# 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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### 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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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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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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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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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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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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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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### 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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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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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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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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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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### 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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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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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 ...

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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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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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### 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 ...

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### 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 ...

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### 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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### 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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### 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 ...

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### 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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### 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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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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### 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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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 $...

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### 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 ...

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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 ...

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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 $L^{p}(M,\tau)$?