All Questions
Tagged with oa.operator-algebras von-neumann-algebras
504 questions
2
votes
1
answer
356
views
The fundamental group of the von Neumann algebra of a free group of infinite rank
It is well-known that that the fundamental group (in the sense of Murray and von Neumann) of the factor $L(F_{\mathbb{N}})$ is
$\mathbb{R} \smallsetminus \{0\}$. I think that by the cutting and ...
- 481
2
votes
1
answer
158
views
Reference request for a type III action of a group on a manifold
Let an action of a group $\Gamma$ on a manifold $M$ such that $L^{∞}(M)⋊Γ$ is a type $III$ factor.
André Henriques posted here the following comment :
I don't know the literature, so I can't point to ...
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:\...
- 1,879
2
votes
1
answer
186
views
Von Neumann algebras with isomorphic sets of partial isometries
Given a von Neumann algebra $M$, let
$$
S(M) = \{u\in M: uu^*u=u\}
$$
be the set of partial isometries in $M$. Given $u,v\in S(M)$, it is well known that $uv \in S(M)$, provided $u^*u$ ...
- 2,263
2
votes
1
answer
141
views
A congruence relation on the projection lattice
This question is a continuation of what I asked here. Tristan Bice showed the following nice result there:
Let $A$ be a von Neumann algebra and $P$ its projection lattice, ordered by $p\leq q\...
- 1,731
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
2
votes
1
answer
324
views
Direct proof a property of hyperstonean spaces
First, let me state some basic facts and definitions for my question. I believe these are well-known among experts working on von Neumann algebras, but let me state them anyway since my question is ...
- 1,586
2
votes
0
answers
93
views
Amenability and the unitary group of an operator algebra
Let $M$ be a von Neumann algebra and $U(M)=\{x\in M: x^*=x^{-1}\}$ be its unitary group. In this post, we equip $U(M)$ only with the relative weak$^*$ topology $\sigma(M,M_*)$. Then, $U(M)$ is a ...
- 2,605
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
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
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
2
votes
0
answers
121
views
Invariant weights associated to algebraic quantum groups
Consider an algebraic quantum group $(A, \Delta)$ in the sense of Van Daele, i.e. a regular multiplier Hopf $^*$-algebra with a positive left integral $\varphi$ and a positive right integral $\psi$.
...
- 175
2
votes
0
answers
68
views
What about the structure theory in Baer *-rings?
In the literature, Baer *-rings are called as the algebraic analogue of von Neumann alegars.
It is well-known that
Theorem. Every von-Neumann algebra is decomposed into a direct sum of the algebras of ...
- 4,058
2
votes
0
answers
176
views
Projections in von Neumann algebra tensor product
Let $\mathcal M\subseteq B(\mathcal H)$ be a von Neumann algebra with normal faithful semifinite trace $\tau.$ Consider the von Neumann algebra $\mathcal N:=L_\infty([0,1])\overline{\otimes}\mathcal M$...
- 1,879
2
votes
0
answers
192
views
Almost periodicity and approximation in tracial von Neumann algebra
Let $N$ be a von Neumann algebra with a faithful normal tracial state $\tau$. For a countable group $G$, let $\sigma: G\rightarrow \text{Aut}(N)$ be a $G$- action on $N$ which preserves the tracial ...
- 73
2
votes
0
answers
136
views
McDuff-to-hyperfinite step in Connes' Injectivity $\Rightarrow$ Hyperfiniteness
In Connes' "Classification of Injective Factors" (1976) the last step in Injectivity $\Rightarrow$ Hyperfiniteness (Thm. 5.1) is the implication 2. $\Rightarrow$ 1., where
$N \cong R$,
a) $...
- 121
2
votes
0
answers
105
views
Comparing two quantities related to the norm of an inner derivation
Let $M$ be a von Neumann algebra sitting in $B(H)$.
Let $U(M)$ denote the unitary group of $M$.
Let $I(M):=\{\tau\in M\,|\,\tau=\tau^*=\tau^{-1}\}$ the set of involutions in $M$.
Let $SAC(M):=\{h\in M\...
- 71
2
votes
0
answers
139
views
Fixed point subalgebra
Suppose that $M$ is a von Neuman algebra and we have an action of a finite group $G$ on $M$. Denote by $M^{G}$ the fixed point subalgebra and suppose that $M^{G}=\mathbb{C}$ (i.e., we have an ergodic ...
- 59
2
votes
0
answers
112
views
Anticommutation of convolution products on trace class operators of quantum groups
This question was originally posted to MathStackExchange.
Let $\mathbb{G}$ be a locally compact quantum group and let $W$ and $V$ be the left and right fundamental unitaries, i.e., they implement the ...
- 59
2
votes
0
answers
113
views
Induction of group von neumann algebra for group homomorphism with amenable kernel
Let $\alpha:H\to G$ be a group homomorphism betwenn discrete countable groups, and assume that the kernel of $\alpha$ is an amenable group, denoted by $K$. I would like to know ...
- 2,630
2
votes
0
answers
99
views
Convergence of Brown measures
For each $n\in \mathbb N$, let $\mathcal M_n$ be a finite von Neumann algebra with a faithful trace $\tau_n$. Fix a non-principal ultrafilter $\omega$ on $\mathbb N$. Let $\mathcal M^\omega$ be the ...
- 1,418
2
votes
0
answers
156
views
Extension of a theorem of Bisch to cyclotomic integers of fixed degree
Theorem 3.2 in this paper (Bisch, 1994) states that for a finite depth ${\rm II}_1$ subfactor of integral index, every intermediate subfactor are also of integral index. As an application, every such ...
2
votes
0
answers
135
views
On crossed product of L^{P} spaces
Let $M$ be a von Neumann algebra with faithful normal state $\varphi$, and $G$ be a group action on $M$ preserving $\varphi$. Then is it true
\begin{align*}
L^{p}(M\rtimes G, \varphi^{M\rtimes G})\...
- 677
2
votes
0
answers
62
views
On $L^{1}(M',\tau')$
Let $M$ be a $\mathrm{II}_{1}$ factor with the faithful trace $\tau$ in standard form. Suppose $\tau'(x')=\tau(Jx'^{*}J)$. If we complete $M$ with $\|\cdot\|_{1}$ norm, i.e., $\|x\|_{1}=\tau(|x|)$, ...
- 677
2
votes
0
answers
100
views
On relative commutant inside crossed product
Let $G$ be a discrete group acting on vN algebra $M$ in standard form. My question what is relative commutant of $M$ and $L(G)$ infact what is $M'\cap (M\rtimes G)$ and $L(G)'\cap(M\rtimes G)$?
- 677
2
votes
0
answers
108
views
On standard form of corners
Let $(M, \tau)$ be a $\mathrm{II_{1}}$ factor equipped with the faithful normal tracial state $\tau$ acts on $L^{2}(M,\tau)$ is in standard form. The question is: if we consider vN algebras $PMP(P\in ...
- 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
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
2
votes
0
answers
710
views
What is spectral multiplicity for multiplication operators in general von Neumann algebra set up?
When two multiplication operators $M_{f}$ and $M_{g}$ acting on $L^2(X,\mu) $and $L^2(Y,\nu)$ are unitary equivalent? How multiplicity function look like here? What is the spectral multiplicity in ...
- 227
2
votes
0
answers
212
views
Tensor product of traces of von Neumann algebras
I am trying to understand how to define tensor product of normal semifinite faithful (in short n.s.f) traces between two von Neumann algebras $(M_1,\tau_1)$ and $(M_2,\tau_2),$ where $\tau_i$ is the n....
- 455
2
votes
0
answers
164
views
An operator valued Egoroff's theorem
The following statements suggests $B(H)$-valued Egoroff's theorem when $H$ is a separable Hilbert space. Probably it will be hold even if a von Neumann algebra $M$ whose predual is separable is ...
- 4,058
2
votes
0
answers
205
views
relative amenability of von Neumann algebra
Let $\cal{M}$ be a finite von Neumann algebra and $\cal{N}$ be a von Neumann subalgebra of $\cal{M}$.
The von Neumann algebra $\cal{M}$ is is amenable relative to
$\cal{N}$ if there exists a norm ...
- 565
2
votes
0
answers
157
views
Support vectors and relative modular operator
I'm studying the relative modular operator and I'm looking for o good text to do it. Until now I'm using Araki's papers but I don't know how to deal with the support of a vector, $s^M(\xi)$, which is ...
2
votes
0
answers
165
views
Rank–nullity theorem for finite von Neumann algebras
The rank-nullity theorem states that for $U, V$ finite dimensional vector spaces and $T:U \to V$ a linear map $$\dim(U) = \dim(im(T)) + \dim(ker(T)) $$
Let $M \subset B(H) $ be a finite von Neumann ...
2
votes
0
answers
101
views
Are there infinitely many amenable Hadamard-Petrescu subfactors?
The complex Hadamard matrices of dimension $n$ are used to build index $n$ subfactors through the commuting square construction. For more details, see the paper Subfactors and Hadamard Matrices by W....
2
votes
0
answers
132
views
Uniqueness of the tensor product decomposition of subfactors
A subfactor $(N \subset M)$ is indecomposable if (for $N_i \subset M_i)$: $$(N \subset M) = (N_1 \otimes N_2 \subset M_1 \otimes M_2) \Rightarrow \exists i \ N_i = M_i$$
Then, a subfactor $(N ...
2
votes
0
answers
412
views
Two Definitions of Non-commutative $L^p$ space
Throughout, let $(\mathcal{M},\tau)$ be a von Neumann algebra $\mathcal{M}$, acting on a Hilbert space $H$, with normal semifinite faithful trace $\tau$.
In the survey article by Pisier and Xu, the ...
- 75
2
votes
0
answers
149
views
Planar algebraic translation of a subfactor property
Let $N \subset M$ be an irreducible finite depth and finite index subfactor.
$M$ is a completely reducible (algebraic) $N$-$N$ bimodule, it decomposes into irreducibles as follows :
$$M=\bigoplus_{...
2
votes
0
answers
481
views
Versions of the spectral theorem
Since any $C^*$-algebra can be represented as an algebra of bounded operators $\mathcal{B(H)}$ on a Hilbert space $\mathcal{H}$, the spectral theorem applies to all $C^*$-algebras:
($*$) $A=\int_{\...
- 585
2
votes
0
answers
158
views
About the classification of infinite depth irreducible finite index maximal subfactors
The Temperley Lieb subfactors $A_{\infty}$ are the first examples of infinite depth irreducible finite index maximal subfactors. We can see these subfactors as coming from the simple Lie group $SU(2)$....
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
2
answers
148
views
Show convergence of net associated to GNS-triplet associated to state on a $C^*$-algebra
Let $A\subseteq B \subseteq B(H)$ be an inclusion of $C^*$-algebras where $H$ is some Hilbert space. We have the following conditions:
B is a von Neumann algebra with $A'' = B$.
The inclusion $A \...
- 175
1
vote
2
answers
282
views
On topology in von Neumann algebras
Suppose $M$ is von Neumann algebra, $A$ is $*$-algebra in $M$, further if $(A)_1$, the unit ball of $A$ is strong operator closed, does it implies $A$ is von Neumann algebra? I started proving this ...
- 227
1
vote
1
answer
611
views
Commutant of a von Neumann algebra as the linear span of unitaries.
I'm reading chapter 4 of Gerard Murphy's C*-algebras book and am confused by a statement in his proof of theorem 4.1.10. In his proof, he says, "$A'$ is the linear span of its unitaries" (where $A'$ ...
1
vote
1
answer
188
views
Uniqueness of the predual of a W*-algebra
Consider the following fragments in Takesaki's "Theory of operator algebras" (volume I):
Question: So, we have an abstract Banach space $F$ with $A \cong F^*$. In Lemma 3.6, one considers ...
- 175
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
1
vote
1
answer
211
views
Tensor product of faithful normal states is faithful
I know that given C*-algebras $A, B$ with faithful states $\omega,\varpi$, the tensor product state $\omega\otimes\varpi$ on the minimal tensor product $A\otimes_{\text{min}}B$ is faithful.
I also ...
- 439
1
vote
1
answer
164
views
Two invariants for type III factors
There are two invariants for the type $III$ factor $M$, namely, $S(M)$ and $T(M)$.
When $S(M)=[0, \infty)$, $M$ is a factor of type $III_{1}$.
My question : how to determine whether $M$ is a factor of ...
- 427
1
vote
1
answer
125
views
On existence of certain operators in von Neumann algebra
Let $M\subset B(\mathcal{H})$ be an infinite dimensional vN algebra in standard form. Fix $\xi\neq 0 \in \mathcal{H}$, does there exist $M\ni x_{\xi}\neq I$ such that $x_{\xi}(\xi)=\xi$?
- 181
1
vote
1
answer
683
views
About separability of von Neumann algebras [closed]
Is a von Neumann algebra always separable in the $\sigma$-weak topology? If not, give a counterexample. Under what conditions will it be separable?
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