# 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

### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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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### 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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### 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 $R$?
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 ...