Questions tagged [von-neumann-algebras]

Subtag of the [oa.operator-algebras] tag for questions about von Neumann algebras, that is, weak operator topology closed, unital, *-subalgebras of bounded operators on a Hilbert space.

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comparison of two projections in a non-factor von Neumann algebra

In a factor $M$, we know that for any two projections $P$ and $Q$ in $M$, either $P\preceq Q$ or $Q\preceq P$ holds true. Here $\preceq$ denotes the Murray-von Neumann subequivalence of two ...
Manish Kumar's user avatar
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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 ...
Andrei Jaikin's user avatar
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Inflating the double dual of a C*-algebra (matrix algebra of double dual)

in this post I would like to discuss the fact that If $A$ is a $C^*$-algebra, then $M_n(A^{**})\cong M_n(A)^{**}$, as mentioned in Brown and Ozawa. I can't really see it. Actually, it is enough for me ...
Just dropped in's user avatar
2 votes
1 answer
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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\...
passerby51's user avatar
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Normal linear functionals on bicommutants of C*-algebras

I am going through the proof of the Sherman-Takeda theorem and Fillmore's book "A User's Guide on Operator Algebras" seems to have a nice approach, but something seems off to me: We need to ...
Just dropped in's user avatar
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What is a type $\text{II}_1$ factor von Neumann algebra?

After finding formal definitions in various texts (see, eg, Witten, Notes On Some Entanglement Properties Of Quantum Field Theory, Rev. Mod. Phys. 90, 45003 (2018), doi:10.1103/RevModPhys.90.045003 ...
Victor V Albert's user avatar
6 votes
2 answers
858 views

What are applications of Jones polynomial on von Neumann algebras?

I have read according list of below papers a basic connection between Jones polynomial and statistical mechanics is that the Kauffman bracket or Kauffman polynomial a polynomial invariant of knots is ...
zeraoulia rafik's user avatar
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Intuition for conformal nets

I was planning on reading the work of Arthur Bartels, Christopher L. Douglas and André Henriques on the 3-category of conformal nets as discussed in these papers: Coordinate-free nets, Conformal ...
Chetan Vuppulury's user avatar
4 votes
1 answer
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When a normal functional is restricted to a vn Neumann sub-algebra

I have already asked this question and no comment(s) received up to now. I am so curious to get feedback concerning the problem. Let $M$ be a vn Neumann subalgebra in $B(H)$. Let $f$ and $g$ be ...
ABB's user avatar
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Weak Hopf algebra structure on twisted group algebra

A (normalized) $2$-cocycle on a finite group $G$ with values in $S^1$ is a map $\sigma:G\times G\rightarrow S^1$ such that $$\sigma(g,h)\sigma(gh,k)=\sigma(h,k)\sigma(g,hk)$$ and $$\sigma(g,e)=\sigma(...
Keshab Bakshi's user avatar
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The double dual of the unitization of a $C^*$-algebra

I am studying the proof that if $A$ is a $C^*$-algebra such that $A^{**}$ is a semidiscrete vN algebra, then $A$ has the completely positive approximation property (CPAP). I was able to handle the ...
Just dropped in's user avatar
8 votes
2 answers
521 views

Are (completely) positive maps approximated by normal (completely) positive maps?

Let $\mathcal{H}$ denote a Hilbert space and $B(\mathcal{H})$ denote the algebra of all bounded operators on $\mathcal{H}$. By recognizing the (Banach) dual of $B(\mathcal{H})$ with the double dual of ...
Manish Kumar's user avatar
3 votes
1 answer
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Reference for "the algebra of multiplication by all measurable bounded functions acts in Hilbert space in a unique manner"

I read a paper of Alain Connes on "Duality between shapes and spectra" and in page 4, he says Due to a theorem of von Neumann the algebra of multiplication by all measurable bounded ...
dohmatob's user avatar
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Noncommutative torus as a von Neumann algebra

Le $\theta$ be irrational. One can define the noncommutative torus $A_{\theta}$ as a universal algebra generated by two unitaries $u,v$ satisfying the relation $vu=e^{2 \pi i \theta} uv$. This is an ...
truebaran's user avatar
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Can one associate a "nice" topos to a von Neumann algebra?

The question here inspires my present question. Reyes proves here that the contravariant functor Spec from the category of commutative rings to the category of sets cannot be extended to the category ...
Jon Bannon's user avatar
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"Non-group" ${\rm II}_1$ factors

Do we have existence/examples/criteria for a ${\rm II}_1$ factor that is not isomorphic to $L(G)$ for any group $G$?
Chilperic's user avatar
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Which complete orthomodular lattices arise from von Neumann algebras?

Let $A$ be a von Neumann algebra. Then a classic observation is that the set of projections $\Pi(A)$ is naturally a complete orthomodular lattice. Question 1: Is the construction $A \mapsto \Pi(A)$ a ...
Tim Campion's user avatar
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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 ...
Sebastien Palcoux's user avatar
11 votes
1 answer
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Are groups with the Haagerup property hyperlinear?

In his 2008 paper Hyperlinear and Sofic Groups: A Brief Guide, Pestov asked (Open Question 9.5) whether every group with the Haagerup property is hyperlinear (or sofic). Has this question been ...
MaoWao's user avatar
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Action of a finite group on a finite factor

Question: Let $G$ be a finite group and let $P$ be a $\rm II_1$ factor. Assume that $G$ acts on $P$ in a trace-preserving manner, such that the crossed product algebra $P \rtimes G$ is a factor. Is $G ...
Beginner Samya's user avatar
8 votes
1 answer
422 views

Is $L(\mathbb{Z}*\mathbb{Z}_{2})$ a free group factor?

This is a reference request for something that is likely to be well-known to operator algebraists. I will not, therefore, include the technical definition of free product of finite von Neumann ...
Jon Bannon's user avatar
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Monotone series of projections converging to 1 in von Neumann algebra

The following statement is being used a lot in the literature, and I wonder how to prove it. Let $M$ be an infinite-dimensional von Neumann algebra (with unit element), show that there is an ...
dreamwave's user avatar
3 votes
2 answers
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Jordan isomorphisms of type I von Neumann algebras

Let $\mathcal M$ and $\mathcal N$ be two von Neumann algebras. A linear map $J:\mathcal M\to\mathcal N$ is said to be a Jordan isomorphism if $J$ is bijective, $*$-preserving and $J(xy+yx)=J(x)J(y)+J(...
A beginner mathmatician's user avatar
2 votes
1 answer
252 views

A question on quantum tori

Let $\mathbb T_\theta^2$ be quantum tori generated by two unitary operators $u,v$. can $u,v$ be finite dimensional?
A beginner mathmatician's user avatar
4 votes
1 answer
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Characterizing the Haagerup property of finite von Neumann algebras via unbounded derivations

A correspondence $_{N} H_{N}$ is a Hilbert space with two normal commuting left/right representations of $N$ in $B(H)$. The following characterization of property (T) for finite von Neumann algebras ...
Adrián González Pérez's user avatar
1 vote
1 answer
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Explanation of $\sigma$-weak topology von a von Neumann algebra [closed]

Let $A$ be a von Neumann algebra. I want to understand the precise meaning of the $\sigma$-weak topology on $A$. What I understand so far is the following: The $\sigma$-weak topology, which we will ...
Shirley Richardson's user avatar
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63 views

Limit of spectral projection of increasing sequence of positive operators

Let $\mathcal M$ be a von Neumann algebra. Suppose $(x_\alpha)\subset \mathcal M$ is bounded increasing net of positive operators converging to a positive operator $x\in\mathcal M.$ Is it true that $\...
A beginner mathmatician's user avatar
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Index of a particular subfactor

If a compact group $G$ acts on vN algebra factor $M\subset B(L^{2}(M))$, what would be the index of subfactor $[M^{G}:M]$? KIndly explain the answer.
sibani's user avatar
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Is there a non-irreducible maximal subfactor other than two-sided TLJ?

A subfactor $N \subseteq M$ is called: irreducible if $N' \cap M = \mathbb{C}$, maximal if for any intermediate subfactor $N \subseteq P \subseteq M$ then $P=\{N,M \}$. The two-sided ...
Sebastien Palcoux's user avatar
2 votes
1 answer
386 views

Proof of uniqueness of predual of von Neumann algebra

I am currently reading Jesse Peterson's lecture notes on von Neumann algebras. I'm confused by lemma 4.4.2. In particular, it seems to me that Hahn Banach theorem here can only conclude the map $\phi$ ...
user151245's user avatar
13 votes
0 answers
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Connes Embedding Conjecture is false [closed]

This preprint from yesterday claims to prove that Connes Embedding Conjecture fails. Since the paper is from outside Operator Algebras (Computer Science/Quantum Computing) and they actually work on ...
Martin Argerami's user avatar
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Action of hyperbolic group on von Neumann algebra

Let $G$ be a hyperbolic group. Let $M$ be a vN algebra in standard form. Can there exist a faithful action of $G$ on $M$ such that \begin{align*} \sigma_{g_n} \rightarrow I \end{align*} for some ...
sibani's user avatar
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On mixing and weak mixing subalgebras of finite von Neumann algebras

Let $M$ be a full $\mathrm{II}_1$ factor. Consider mixing and weak mixing subfactors $B$ and $C$ of $M$. Are $B$ and $C$ full?
sibani's user avatar
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Property gamma for type III factors

I am struggling to find a definition which uses centralizing sequences for property gamma in type III factors? If $M$ is type $\mathrm{III}$ factor, what is the exact definition of property gamma ...
sibani's user avatar
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2 votes
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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})\...
user136400's user avatar
2 votes
1 answer
525 views

Ultrapower of an ultrapower of von Neumann algebras

Let $M$ be a $\mathrm{II}_1$ factor, fix a ultrafilter $\omega$, we know the ultrapower $M^{\omega}$, is again $\mathrm{II}_1$ factor. The question is that what is the ultrapower of $M^{\omega}$, i.e.,...
sibani's user avatar
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5 votes
2 answers
832 views

Unusual crossed product constructions being factors

Let $A$ be an abelian von Neumann algebra and $G$ a countable group acting on $A$. In the literature we meet usually two kinds of crossed product $A \rtimes G$ being a factor: if the action is (...
Sebastien Palcoux's user avatar
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On prime factors

Let $M$ be a prime $\mathrm{II}_1$ factor. Let $N$ be a non hyperfinite finite index subfactor $N$, is $N$ prime factor?
sibani's user avatar
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7 votes
0 answers
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Abstract characterization of group von Neumann algebra (II1 factor)

The group von Neumann algebra $L\Gamma$ is a factor if and only if the group $\Gamma$ is ICC (i.e. infinite conjugacy class property). Moreover if $\Gamma$ is nontrivial then $L\Gamma$ is a $\mathrm{...
Sebastien Palcoux's user avatar
0 votes
1 answer
287 views

On conditional expectation from tensor products

Let $M$ be a $\mathrm{II}_{1}$ factor. Does there exist a conditional expectation from $M^{\otimes 2}$ to $M$ preserving the trace $\tau^{\otimes 2}$?
user136400's user avatar
3 votes
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84 views

Finite index subfactors of hyperfinite type $\mathrm{III}_{\lambda}$ factors

Let $M$ be a hyperfinite type $\mathrm{III}_\lambda$ factor. $N$ be a finite index type $\mathrm{III}$ subfactor, is it true $N$ is hyperfinite type $\mathrm{III}_{\lambda}$?
user136400's user avatar
8 votes
1 answer
223 views

Is the invariant subalgebra of the von Neumann algebra $L(F_k)$ isomorphic to $L(F_k)$?

Let the symmetric group $G=S_{k}$ act on the von Neumann algebra of the free group $L(F_k)$ via permuting its generators. Is the fixed point algebra under the action isomorphic to the whole algebra, i....
user136400's user avatar
3 votes
0 answers
158 views

Given non-type-I subfactors $R \subset S$, must $S$ have a projection that meets no projection in $R$ except $1$?

Let $R \subset S$ be distinct non-type-I von Neumann factors; say two projections $P, Q \in S$ "meet" if they have a common non-null subprojection (i.e. if $P \wedge Q \neq 0$), and call $P$ "$R$-...
Doug McLellan's user avatar
0 votes
0 answers
106 views

On an application dominated convergence theorem in vN algebras

$M$ be a $\mathrm{II}_{1}$ factor equipped with the faithful normal trace $\tau$ in the standard form. Let $\tau(Jx'J\eta)=0,\forall x' \in M'$ and fixed $\eta$ in $L^{1}(M,\tau)$. Is it true $\eta=0$?...
user136400's user avatar
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., $\|x\|_{1}=\tau(|x|)$, ...
user136400's user avatar
4 votes
0 answers
112 views

On existence of property gamma of C star simple group von Neumann algebra

We call a finite vN algebra $M$ with trace $\tau$ has property Gamma iff there exist unitary $u_{n}\rightarrow 0$ weakly and $\|xu_{n}-u_{n}x\|\rightarrow 0$. My question can we have C*-simple ...
sibani's user avatar
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2 votes
0 answers
90 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 $L(G)'\cap(M\rtimes G)$?
user136400's user avatar
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 $x_{\xi}(\xi)=\xi$?
sibani's user avatar
  • 181
31 votes
0 answers
893 views

Is this representation of Go (game) irreducible?

This post is freely inspired by the basic rules of Go (game), usually played on a $19 \times 19$ grid graph. Consider the $\mathbb{Z}^2$ grid. We can assign to each vertex a state "black" ($b$), "...
Sebastien Palcoux's user avatar
0 votes
0 answers
88 views

On invertibility of ergodic averages

Let $x$ be invertible unbounded operator affiliated operator to the $\mathrm{II_{1}}$ factor $(M,\tau)$. Under which condition on $x$, the iterates also $1+\sigma(x)+\cdots+\sigma^{n}(x)$ are ...
sibani's user avatar
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