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.
436
questions
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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 ...
3
votes
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(...
2
votes
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?
4
votes
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 ...
1
vote
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 $\...
0
votes
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.
3
votes
0answers
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 ...
1
vote
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 $\phi$ ...
12
votes
0answers
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 ...
0
votes
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 ...
0
votes
0answers
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?
1
vote
0answers
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 ...
2
votes
0answers
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})\...
2
votes
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.,...
2
votes
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 (...
0
votes
0answers
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?
6
votes
0answers
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{...
0
votes
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 $\tau^{\otimes 2}$?
3
votes
0answers
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 $\mathrm{III}_{\lambda}$?
7
votes
0answers
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....
3
votes
0answers
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 $P$ "$R$-...
0
votes
0answers
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 $\eta=0$?...
2
votes
0answers
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., $\|x\|_{1}=\tau(|x|)$, ...
3
votes
0answers
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 ...
1
vote
0answers
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 $L(G)'\cap(M\rtimes G)$?
0
votes
0answers
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 ...
1
vote
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 $x_{\xi}(\xi)=\xi$?
28
votes
0answers
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" ($b$), "...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
3
votes
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
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
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 ...
1
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0answers
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 $\mathcal ...
4
votes
0answers
83 views
type III$_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
votes
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 ...
2
votes
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 $\...
1
vote
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
votes
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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votes
0answers
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 ...
3
votes
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 $R$?
1
vote
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 ...
1
vote
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
votes
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
votes
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
votes
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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votes
0answers
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 $L^{p}(M,\tau)$?
