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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Image of $PB_n$ in Temperley-Lieb algebra
Fix a ring $R$ (say $\mathbb{Z}[t, t^{-1}]$) and choose some $\delta\in R$. We can then define the Temperley-Lieb algebra $TL_n$ as being generated by
$U_1, \dots, U_{n-1}$ and subject to the ...
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"Project" an operator outside of a von Neumann Algebra into it
Suppose $W$ is a proper von Neumann Algebra contained in $B(H)$ and the identity in $W$ is the identity mapping of $H$ (namely, $W$ does not have non-trivial null space).
Given a self-adjoint $T\in W$...
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non centrally free actions of ameanable groups on the hyperfinite III_1 factor
Let $R$ be a hyperfinite $\mathit{III}_1$ factor, and let $Out(R)$ be its set of automorphisms modulo inner automorphisms. There is a canonical and important homomorphism $\phi:\mathbb R\to Z(Out(R))$ ...
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Measurability of a net
Let $(f_\epsilon)_{\epsilon>0}$ be a family of positive measurable functions on $L_p(\mathbb R)$ where $1<p<\infty.$ Assume that the pointwise supremum $f^*(x)=\sup_{\epsilon>0}|f_\epsilon(...
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Monotone convergence theorem for increasing net of positive functions
Suppose that we have $(\Omega,\mu)$ a $\sigma$-finite measure space. I have the following question.
(Assume that $(f_i)_{i\in I}$ be an increasing net of positive measurable functions such that $f_i\...
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Semi-commutative von Neumann algebras
Suppose $\Omega$ is a $\sigma$-finite measure space with measure $\mu.$ Let $\mathcal M\subseteq B(H)$ be a von Neumann algebra.
Can an element of $L_\infty(\Omega)\overline{\otimes}\mathcal M$ be ...
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Takesaki lemma 1.16 (volume II, chapter VII)
I am trying to understand the proof of the implication $(i)\implies (ii)$ in Takesaki's book "Theory of operator algebras II", chapter VII, which says the following:
The relevant setting ...
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What about the structure theory in Baer *-rings?
In the literature, Baer *-rings are called as the algebraic analogue of von Neumann alegars.
It is well-known that
Theorem. Every von-Neumann algebra is decomposed into a direct sum of the algebras of ...
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1
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Impact of annihilators in C*-algebras
Let $A$ be a unital C*-algebra. Let $S\subseteq A$. We put $$\operatorname{Ann}_r(S)=\{a\in A : \forall s\in S,~ ~as=0\}$$
Suppose that $A$ satisfies the following property:
For every $S\subseteq ...
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Projections in von Neumann algebra tensor product
Let $\mathcal M\subseteq B(\mathcal H)$ be a von Neumann algebra with normal faithful semifinite trace $\tau.$ Consider the von Neumann algebra $\mathcal N:=L_\infty([0,1])\overline{\otimes}\mathcal M$...
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Takesaki volume II chapter VII lemma 1.15
Consider the following fragments from Takesaki's second volume "Theory of operator algebras" in chapter VII on weight theory, lemma 1.15. Here $\mathcal{M}$ is a von Neumann algebra and $\...
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Relating different definitions of dual of a compact quantum group
Let $\mathbb{G}$ be a compact quantum group in the sense of Woronowicz. We can look at its associated dense Hopf$^*$-subalgebra $\mathbb{C}[\mathbb{G}]$. Hence, in the framework of multiplier Hopf $*$-...
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Types of normal states in the injective III$_1$ factor
The injective type III$_1$ factor is isomorphic to the Araki-Woods factor $R_{\infty}$.
I wonder how many types of normal states on $R_{\infty}$.
Haagerup mentions that there exist two different ...
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Defining the abstract tensor product of W*-algebras via a universal property
I am playing around a bit with $W^*$-algebras, and I'm trying to come up with a definition for the $W^*$-algebraic tensor product. Here is my first attempt:
It is easy to show that such an object ...
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Decomposition of an element as a difference of positive elements in the definition subalgebra of a weight (Takesaki)
I originally asked this on MSE, but did not get an answer there.
Let $M$ be a von Neumann algebra. Let $\varphi: M_+ \to [0, \infty]$ be a weight on $M$. Consider
\begin{align*}&\mathfrak{p}_\...
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Almost periodicity and approximation in tracial von Neumann algebra
Let $N$ be a von Neumann algebra with a faithful normal tracial state $\tau$. For a countable group $G$, let $\sigma: G\rightarrow \text{Aut}(N)$ be a $G$- action on $N$ which preserves the tracial ...
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A list of unital Banach *-algebras whose left annihilators are principal
Let $A$ be a unital Banach *-algebra. For a given subset $S\subseteq A$, the left annihilator of $S$, denoted by Ann($S$), is given by Ann$(S)=\{x\in A: xS=0\}$. Let us say Ann($S$) is principal if ...
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The center of a representation von Neumann algebra, and finite index subgroups
Consider a (countable) group $G$, a subgroup $H\leq G$ of finite index, and a unitary representation $\pi:G\to \mathcal{U}(\mathcal{H})$.
If the center of the von Neumann algebra $\pi(H)''$ is finite ...
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$L_1$-subspace of the predual of a von Neumann algebra
If $M$ is a type $II$ von Neumann algebra, then the predual has a complemented subspace isometric to $L_1(0,1)$. It follows from the existence of expectation. However, I don't know whether such a ...
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Families of representations of von Neumann algebras
Let $A$ be a von Neumann algebra and let $H$ be a (separable) Hilbert space.
It is known (see e.g., Section IV, Thm. 5.5 of Takesaki I) that there exists a Hilbert space $K$ such that $A \subset \...
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Extending a $\sigma$-weakly continuous map: Takesaki IV.5.13
Consider the following fragment from chapter IV in Takesaki's book "Theory of operator algebra I":
Why is the boxed line true? Takesaki argues that
$$\theta_0: \mathscr{M}_1\otimes_{\...
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Predual theorem proof in Takesaki's volume I
Consider the following fragment from Takesaki's book "Theory of operator algebra I" (Section III.3 ,p133-134).
Why is the boxed line true? I can see that $\epsilon: \widetilde{A}\to A$ is ...
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Takesaki: Lemma about enveloping von Neumann algebra
Consider the following lemma with proof from Takesaki's book "Theory of operator algebra I" (p121):
It appears to me that Takesaki claims at the end of the proof that $\pi(A)_1$ is $\sigma$-...
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Takesaki's proof of the Kaplansky density theorem
Consider the following fragment from Takesaki's book "Theory of operator algebra I":
Why is the boxed sentence true? It looks like they replace $A$ by its strong$^*$-closure. Is this ...
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Tomita–Takesaki theory and subfactors
Let $M$ be a von Neumann algebra acting on a Hilbert space $H$. Let $\Omega$ be a cyclic and separating vector in $H$. Let $J$ and $\Delta$ be the corresponding modular conjugation and modular ...
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Matrix units in von Neumann algebras, and $K_0$ groups
This question arises from trying to understand the proof of Lemma 3.1.4 in De Commer, Martos, and Nest - Projective representation theory for compact quantum groups and the quantum Baum–Connes ...
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Tensor product of a von Neumann algebra and $L_\infty $
Let $R$ be the hyperfinite $II_1$-factor. We know that $R$ is isomorphic to $R\otimes R$. So, $L_\infty(0,1) \otimes R$ is a von Neumann subalgebra of $R$.
I am not sure whether it is sure for any ...
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Sequences in von Neumann algebras
Let $(x_n)$ be a sequence in a von Neumann algebra $M$ or its predual $M_*$.
Is there a hyperfinite von Neumann subalgebra $N$ of $M$ such that $(x_n)\subset N$ or $N_*$?
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McDuff-to-hyperfinite step in Connes' Injectivity $\Rightarrow$ Hyperfiniteness
In Connes' "Classification of Injective Factors" (1976) the last step in Injectivity $\Rightarrow$ Hyperfiniteness (Thm. 5.1) is the implication 2. $\Rightarrow$ 1., where
$N \cong R$,
a) $...
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Takesaki theorem 2.6
I originally posted this question on MSE and didn't get a satisfactory answer, even after putting a bounty on it. Hence, I thought I should ask here:
Consider the following theorem in Takesaki's book &...
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Following a calculation of entropy in the first quantization scheme
I'm trying to follow the computations of example 5.1 in this paper. To begin with they have a symplectic Hilbert space $(\mathcal{K},\tau,\sigma)$, where $(\mathcal{K},\tau)$ is a separable Hilbert ...
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Comparing two quantities related to the norm of an inner derivation
Let $M$ be a von Neumann algebra sitting in $B(H)$.
Let $U(M)$ denote the unitary group of $M$.
Let $I(M):=\{\tau\in M\,|\,\tau=\tau^*=\tau^{-1}\}$ the set of involutions in $M$.
Let $SAC(M):=\{h\in M\...
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Measurable structures for direct integrals
I'm working with the notion of direct integrals as in Dixmier. Briefly: Given a measurable space $X$ and a family of separable Hilbert spaces $(H_x)_{x\in X}$, a measurable structure is a subspace $Y$ ...
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Premeasurability of affiliated operators for type $\textrm{III}$ von Neumann algebras
$\DeclareMathOperator\dom{dom}$If $M\subset B(H)$ is a semifinite von Neumann algebra with faithful, normal, semifinite trace $\tau$, then a closed operator $T:H\rightarrow H$ intertwining the action ...
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Uniqueness of the predual of a W*-algebra
Consider the following fragments in Takesaki's "Theory of operator algebras" (volume I):
Question: So, we have an abstract Banach space $F$ with $A \cong F^*$. In Lemma 3.6, one considers ...
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About $\sigma$ strong$^*$-functionals and seminorms
I'm reading the book "Theory of operator algebras" by Takesaki. In this book, the $\sigma$-strong$^*$ topology on the space $B(H)$ (bounded operators on the Hilbert space $H$) is defined (...
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Fixed point subalgebra
Suppose that $M$ is a von Neuman algebra and we have an action of a finite group $G$ on $M$. Denote by $M^{G}$ the fixed point subalgebra and suppose that $M^{G}=\mathbb{C}$ (i.e., we have an ergodic ...
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Anticommutation of convolution products on trace class operators of quantum groups
This question was originally posted to MathStackExchange.
Let $\mathbb{G}$ be a locally compact quantum group and let $W$ and $V$ be the left and right fundamental unitaries, i.e., they implement the ...
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Closability of a natural bimodule map between cyclic correspondences of von Neumann algebras
Let $M$ and $N$ be von Neumann algebras, and $\mathcal{H}$ a cyclic $M-N$ correspondence with unit cyclic vector $\xi$. For which $\eta\in \mathcal{H}$ is the bimodule map extending $\xi\mapsto \eta$ ...
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Normal states on a type III$_1$ factor
Let $M$ be a type III$_1$ factor. Suppose $\rho$ is a normal state on $M$, given any $c\in [0,2]$, can we find a normal state $\rho'$ on $M$ such that $\|\rho-\rho'\|=c$? Or can we find a sequence of ...
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Two invariants for type III factors
There are two invariants for the type $III$ factor $M$, namely, $S(M)$ and $T(M)$.
When $S(M)=[0, \infty)$, $M$ is a factor of type $III_{1}$.
My question : how to determine whether $M$ is a factor of ...
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Existence of quantum states given reduced states on subsystems
Suppose $$\mathcal{H}=\bigotimes_{i\in I} \mathcal{H}_i$$ is a tensor product of Hilbert spaces, where $I$ is some index set. Given a $J\subset I$, let $$\mathcal{H}_J=\bigotimes_{i\in J} \mathcal{H}...
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Strong contractibility of unitary group of properly infinite von Neumann algebras
In the introduction of their 1993 paper (see reference below), Popa and Takesaki write
As it turns out, in these topologies [the weak and strong topology] $U(\mathscr{H})$ is again contractible (cf.
[...
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Twisted crossed product von Neumann Algebras
I asked a question over on Math.stackexchange a few days ago, but it didn't get much activity. Hopefully this question isn't considered too elementary by the standards of Mathoverflow. Here is what I ...
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Direct integral decomposition relative to a given measure space
It is well known that a separable Hilbert space $H$ decomposes as a direct integral in the presence of an abelian von Neumann algebra $\mathscr A\subseteq B(H)$.
More precisely, and quoting from ...
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Show convergence of net associated to GNS-triplet associated to state on a $C^*$-algebra
Let $A\subseteq B \subseteq B(H)$ be an inclusion of $C^*$-algebras where $H$ is some Hilbert space. We have the following conditions:
B is a von Neumann algebra with $A'' = B$.
The inclusion $A \...
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Opposite $C^*$ algebras
$\DeclareMathOperator\op{op}$Let $A$ be a $C^*$-algebra. We know that $A$ admits a natural operator space structure, namely the operator space structure induced by any faithful $*$-representation of $...
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Covariant representations and crossed products of von Neumann algebras
Let $(M,G,\alpha)$ be a $W^\ast$-dynamical system with $G$ locally compact abelian (I am mostly interested in the case $G=\mathbb{R})$. A covariant representation of $(M,G,\alpha)$ is a pair $(\pi,u)$ ...
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Factor states on C*-algebras
Which C$^*$-algebras admit factor states for which the von Neumann algebra it generates in the corresponding GNS representation is a type III$_1$ factor? For example, do all purely infinite algebras ...
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Dual operator space
Suppose $E$ is an operator space and $E^*$ is the dual operator space. It is well known that the matricial norm structure on $E^*$ is given by the formula $\|[f_{ij}]_{i,j=1}^n\|_{M_n(E^*)}:=\sup\{\|...
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