All Questions
Tagged with oa.operator-algebras von-neumann-algebras
504 questions
7
votes
0
answers
268
views
Enveloping von Neumann algebra of Clifford algebra
As explained in the book "Spinors in Hilbert Space" by Plymen and Robinson, if $V$ is a complex (separable) Hilbert space with a real structure, and $\mathrm{Cl}(V)$ the corresponding Clifford algebra,...
- 9,853
0
votes
0
answers
54
views
On cyclicity of a module
Let $A$ be a $\text{ von Neumann algebra }$, $\mathcal{H}$ is a cyclic $A$ module, $G$ be a finite group acting on $A$, is $\mathcal{H}$ cyclic module over fixed point subalgebra of the action? ...
- 677
0
votes
0
answers
68
views
On existence of sequence of unitaries in $II_{1}$ factor $M$
Let $M$ be a $\mathrm{II}_{1}$ factor acting on $L^{2}(M, \tau)$ in standard form, let $\{e_{n}:n \in \mathbb{N}\}$ be fixed orthonormal basis of $L^{2}(M, \tau)$, does there exist sequence of ...
- 677
1
vote
0
answers
118
views
Some doubt on crossed product von Neumann algebras
There are two definitions in different books. Let $G \curvearrowright M$, then there is the definition of forming Group ring $M[G]$, define product and addition then make its algebra. represent the ...
- 677
2
votes
1
answer
83
views
On existence of fixed point operator
Let $M$ be an infinite dimensional non-type $I$ factor, given $\xi$ in $\mathcal{H}$, does there exist a not identify operator $x$ in $M$ such that $x\xi=\xi$, I have tried with taking projection $P_{\...
- 677
-1
votes
2
answers
640
views
Invariance of spectrum under conjugation
Let $T$ be a self-adjoint invertible operator on $\mathcal{H}$ with a continuous spectrum, means the spectral measure is nonatomic. For which class of invertible operators $V$( with continuous ...
- 677
1
vote
1
answer
125
views
On commutant of $II_{1}$ factors
Suppose $M$ is $II_{1}$ factor but need not be in standard form. Under what condition (on $M$ or Hilbert space) is the commutant $M'$ of $M$ again $II_{1}$ factor on the Hilbert space acted by $M$?
- 677
1
vote
0
answers
160
views
Projections in tensor product of vN algebras
Can we write any projection in the tensor product vN algebra $M\otimes N$ in terms of limits of projections $p\otimes q$, where $p$ and $q$ are projections in M, N or somewhat relate the projections ...
- 677
1
vote
1
answer
96
views
On boundedness of sequence of operators in vN algebra
Let $x_{n}$ be a sequence of operators in vN algebra $M$, $\Omega$ is a cyclic vector for $M$, if $x_{n}\Omega$ converges in $\mathcal{H}$, can we say there exist a subsequence $\{y_{n}\}$ of $\{x_{n}\...
- 677
2
votes
0
answers
66
views
Separating vector on dense subalgebra
Suppose $M$ be a vN algebra and $U$ be a S.O.T dense self-adjoint subalgebra of $M$ has separating vector, does $M$ have? If not give a counterexample. Or there is a condition on M like type II_{1} or ...
- 677
0
votes
1
answer
372
views
A question on standard form in von Neumann algebra
Let $M$ be a vN algebra (represented GNS space with respect to state) in standard form. Under which condition we can say a subalgebra $B$ of $M$ is also in standard form? If there exist $\varphi$ ...
- 677
3
votes
1
answer
344
views
Commutant of subalgebra of tensor product
Consider the von Neumann subalgebra of $M\otimes M$ by $ B= \mathrm{vN} \{T\otimes T: T\in M\}$. What is the commutant of B?
- 677
0
votes
0
answers
77
views
On cyclicity of fixed point algebra of flip automorphism
Let $M$ be a von Neumann algebra having a cyclic vector in $\mathcal{H}$, is the fixed point subalgebra under the flip automorphism on $M\otimes M$ has a cyclic vector in $\mathcal{H}\otimes \mathcal{...
- 677
1
vote
0
answers
68
views
Studying fixed point algebra under group action
If $M$ is in standard form, consider the action of a finite group on $M$, does the fixed point subalgebra under the action is in standard form? What we can say if $M$ is hyperfinite $\mathrm{II}_{1}$ ...
- 677
1
vote
1
answer
126
views
Subalgebras of $II_{1}$ factor
Let $M$ be a type $II_{1}$ factor, Let $B$ is an infinite dimensional nonabelian subalgebra. Is it true that $B$ always type $II_{1}$ ?
- 677
2
votes
1
answer
67
views
Analogue of spectral values of automorphisms in vN algebra
Is there any analog of studying spectral properties of automorphisms of von Neumann algebra? Does it make sense, if anybody knows please give a reference.
- 677
1
vote
2
answers
354
views
Regarding Haagerup $L^{P}$ spaces
There is a definition in Haagerup's paper on $L^{P}$ spaces for weights, my question is after putting the norm is it become semifinite $L^{P}$ space on the crossed product? I am not clear please help. ...
- 677
6
votes
2
answers
598
views
Why are ultraweak *-homomorphisms the `right' morphisms for von Neumann algebras (and say, not ultrastrong)?
Both the ultraweak and ultrastrong topologies are intrinsic topologies in the sense that the image of a continuous (unital) $*$-homomorphism between von Neumann algebras (in either topology) is a von ...
- 159
0
votes
0
answers
143
views
On $s$-numbers in finite von Neumann algebra
$T$ is an operator in $M$, $M$ is finite von Neumann algebra. There is a notion of singular value function that is ($s$-numbers). My question is: what is $s$-number for tensor product of two operators ...
- 677
2
votes
1
answer
368
views
On diagonal part of tensor product of $C^*$-algebras
Suppose we have a $C^*$-algebra $\mathcal{U}$, Consider the $C^*$-subalgebra generated by elements of the form $a\otimes a$, what is it isomorphic to? Is it isomorphic to $\mathcal{U}$ itself?
- 677
3
votes
1
answer
206
views
Ultraproduct of non-commuative $L^p$-spaces
Let $1<p<\infty.$ Let $I$ be a non-empty set and $\mathcal{U}$ be an ultrafilter over $I.$ Let $M_i$ be von Neumann algebras equipped with normal faithful semifinite traces $\tau_i,$ $i\in I.$ ...
- 1,879
11
votes
2
answers
490
views
Actions of locally compact groups on the hyperfinite $II_1$ factor
Let $R$ be the hyperfinite $II_1$ factor, and let $G$ be a locally compact group.
(1) Does there always exist a continuous, (faithful) outer action of $G$ on $R$?
(2) If so, how does one ...
- 43.2k
3
votes
2
answers
241
views
Polar decomposition of tensor product of operators in von Neumann algebra
If $T=V|T|\text { and } S=W|S|$ is the polar decomposition of $T$. Is it true that the polar decomposition of $T\otimes S$ is $T\otimes S=(V\otimes W)(|T| \otimes |S|)$. If $T$ and $S$ are self-...
- 677
3
votes
2
answers
264
views
Ultraweak topology in abelian von Neumann algebras
Let $A$ be an abelian von Neumann algebra acting on the (not necessarily separable) Hilbert space $\mathcal{H}$ (with identity $I$). From the Gelfand-Neumark theorem, there is a compact Hausdorff ...
- 159
0
votes
2
answers
294
views
Computing multiplicity function for self adjoint operator with nonatomic spectral measure
Suppose $T$ is a self-adjoint operator in $B(H)$ with $\sigma(T)$ a spectrum of $T$. $\mu$ is a spectral measure. For the operators having a generally continuous spectrum how to calculate the ...
- 677
1
vote
0
answers
83
views
Sequence of unitaries in type III von Neumann algebra
Consider a type III von Neumann algebra $\mathcal{M}$ and an isometry $w$. How does one show that there exists a sequence of unitaries $u_n\in\mathcal{M}$ that converge strongly to $w$?
For instance,...
10
votes
0
answers
120
views
Morita equivalence for graded von Neumann algebras
I am interested in understanding Morita equivalence of $Z_2$-graded von Neumann algebras. In the ungraded case, Rieffel showed that all Type I factors are Morita-equivalent, while for Type III factors ...
- 1,579
-1
votes
1
answer
180
views
On spectral multiplicity of left shift operators
Let $U$ be an operator defined on $l^{2}(\mathbb{Z})$ by $U(e_{n})=e_{n-1}$, where $e_{n}$ is an orthonormal basis of $l^{2}(\mathbb{Z})$. $U$ is a left shift operator. Since $U$ is unitary operator ...
- 227
6
votes
1
answer
494
views
Property $\Gamma$ in terms of Correspondences
A type $II_{1}$ factor $M$ with trace $\tau$ has Property $\Gamma$ if for every finite subset $\{ x_{1}, x_{2},..., x_{n} \} \subseteq M$ and each $\epsilon >0$, there is a unitary element $u$ in $...
- 7,047
5
votes
0
answers
119
views
Pimsner-Popa basis dealing with higher relative commutants
Let $(N \subseteq M)$ be a finite index unital inclusion of ${\rm II}_1$ factors. Let $e_1$ be the Jones' projection.
A finite subset $\{\lambda_i, i \in I\} \subset M $ is called a (right) Pimsner-...
6
votes
1
answer
397
views
Real rank 0 implies stable rank 1 on $C^\ast$-algebras?
A $C^\ast$ algebra has defined stable rank (https://www.univie.ac.at/nuhag-php/bibtex/open_files/2079_Rieffel-StableRank.pdf) and real rank (https://core.ac.uk/download/pdf/82123484.pdf), which are ...
- 233
3
votes
1
answer
252
views
What is the story behind this Hilbert space in the definition of Hilbert Modules
Here is Deflnition 1.5 of Hilbert module in "L^2-invariants: theory and applications to geometry and K-theory", Springer-Verlag, 2002, by W. Lück:
A Hilbert $\mathcal N(G)$-module $V$ is a Hilbert ...
- 2,118
7
votes
1
answer
491
views
Projections in the tensor product of von Neumann algebras
This question seems elementary, but I have already asked an expert who does not know the answer, so I would like to post here.
Let $M$ and $N$ be von Neumann algebras, and let $M\bar{\otimes}N$ be ...
- 619
5
votes
1
answer
208
views
Cartan subalgebras in the group algebras of virtually abelian groups
Let $G$ be a virtually abelian group. Are there any general results on the existence or non-existence of Cartan subalgebras in the generated group $C^*$-algebra or group von Neumann algebra?
- 741
6
votes
2
answers
248
views
Extension of a von Neumann algebra by a von Neumann algebra
I asked this question at MSE now I repeat it at MO:
Let $A,B,C$ be $3$ unital $C^*$ algebras. Assume that we have the following short exact sequence of $C^*$-algebras:
$$0\to A\to C\to B\...
- 356
1
vote
0
answers
83
views
Are these kinds of "crossed product" studied?
Let $M$ be a von Neumann algebra acting in a Hilbert space $H$, and let $\rho$ be a representation of a group $G$ on a Hilbert space $K$. Define $M\rtimes_\rho G$ to be a von Neumann algebra acting in ...
- 2,118
1
vote
0
answers
74
views
About crossed product of the group von Neumann algebra
Let $G$ be a group with a normal subgroup $N$. If $K$ is a field, then it is well known in ring theory that $K[G]\cong K[N]*(G/N)$ ($*$ stands for the crossed product). Do we have such an isomorphism ...
- 2,118
3
votes
1
answer
388
views
Fundamental group and group measure space construction
Let $N$ be a type ${\rm II}$ factor, with trace $\tau$. Consider its fundamental group$$ \mathcal{F}(N)= \{ \tau(p)/\tau(q) \ | \ p,q \text{ non-zero finite projections in } N \text{ and } pNp \simeq ...
6
votes
0
answers
232
views
Group $C^*$ vs group von-Neumann algebras
Let $\Gamma$ be a countable (discrete) group (in what follows, make additional assumptions as you wish). Let $C^*_r(\Gamma)$ and $W^*_r(\Gamma)$ be the reduced $C^*$-algebra respectively the reduced ...
- 9,853
5
votes
1
answer
306
views
Cartan subalgebra and group measure space construction
Let $N$ be a ${\rm II}_1$ factor. A maximal abelian self-adjoint subalgebra (MASA) is a $*$-subalgebra $A \subset N$ such that $A' \cap N = A$. It is called a Cartan subalgebra if moreover $\mathcal{N}...
1
vote
1
answer
172
views
Examples of isomorphic W* algebra with non-homeomorphic weak topology
Due to the uniqueness of the predual, a W* algebra, when realized as a von Neumann algebra in any way, always has a unique, well-defined ultraweak (or $\sigma$-weak) topology. The same can be said ...
- 1,586
3
votes
2
answers
207
views
Commutative direct summands of C*-algebras
I have a question about commutative direct summands of $C$*-algebras.
Let $A$ be a $C$*-algebra (with unit) and suppose that its bidual $A^{**}$ has a commutative direct summand, that is, $A^{**}=B\...
- 633
1
vote
1
answer
193
views
When is $\inf_{n\geq0}x^n\neq0$?
Let $M$ be a von Neumann algebra acting on a Hilbert space $H$. Let $x$ be a positive element of $M$ with $\|x\|=1$. So, $(x^n)_{n\in\mathbb N}$ is a decreasing sequence of positive elements and $y:=\...
- 2,118
1
vote
0
answers
185
views
Unitary element of the group algebra
Let $G$ be a torsion free group. Are unitary elements of $\mathbb CG$ studied? By unitary element I mean an element $\alpha$ in $\mathbb CG$, such that $\alpha^*\alpha=1$? Do the triviality of unitary ...
- 2,118
3
votes
0
answers
109
views
Does this element belong to $\mathbb CG$?
Let $G$ be a torsion-free group. Let $\alpha$ be a symmetric element of $\mathbb CG$, i.e. $\alpha^*=\alpha$, with $\|\alpha\|_1=\sum|\alpha(g)|<1$, so $\beta:=\sum_{n\ge 0}(-1)^n\alpha^n$ is an ...
- 2,118
32
votes
3
answers
2k
views
What does it mean for a category to admit direct integrals?
Given an infinite countable group $G$, the category of unitary representations of $G$ admits direct integrals.
Namely, given a measure space $(X,\mu)$ and a measurable family of unitary $G$-reps $(H_x)...
- 43.2k
7
votes
1
answer
732
views
To what extent can a von Neumann algebra be determined by its projection lattice structure?
Let $ M, N $ be von Neumann algebras, $ P $ (resp. $Q$) the projection lattice of $M$ (resp. $N$). Any isomorphism $ \varphi : M \to N $ on the level of involutive algebras induces an isomorphism $ \...
- 1,586
2
votes
1
answer
708
views
Topology of state space in von Neumann algebras
What are the sufficient conditions for a von Neumann algebra to have a first countable set of states with respect to the weak * operator topology?
- 123
2
votes
0
answers
201
views
An example of non trivial projections in a group von Neumann algebra
Let $G$ and $\text{vN}(G)$ be a torsion free group and its group von Neumann algebra. Is there a characterization of non trivial projections in $\text{vN}(G)$? If not, is a certain class of them ...
- 2,118
3
votes
1
answer
192
views
An analytical zero divisor
Let $G$, $\mathbb C[G]$ and $\text{vN}(G)$ be a torsion free group, it's group ring and group von Neumann algebra, resp.. Let $0\neq\alpha\in\mathbb C[G]$ and $0\neq p\neq1$ is a projection in the ...
- 2,118