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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35 views

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 ...
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2answers
53 views

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(...
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1answer
137 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?
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1answer
119 views

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 ...
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1answer
73 views

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 ...
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47 views

Support of commuting elements in a von Neumann algebra

Let $\mathcal M$ be a semifinite von Neumann algebra and $a$ be an unbounded positive self-adjoint operator affiliated to $\mathcal M$. Suppose $x\in\mathcal M$ is such that $x$ commutes with all ...
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37 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 $\...
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1answer
53 views

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.
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101 views

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 ...
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1answer
196 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 $\phi$ ...
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2k views

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 ...
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1answer
134 views

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 ...
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77 views

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?
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103 views

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 ...
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109 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})\...
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1answer
331 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.,...
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1answer
320 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 (...
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502 views

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?
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291 views

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{...
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1answer
141 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 $\tau^{\otimes 2}$?
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68 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 $\mathrm{III}_{\lambda}$?
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125 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....
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144 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 $P$ "$R$-...
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89 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 $\eta=0$?...
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54 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., $\|x\|_{1}=\tau(|x|)$, ...
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90 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 ...
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73 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 $L(G)'\cap(M\rtimes G)$?
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36 views

On conditional expectation onto subalgebras of crossed product

Let $G$ be a discrete group, Let $G$ acts on vN algebra type $\mathrm{III}$ i.e. $(M,\varphi)$ preserving the non-tracial state $\varphi$. Let $A$ be a subalgebra of $M\rtimes G$, Does always exist ...
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1answer
118 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 $x_{\xi}(\xi)=\xi$?
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718 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" ($b$), "...
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85 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 ...
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64 views

On extension of automorphism of M to automorphism of crossed product

Let $\sigma$ $\in$ $Aut(M)$, where $M$ is a vN algebra. If a discrete group $G$ acts on $M$ via $\alpha$, is it possible to extend $\sigma$ to the automorphism of $M\rtimes_{\alpha} G$? Special ...
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1answer
127 views

On analogue of ratio in operator algebras

For functions spaces we have ratio of two functions namley $\frac{f}{g}$, Can ratio of two operators in von Neumann algebra make sense? If at all it makes sense, what will be the proper definition of ...
3
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138 views

Odometer actions of groups

If a group $G$ acts on a Cantor set $(X,\mu)$ by odometers, my question is: what is the explicit automorphism $\alpha_{g}$ for the extended Koopman action on $L^{\infty}(X,\mu)$, for $g$ $\in$ $G$? I ...
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121 views

On examples of action of C-star simple group on von Neumann algebra

Can there exist a faithful action of a $C^{*}$-simple group $G$ on a von Neumann algebra $(M,\varphi)$ equipped with faithful normal state $\varphi$ such that action preserves the state $\varphi$ and ...
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1answer
189 views

A certain class of representations

Let $g$ be a non-identity element in a torsion-free amenable group, does there exist a finite-dimensional unitary representation $\pi$ with $\pi(g)\neq 1$? (The word "finite-dimensional" was ...
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141 views

Noncommutative analogue of Radon-Nikodym derivative

Let $\mathcal M$ be a noncommutative probability space i.e. there is a trace $\tau$ on $\mathcal M$ such that it is normal, faithful and $\tau(1)=1.$ Let $\tau_1$ be another normal trace on $\mathcal ...
4
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0answers
83 views

type III$_1$ states

Given a von Neumann algebra that is a type III$_1$ factor with the state $\omega$ and any $\epsilon>0$ is it always possible to find a projection or a partial isometry in the algebra such that its ...
3
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1answer
78 views

Converegence of modulus in nocommutative $L_p$-spaces

Let $1\leq p<\infty.$ Let $\mathcal M$ be a von Neumann algebra equipped with a normal semifinite faithful trace $\tau.$ Let $L_p(\mathcal M,\tau)$ be the associated noncommutative $L_p$-space. Let ...
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1answer
99 views

Extending $C^*$-norms from $*$-subalgebras

Let $A$ be a unital $*$-algebra, and $B$ a unital $*$-subalgebra of $A$. In addition, assume that there exists a $B$-$B$-sub-bimodule $C \subset A$, such that $$ A \simeq B \oplus C, $$ where $\...
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0answers
96 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 ...
2
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1answer
100 views

Normal $*$-homomorphism

Let $\pi:\mathscr M\to\mathscr M$ be a normal $*$-homomorphism between a von Neumann algebra $\mathscr M.$ Assume $\mathscr M$ has a normal semifinite faithful trace. Does $\pi$ extend as a bounded ...
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41 views

On weakly equivalent actions

If $G\curvearrowright (X,\mu)$ and $G\curvearrowright (Y,\nu)$ are weakly equivalent pmp actions(standard definition in literature),( where $G$ is discrete group and $\mu$, $\nu$ are probability ...
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2answers
307 views

Extension of trace on von Neumann subalgebra

Let $R$ be a finite von Neumann algebra and $S$ be a von Neumann subalgebra of $R$. Does every normal tracial state on $S$ extend to a normal tracial state on $R$?
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1answer
141 views

Separability of von Neumann algebra

In a lecture note, from where I am studying theory of von Neumann algebras, the author has commented that the following are equivalent. Let $A$ be a von Neumann algebra. $A$ is SOT separable. $A$ is ...
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0answers
86 views

On a doubt on spectral measure on Gelfand spectrum

In the lecture notes of Peterson https://math.vanderbilt.edu/peters10/teaching/spring2013/vonNeumannAlgebras.pdf he proves the following theorem THEOREM 2.7.5: Let $\mathcal H$ be a Hilbert space and $...
6
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1answer
168 views

Equivalence of $\sigma$-weak topology to another topology

Let $\mathcal H$ be a Hilbert space. Define a topology $\tau_1$ on $B(\mathcal H)$ by the family of seminorms $x\mapsto |Tr(xa)|,$ $a\in L^1(B(\mathcal H)).$ Here $B(\mathcal H)$ denotes the set of ...
5
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1answer
273 views

Variations on Kaplansky Density

Let $A$ be a $C^*$-algebra and $\pi:A\rightarrow B(H)$ a faithful $*$-representation, so $M=\pi(A)''$ is a von Neumann algebra and $A\rightarrow M$ is an inclusion. von Neumann's Bicommutant Theorem ...
3
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0answers
166 views

Is there a noncommutative version of von Neumann's ergodic theorem? [closed]

The two most celebrated ergodic theorems are Birkhoff's ergodic theorem and von Neumann's ergodic theorem. E. C. Lance in his remarkable work (Ergodic Theorems for Convex Sets and Operator Algebras) ...
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33 views

Extensions of group actions from von Neumann algebra $\mathrm {II_{1}}$ factor $M$ to its $L^{p}(M,\tau)$

Let $G$ be a countable discrete group acting on $M$, does the action of $G$ extend uniquely to the action on $L^{p}(M,\tau)$?

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