All Questions
Tagged with operator-algebras or oa.operator-algebras
2,153 questions
4
votes
1
answer
250
views
Can we solve the FGF problem by finding an appropriate action?
If we can find an action of the free group $\mathbb{F}_2$ on a measure space $X$ such that the crossed product $M=L^∞(X)⋊\mathbb{F}_2$ is a ${\rm III}_1$ factor with core isomorphic to $L(\mathbb{F}_2)...
4
votes
0
answers
333
views
Baum Connes Conjecture [closed]
I have recently decided on a topic for my master thesis. I want to compare the Baum Connes conjecture as it is formulated in topology to the conjecture as it is formulated in functional analysis. I ...
- 49
2
votes
2
answers
1k
views
Does a conditional expectation from a von Neumann algebra to its center exist?
In a finite von Neumann algebra, the unique tracial state serves as one, then for a general von Neumann algebra, does it exist?
- 1,528
7
votes
1
answer
347
views
Separability of the C*-algebra in the definition of K-homology
There are (at least) two approaches to K-homoology: one is via the so called dual algebra which is due to Paschke. The second is via the Fredholm modules and is due to Kasparov. In Nigel Higson's book ...
- 9,330
7
votes
4
answers
1k
views
Quantum channels as categories: question 1.
A quantum channel is a mapping between Hilbert spaces, $\Phi : L(\mathcal{H}_{A}) \to L(\mathcal{H}_{B})$, where $L(\mathcal{H}_{i})$ is the family of operators on $\mathcal{H}_{i}$. In general, we ...
- 283
2
votes
2
answers
1k
views
the spectrum of matrix with positive entries
It is well known that a matrix which all entries are positive real numbers, has a positive eigenvalue.(see algebraic topology, by Allen Hatcher). Now is the following generalization, true?
Let A ...
- 356
2
votes
1
answer
370
views
On the second dual of $C[0,1]$
I have two questions on the second dual of $C[0,1]$:
R. D. Mauldin ([1]) proved that: For a given bounded linear functional
$T: C[0,1]^*\to \mathbb{C}$ there is a bounded function $\psi$ defined on ...
- 4,058
4
votes
1
answer
328
views
Tomita Takesaki theory and boundeness of $S$
Let $M$ be a von Neumann algebra, $\xi$-separating and cyclic vector for $M$. Let $S$ be antilinear operator acting as $x \xi \mapsto x^* \xi$ where $x \in M$. Then one can show that $S$ is closable ...
- 9,330
2
votes
1
answer
926
views
Eigenvalues and Compact Resolvent
For $A$ an unbounded (densely defined) operator on a separable Hilbert space, what conditions on its eigenvalues will show that, for $\lambda \notin $spec$(A)$, we have that $(A-\lambda)^{-1}$ is a ...
- 465
10
votes
1
answer
1k
views
Separating vectors for C$^*$-algebras
(I asked this on math.stackexchange, without response).
Let $A$ be a C$^*$-algebra, concretely acting on a Hilbert space $H$. Suppose that $\xi_0\in H$ is cyclic and separating for $A$ (that is, the ...
- 18.7k
10
votes
3
answers
1k
views
subgroup of SU(N) with maximal manifold dimension
Given the group SU(N) of NxN unitary matrices, does there exist a subgroup S
with a manifold dimension larger than the SU(N-1) manifold dimension and
smaller than the SU(N) one? S should not ...
- 1,207
6
votes
1
answer
321
views
Kernel of the natural map between group $C^*$-algebras
Let $\Gamma$ be a discrete group. We can form two $C^*$-algebras: the universal (or full) and reduced, to be denoted by $C^*_u(\Gamma)$ and $C^*_r(\Gamma)$ (respectively). Both of them are completions ...
- 9,330
5
votes
1
answer
239
views
second dual of minimal tensor products of $C^*$-algebras
Let $A$ be a unital $C^*$-algebras and $K(H)$ is $C^*$-algebras of compact operators on separable Hilbert space $H$. Is it true that $(A \otimes K(H))^{**}= A^{**} \overline{\otimes}B(H)$?
- 85
2
votes
1
answer
171
views
algebraic version and polar decomposition
I have been thinking about polar decomposition of $C^*$-algebras. I could not find a proper reference where it says: Let $A$ be a $C^*$-algebra, and $b$ an invertible element of $A$ with modulus $\...
4
votes
0
answers
187
views
Gaussian actions with no Bernoulli part
In an unrelated research project I came upon an example of a mixing unitary representation $\pi: \mathbb{F}_{\infty}\to B(\mathsf{H})$ of the free group on infinitely many generators, such that no ...
- 5,005
6
votes
1
answer
976
views
Approximate units from strictly positive elements in $C^{*}$-algebras.
The existence of a countable approximate unit in a $C^{*}$-algebra $B$ is equivalent to the existence of a strictly positive element $h\in B$. There are several ways to construct an approximate unit ...
- 213
5
votes
1
answer
310
views
C*-Algebras: Dynamics vs. Derivations
Problem
Given a C*-algebra $\mathcal{A}$.
Consider dynamics $\tau:\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$ and $\tau':\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$.
(More precisely, strongly continuous one-...
- 598
4
votes
1
answer
135
views
Is the module action $M\times M^*\to M^*$ jointly continuous?
Let $M$ be a W*-algebra and consider the following map:
$$\gamma: M\times M^*\to M^*: (a,f)\to af$$
where $af(b)=f(ba)$. Let us consider $M$ under the weak topology $\sigma(M,M^*)$ and $M^*$ under ...
- 4,058
10
votes
0
answers
300
views
Mackey's Program on Algebraic Ergodic Theory
I knew about Mackey's Program from Arnold's book Random Dynamical Systems and it referred to K. Schmidt's book Algebraic Ideas in Ergodic Theory, which was published in 1990. However, that is the ...
- 241
5
votes
0
answers
314
views
C$^*$-algebras in which the spectral radius is comparable to the norm
For every commutative C$^*$-algebra the spectral radius is equal to the norm. My question is:
For which C$^*$-algebras $\mathcal A$ does there exist a constant $C>0$ such that $$C\|a\| \leq ...
- 3,984
1
vote
2
answers
398
views
A question on unbounded operators
Assume that $H$ is a separable Hilbert space.
Is there a polynomial $p(z)\in \mathbb{C}[x]$ with $deg(p)>1$ with the following property?:
Every densely defined operator $A:D(A)\to D(A),\;D(A)\...
- 356
5
votes
1
answer
277
views
Power's Theorem for irreducible representations
Let $A_{\alpha}\subset B(H)$ be a bunch of unital C*-algebras acting on a Hilbert space $H$ given together with their character spaces $M(A_{\alpha})$'s. A very nice theorem of Stephen C. Power ...
- 83
4
votes
0
answers
224
views
Equations in finite subgroups of unitary groups
Let $n$ be an integer. Andreas Thom mentioned that Camille Jordan showed that there exists some $m \in {\mathbb N}$ (depending on $n$), such that for any pair of $n \times n$-unitaries $u,v \in U(n)$ ...
- 24.8k
2
votes
1
answer
307
views
When the reduced $C^*$-algebra of $\Gamma$ admits character then $\Gamma$ is amenable [closed]
Suppose that $C^*_r(\Gamma)$ admits some character (homomorphism into $\mathbb{C}$)-here $\Gamma$ is discrete group and $C^*_r(\Gamma)$ is the closure of the image of the group ring $\mathbb{C}\Gamma$ ...
- 9,330
5
votes
0
answers
149
views
Algebras involved to define spectral triple
A spectral triple consist from $(A,H,D)$ where $A$ is some unital $*$-subalgebra of $B(H)$ and $D$ is unbounded operator with compact resolvent such that for all $a\in A$ the commutator $[D,a]$ is ...
- 9,330
1
vote
1
answer
109
views
Continuous factors for invertible simple tensors
Our following question is motivated by this very interesting answer
Assume that $A$ is a $C^{*}$ algebra. Put $X=\{a\otimes b \mid a,b \in G(A)\}$ where $G(A)$ is the space of all ...
- 356
5
votes
0
answers
50
views
Non-existence of projections in crossed product
If $X$ is a smooth manifold on which a Lie group $G$ acts properly and cocompactly (meaning $X/G$ is compact), then one can find a compactly support cut-off function $c:X\rightarrow\mathbb{R}$ such ...
- 1,903
1
vote
1
answer
106
views
Norm of a cb-homomorphism restricted to a generating operator space
Let $\mathcal A \subset B(H)$ be an operator algebra and $\varphi: \mathcal A \rightarrow B(K)$ a completely bounded homomorphism. Suppose $\mathcal M \subset \mathcal A$ is an operator space such ...
- 3,984
3
votes
0
answers
101
views
Does von Neumann density imply strong additivity of a conformal net?
Let $\mathcal A$ be a conformal net, and let $\mathcal J$ be the set of all proper open sub-intervals of $S^1$.
We say that $\mathcal A$ satisfies von Neumann density, if for any representation $\pi$...
- 585
3
votes
1
answer
274
views
Finite codimensional subvector space of $C^{*}$ algebras which contains no invertible elements
Assume that $A$ is a unital $C^{*}$ algebra. Is there a subvector space $Y\subset A$ of finite codimension which does not contain any invertible element?
Let $n(A)$ be the infimum of such ...
- 356
12
votes
2
answers
1k
views
Do Burnside Group Factors have Gamma?
The Free Burnside group $G=B(2,665)=\langle a,b|g^{665} \rangle$ is infinite, by the work of Adyan and Novikov. Furthermore, the centralizer of any nonidentity element in $G$ is finite cyclic, and so ...
- 7,067
5
votes
1
answer
577
views
Is there a trivial construction of the trace on the Jones basic construction?
Let $N$ be a type $II_{1}$-factor with trace $\tau$, and $B$ a von Neumann subalgebra. The existence of the semifinite trace on the Jones basic construction $\langle N, e_{B} \rangle$ is reasonably ...
- 7,067
3
votes
2
answers
700
views
positive element in C* tensor product
Let A, B be two C*-Algebras and $A\otimes B$ denote their minimal tensor product(I don't know whether C*-norm matters or not, but for simplicity we can assume that one of them is nuclear so all C*-...
- 411
3
votes
1
answer
220
views
Existence of state on a C*-algebra satisfying $|\tau(ab)|=\|ab\|$
(This is a repost of a question from math.SE, https://math.stackexchange.com/questions/1240966/existence-of-state-on-a-c-algebra-satisfying-tauab-ab)
Let $a,b$ be elements of a unital C*-algebra $A$ ...
- 3,115
20
votes
2
answers
805
views
Nonseparable disintegration theory: references
I mean a theorem of the following kind. Let $A$ be a C*-algebra, and let $\pi: A\to B(H)$ be its representation. Then there exist a set $P$ with a positive measure $\mu$, a field of Hilbert spaces ...
- 1,152
6
votes
1
answer
446
views
A Question About the Elliott-Natsume-Nest Proof of Bott Periodicity
In Wegge-Olsen’s book K-Theory and C$ ^{*} $-Algebras, there is an outline of a proof of Bott Periodicity (the proof is due to George Elliott, Toshikazu Natsume and Ryszard Nest). The first step of ...
user36116
4
votes
0
answers
121
views
Almost order zero approximations- separability and localizations
I'm currently reading the paper The nuclear dimension of $C^*$-algebras by Winter and Zacharias.
I'm trying to understand the proof of:
Proposition 3.2.: Let $A$ be a $C^*$-algebra with $dim_{nuc}A=...
- 81
4
votes
1
answer
157
views
Terminology: jointly completely bounded?
This question has a subjective component but I would like answers that try to stick to concrete observable facts, such as which papers use which terminology. However, the informed impressions of those ...
- 25.8k
0
votes
1
answer
111
views
If $A$ is a $C^*$-algebra, then $H^1 (A, D(A)) = \{ 0 \}$ (first cohomology group )?
If $A$ is a $C^*$-algebra, then $H^1 (A, D(A)) = \{ 0 \}$ (first cohomology group )?
We don't know that is an open problem or it has counterexample...
- 19
7
votes
1
answer
703
views
A Question About Pure States, Support Projections and Central Covers
I am trying to study the paper Consistency of a Counterexample to Naimark’s Problem by Charles Akemann and Nik Weaver, and there is a claim in Lemma 1 of the paper that I am stuck at, which is as ...
user36116
4
votes
2
answers
241
views
Closure of all operator of order $0$ in $B(L^2(R^n))$
The following question arose as I was playing around a little bit with pseudo-differential operators and K-theory and so on.
Let $H^s$ be the Sobolev space of s-times weakly differentiable functions $...
- 2,998
7
votes
3
answers
498
views
Sums of unitaries with small norm in full group $C^*$-algebras
Suppose $G$ is a finitely generated group, with given generating set $S={g_1, \dots, g_n}$. (Assume that if $g\in S$, then $g^{-1}\notin S$. (EDIT: Also assume that $S$ is minimal in the sense that ...
- 2,361
4
votes
1
answer
154
views
Point-ultraweak limit of *-homomorphisms/cpc order zero maps
Suppose we have the following:
A C*-algebra $A$ and a von Neumann algebra $M$ (we can assume that $M$ is $\mathcal B(H)$).
A sequence of *-homomorphisms $\phi_i\colon A\to M$
an ultrafilter $\mathcal ...
6
votes
1
answer
245
views
Comparing cardinalities of the spectrum of two masas in $B(H)$
If I imagine that (the self-adjoint part of) a C*-algebra $A$ represents the algebra of observables of some quantum system, then certain perspectives on algebraic quantum theory would ask me to ...
- 5,407
4
votes
1
answer
250
views
Nuclearity noncommutative torus
I read that the Noncommutative torus (rotation algebra) is nuclear when $\theta\in\mathbb{R}\setminus\mathbb{Q}$. Unfortunately, I haven't found a proof. Could someone give me a reference and/or an ...
- 743
6
votes
2
answers
648
views
Normalizer of a von Neumann algebra
Let $(A,H)$ be a von Neumann algebra in standard form (which means that $H=L^2A$), and
recall that the automorphism group $Aut(A)$ acts on both $A$ and $H$.
Let
$$N:=\{u\in U(H): uAu^*=A\}$$
be the ...
- 43.2k
9
votes
5
answers
870
views
Abelianization of GL(H)
This is related to Theo's question about the abelianizations of finite dimensionsal Lie groups.
I am interested in a specific (infinite-dimensional) case of the above question. Let H be an infinite-...
- 629
1
vote
3
answers
209
views
Extending GUE to a measure on operators?
Let $\mathcal H_n$ denote the space of Hermitian $n\times n$ matrices and let $\mu_{GUE}$ denote the following measure on $\mathcal H_n$:
$$
\mu_{GUE}(dM) = \exp\left( -\frac{n}{2}\text{tr}\ M^2\right)...
- 1,329
10
votes
1
answer
776
views
Saito-Wright definition of Rickart C*-algebras
A C*-algebra is Rickart if for each $x\in A$ there is a projection $p\in A$ so that
$R(x)=pA$.
Here the right-annihilator $R(S)$ of $S\subset A$ is defined
as $$R(S)=\{a\in A\mid xa=0\, \forall x\...
- 810
5
votes
1
answer
2k
views
definition of operator valued integral with spectral measure
I am trying to make sense of some operators that come up on Buchholz and Summers' work on warped convolutions (two works on arxiv: 2008 and 2011).
There, they work on a Hilbert space $H$ and on the ...
- 342