(K,M_n)$?
Edit: closed convex hull added.
I am trying to understand the state space of $C(K,M_n)=C(K)\otimes M_n$ for $K$ a compact space.
My guess would be that these are the closed convex hull of states on $C(...
- 598
7
votes
0
answers
192
views
Reduced group C*-algebra $C^*_r(\mathbb{Z}/2*\mathbb{Z}/2)$: norm of specific elements
Consider the free product of $\mathbb{Z}/2$ with itself with generators
$$
\mathbb{Z}/2*\mathbb{Z}/2=\langle u,v\mid u^2=1=v^2\rangle
$$
and regard its group $C^*$-algebra
$$
C^*(\mathbb{Z}/2*\mathbb{...
- 598
1
vote
0
answers
178
views
A locally convex $C^*$ algebra without zero divisor
Let we have a locally convex $C^*$ algebra $A$. That is $A$ is a TVS equipped with an algebra and an involution structure such that all operations are continuous. Moreover the topology on $A$ ...
- 356
3
votes
0
answers
219
views
Can any POVM be induced by a quantum instrument?
I suspect this is the obvious result of something in operator algebras, but that's far outside my field.
Recall that a projection-valued measure is a map $E$ from a sigma-algebra $\mathcal{F}$ on ...
- 854
1
vote
0
answers
88
views
2-positivity to 3-positivity
Let $B\in M_3(\mathbb{C})$ and $S_3=
\begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0 \\
\end{pmatrix}
$. Suppose $I_3\otimes I_2+B\otimes X+B^*\otimes X^*\geq 0$ for all $2$...
- 231
-1
votes
1
answer
164
views
Closure of the point spectrum of an unbounded diagonalizable operator
Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to $H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...
- 969