All Questions
Tagged with operator-algebras or oa.operator-algebras
2,152 questions
0
votes
0
answers
79
views
Projections to orthogonal complements of conditional expectations
For a conditional expectation from a C^* algebra A to a subalgebra B, we can form a positive projection $P:A\to A$ with image equal to $B$. Question: is $Id - P:A\to A$ a positive map?
- 1,143
2
votes
0
answers
157
views
Why do von Neumann algebras possess identity?
My starting point is that a von Neumann algebra is a $C^*$-algebra with a predual. The usual approaches to showing the existence of identity involve spectral theory (for approximate identity), but ...
- 433
6
votes
0
answers
253
views
Are bounded groups of thin operators on Hilbert space similar to groups of unitaries?
QUESTION. Let $G$ be a group of bounded operators on $\ell^2$, satisfying $\sup_{x\in G} \lVert x\rVert <\infty$, whose elements are all of the form "identity+compact" (sometimes called &...
- 25.8k
14
votes
0
answers
220
views
Unitary group of a von Neumann algebra: is it a retract of $U(H)$?
Let $M\subset B(H)$ be a properly infinite von Neumann algebra (the case I care about is $M=$ hyperfinite $\mathrm{III}_1$).
Consider the unitary groups $U(M)$ and $U(H)$ in their strong operator ...
- 43.2k
-2
votes
1
answer
143
views
Relationship between noncommutative torus for different values of theta [closed]
Let $u,v\in B(L_2(\mathbb T))$ defined as $u(f)(z)=zf(z)$ and $v(f)(z)=f(ze^{-2\pi i\theta})$ for $z\in\mathbb T$ where $\theta\in\mathbb R\setminus\mathbb{Q}$. Denote the $C^*$ algebra generated by $...
- 25.8k
2
votes
1
answer
170
views
Ultralimit of $w^*$-continuous maps
Let $\omega$ be a free ultrafilter on $\mathbb N.$ Let $(\mathcal M_n)$ be a sequence of finite von Neumann algebras. Let $\mathcal N$ be another finite von Neumann algebra and we have maps $\phi_n:\...
CommunityBot
- 1
4
votes
1
answer
270
views
Kaplansky inverse element theorem on group C-star algebra
In a class talking about $C^*$ algebra and (higher) index theory, I heard a theorem
(related to Kaplansky, proved?), that is
Suppose $\Gamma$ is a group (admitting Haar measure if necessary) while $\...
- 193
10
votes
1
answer
428
views
Direct sums of operator spaces
I am interested in the $\ell^1$ analogue of direct sums for Operator spaces, e.g. Operator Space Dictionary. Briefly, and operator space is either a concrete subspace of $B(H)$, the operators on a ...
- 18.7k
1
vote
1
answer
207
views
Weak-star convergence implies trace-norm convergence
By definition, if bounded operators $a_i$ converge to $0$ in the weak*-star topology, then $\operatorname{tr} a_it \to 0$ for any trace-class $t$.
Does this also hold for the trace-norm instead of the ...
- 42.8k
1
vote
0
answers
111
views
Inclusion of finite dimensional C*-algebras and relative commutants of subfactors
Given a subfactor $N\subset M$ with finite Jones index, the inclusion of relative commutants $N^{\prime}\cap M\subset N^{\prime}\cap M_1$ (here, $M_1$ is the basic construction of $N\subset M$) is a ...
- 393
3
votes
0
answers
200
views
What are the first non-maximal non-group-subgroup simple irreducible subfactors?
Definition: For an irreducible (finite index) subfactor $(\mathcal{N} \subset \mathcal{M})$, an intermediate $(\mathcal{N} \subset \mathcal{P} \subset \mathcal{M})$
is normal if the biprojections $e_{\...
- 2,821
1
vote
0
answers
79
views
Doubts on convergence of series of operators
Given an operator algebra of bounded operators $\mathcal{A}$ acting on a Hilbert space $\mathbb{H}$, I am interested in the algebra of tensor products $\mathcal{A}^{N} = \otimes_{k=1}^{N} \mathcal{A}...
- 33
2
votes
0
answers
118
views
Depth of the reduced subfactor
Suppose $N\subset M$ is a finite depth subfactor with $[M:N]<\infty$. Consider the reduced subfactor $pNp\subset pMp$ for some projection $p\in N$. How to calculate the depth of $pNp\subset pMp$ in ...
- 393
5
votes
1
answer
165
views
Approximation from below of positive elements in tensor product of von Neumann algebras
Let $\mathcal M$ and $\mathcal N$ be two von Neumann algebras.
If $x$ is a positive element of $\mathcal M$ and $y$ is a positive element of $\mathcal N$, it is known that $x\otimes y$ is a positive ...
- 2,471
4
votes
0
answers
145
views
Infinite dimensional homology theory for submanifolds of Hilbert and Banach spaces
Is there a version of homology theory for spaces for which explicitly infinite dimensional "cells" are allowed?
The spaces in question include e.g.
\begin{equation}
X = (x: x \in l_2: p_i(x) ...
- 593
1
vote
1
answer
142
views
An inner product and projection property in RKHS
I'm reading an article regarding the condition for the existence of a solution to a constrained Pick interpolation problem, and the author wrote the following equality - for background, we have the ...
- 2,830
3
votes
0
answers
60
views
Noncommutative maximal weak $L_1$ norms with respect to sub algebra
Let $(\mathcal M,\tau)$ be a von Neumann algebra with normal finite faithful trace $\tau.$ For any sequence $(x_n)_{n\geq 1}\in \mathcal M$ define $\|(x_n)\|_{\Lambda_{1,\infty}(\mathcal M;\ell_\infty)...
- 1,879
20
votes
8
answers
12k
views
Can a self-adjoint operator have a continuous set of eigenvalues?
This should be a trivial question for mathematicians but not for typical physicists.
I know that the spectrum of a linear operator on a Banach space splits into the so-called "point," "continuous" ...
- 41.1k
7
votes
0
answers
159
views
Maps in the Künneth theorem for K-theory of C*-algebras
The following is named the Künneth theorem for tensor products in the book by Blackadar on K-theory for operator algebras: If $A$ and $B$ are C*-algebras and $A$ is in the bootstrap class, then there ...
- 2,998
4
votes
1
answer
710
views
Abelian subfactors, a relevant concept?
Through the questions below, this post asks whether the concept of abelian subfactor is relevant.
Remark : here abelian qualifies an inclusion of II$_1$ factors $(N \subset M)$, $N$ is not an abelian ...
- 2,821
6
votes
1
answer
175
views
Lifting a semicommutative von Neumann algebra into an algebra of operator-valued functions
Given a complete probability space $\Omega$ and a von Neumann algebra $\mathcal M$, it is well-known that the tensor product von Neumann algebra $\mathrm L^\infty(\Omega)\overline\otimes\mathcal M$, ...
- 2,830
11
votes
2
answers
638
views
von Neumann algebras as C*-algebras with multiplicative conditional expectation $A^{**}\to A$
Let $A$ be a C*-algebra. We identify $A$ with its canonical image in the bidual $A^{**}$. Consider the following conditions:
(1) $A$ is a von Neumann algebra.
(2) There is a multiplicative ...
- 2,830
4
votes
0
answers
211
views
Irreducible representations of $\mathrm{UHF}_n$
I have a question about the irreducible representations of the $C^*$-algebra $\mathrm{UHF}_n = \bigotimes_{k=1}^\infty M_n$.
For every sequence of unit vectors $(\xi_k)$ in $\mathbb C^n$ there is a ...
- 759
3
votes
0
answers
116
views
Automorphisms of the injective envelope
Let $A$ be a separable $C^∗$-algebra and $(I(A),\kappa)$ be its injective envelope. WLOG assume that $I(A)$ is a monotone complete $C^*$-algebra, and $\kappa:A\to I(A)$ is the identity map.
Let $\...
- 2,605
7
votes
3
answers
697
views
Jordan-Hölder theorem for subfactors?
All the subfactors $(N\subset M)$ are irreducible and finite index inclusions of II$_1$ factors.
First recall that in this paper, D. Bisch characterizes the Jones projections $e_K$ of the ...
- 2,821
1
vote
1
answer
97
views
Choosing a net of projections from a given collection
Let $\mathcal M$ be a von Neumann algebra. Let $S$ be a nonempty set. Consider $\Omega$ to be the collection of all finite subsets of $S$. Suppose for all $\omega\in\Omega$ we have $I_\omega$ a ...
- 5,031
2
votes
1
answer
101
views
Hyperexpectations from injective subfactors of a type $II_1$ factor
Let $M$ be a type $\rm{II}_{1}$ factor with trace $\tau$, acting by the GNS representation on $B(L^{2}(M,\tau))$. Let $R\subset M \subset B(L^{2}(M,\tau))$ be a hyperfinite $\rm{II}_{1}$ subfactor of $...
- 4,351
3
votes
0
answers
304
views
Is the fundamental group of a maximal subfactor always $\mathbb{R}_{+}^{*}$?
The fundamental group $\mathcal{F}(N \subset M)$ of a unital inclusion of II$_{1}$ factors $N \subset M$ is defined as : $\mathcal{F}(N \subset M) =\{t >0 \ | \ (N \subset M)^{t} \simeq (N \...
6
votes
1
answer
195
views
Is there an element in an infinite tensor power of a C*-algebra that is invariant under finite permutations?
Let $A$ be a C*-algebra, and consider the infinite tensor power $A^{\otimes {J}}$, where $J$ is infinite (we consider the minimal or maximal tensor product). To any finite permutation, which is a ...
5
votes
1
answer
158
views
Backwards stable factors
A factor $R$ is called stable if $M_n(R)\cong R$ for all $n>0$. For the sake of this question, we call a factor backwards stable if $R\cong M_n(S)$ implies $S\cong R$ where $S$ is allowed to be any ...
- 759
2
votes
0
answers
119
views
Random matrices may be asymptotically free but never free themselves?
It is well known that independent $N\times N$ unitarily-invariant random matrices (or independent families of random matrices) may be asymptotically free as $N\to \infty$ with respect to the ...
- 21
4
votes
1
answer
172
views
The centralizer of a normal state on a type III$_1$ factor
Let $M$ be any type III$_1$ factor.
Does there must exist a normal state $\rho$ on $M$ such that the centralizer $M_{\rho}$ of $\rho$ is a factor?
- 35.4k
5
votes
0
answers
72
views
Braided monoidal category of (generalized) operator algebras
In the paper "Quantum Collections" Kornell (2016) proved that the category of $W^*$ (von Neumann) (and iirc also $C^*$) algebras, equipped with a suitable choice of tensor product, forms a ...
- 1,084
2
votes
0
answers
147
views
The injective envelopes of UHF algebras
The following is perhaps trivial common folklore for the knowledgeable people in the field: Let $H=\ell^2$. Let $V\subseteq B(H)$ be the Cartan factor of type IV, the self-adjoint operator space that ...
- 2,605
1
vote
0
answers
86
views
Inner product on Standard form of von Neumann algebra
Let $(M, H, H_+,J)$ be a standard form of a von Neumann algebra $M$ acting on a complex Hilbert space $H$ endowed with a self-dual cone $H_+$. Is it true that
$$\langle x,yz\rangle=\langle zx,y\rangle$...
- 131
19
votes
3
answers
711
views
Almost isometric linear maps
Say that a linear map $\varphi : B(\mathcal H) \rightarrow B(\mathcal H)$ is $\epsilon$-almost isometric if
$$ 1 - \epsilon \leq \lVert\varphi(a)\rVert \leq 1+\epsilon, \quad \forall a\in B(\mathcal H)...
- 12.9k
9
votes
0
answers
240
views
What is known about when $vN(G)$ is a factor, for a locally compact group $G$?
When $G$ is a discrete group, it is an elementary result in the theory of von Neumann algebras that the group von Neumann algebra $vN(G)$ is a factor if and only if $G$ is an ICC group.
What is known ...
- 146
3
votes
0
answers
258
views
Von Neumann algebras as complemented subspaces
Question: Does there exist a non-injective von Neumann algebra $M\subseteq B(H)$, which is a complemented Banach subspace of $B(H)$?
According to an MO post, this problem was still open as of 2013. I'...
8
votes
2
answers
519
views
Orthogonal basis of ${\bf Sym}_n(\mathbb R)$, made of orthogonal matrices
My question is motivated by this one, but within real matrices instead of complex ones.
${\bf Sym}_n(\mathbb R)$ is a vector space of dimension $N=\frac{n(n+1)}2$. Equipped with the scalar product $\...
- 156k
2
votes
1
answer
170
views
Defining states on von Neumann algebras from filters on the projection lattices
Let $M$ be a von Neumann algebra, $P(M)$ be its projection lattice, and $\mathcal{F}$ a proper filter on $P(M)$. Does there exist a state $\varphi$ (not necessarily normal) s.t. $\varphi(p) = 1$ for ...
- 42.8k
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) ...
- 42.8k
1
vote
1
answer
180
views
Conditioning a $\mathrm{C}^*$-algebra state with infinite precision
This question (and a second part) have been asked at MSE and gone through two bounties without an answer. I have been beating my head at it for a while without success.
Let $\mathcal{A}$ be a unital $\...
- 42.8k
5
votes
1
answer
165
views
Is norm-continuous representation factored through a Lie quotient group?
I asked this 11 days ago at MSE, but there was no answer, I hope people here could help.
Let $G$ be a locally compact group, and $X$ a Hilbert space. A unitary representation $\varphi:G\to B(X)$ is ...
- 7,426
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 $\...
- 380
-3
votes
1
answer
325
views
Is the category of $Z^* $ algebra equivalent to the category of $C^*$ algebras
A $Z^*$ algebra is a $C^*$ algebra whose all positive elements are zero divisor.
They form a category with usual structures.
Question. Is this category equivalent to the category of $C^*$ algebras?
...
- 356
12
votes
4
answers
877
views
Can you describe the image of the exponential map $B(H)\to B(H)$?
James Tener asks at the 20-questions seminar:
The exponential map $\exp:B(H)\to B(H)$ is just defined by its Taylor series. Can you describe its image?
- 6,357
6
votes
0
answers
207
views
What is the standard groupoid model of the Cuntz algebra?
I know that the Cuntz algebras $\mathcal{O}_n$, $n=1,2,...,\infty$, have groupoid models. I.e. they can be realised as groupoid C*-algebras. Can you describe the standard groupoid model for $\mathcal{...
- 191
2
votes
1
answer
163
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 ...
- 23k
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
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