Questions tagged [von-neumann-algebras]

A von Neumann algebra is a unital *-subalgebra of the algebra of bounded operators on a Hilbert space, closed in the weak operator topology. Also called a $W^*$-algebra, and may be regarded as a non-commutative generalization of $L^\infty$ space. These algebras are extensively used in knot theory, non-commutative geometry, and quantum field theory.

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Projections in a diffuse von Neumann algebra

If a von Neumann algebra $M$ has no minimal projections, we call it a diffuse von Neumann algebra. Suppose $e$ is a projection in a diffuse von Neumann algebra $M$. Since $M$ is diffuse, we can find ...
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Normality implies lower $w$-semicontinuity

I got stuck with the following problem while following Section $10.14$ of the book 'Lectures on von Neumann algebras' by Stratila and Zsido. Problem: Let $\mathcal{M}$ be a von Neumann algebra. Show ...
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noncommutative Holder inequality

I am learning the noncommunicative holder inequality. It is based on operator theory. I have two problems when i try to understand the proof of noncommunicative holder inequality Let $M$ be a von ...
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Conjugation with a unitary is $\sigma$-weakly continuous.

Let $M \subseteq B(H)$ be a von Neumann algebra. Let $U: H \to K$ be a unitary where $K$ is another Hilbert space and consider a faithful $*$-representation $$\pi: M \hookrightarrow B(K): m \mapsto ...
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Free $*$-algebras and inclusion [closed]

Let $A, B, C$ be $C^*$-algebras. Consider the free $*$-algebra generated by $A\sqcup B\sqcup C.$ Then we add relations $\{\text{all relations of A}\}\cup \{\text{all relations of B}\}\cup \{\text{all ...
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Slice maps on a von Neumann algebra completely positive?

Let $M\subseteq B(H)$ and $N\subseteq B(K)$ be von Neumann algebra and let $\omega$ be a positive (normal) functional on $M$. Consider the von Neumann algebraic tensor product $M \boxtimes N \...
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Comparision of two normal states

Suppose $\rho_1$ and $\rho_2$ are two normal states on a type III factor. If $\rho_1$ is not dominated by $k\rho_2$, where $k$ is a positive constant. Then there exits a nonzero projection $e$ such ...
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20 views

Minimal operator space structures on $C^*$ algebras

Let $\Omega$ be a $\sigma$-finite measure space with measure $\mu.$ Then the natural operator space structure on $L_\infty(\Omega)$ is defined to be the operator space structure induced by the ...
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A von neumann algebra has no minimal projections

Suppose that $M$ is a von Neumann agebra with no minimal projections. Let $p$ be a nonzero projection in $M$ and $\rho$ be a normal state on $M$. For any $\epsilon>0$, can we find a projection $e$ ...
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Does it hold $L_\infty(G_1 \times \ldots \times G_n) \simeq \overline{\otimes}_{1 \leq j \leq n} L_\infty(G_j)$?

I am thinking about analysis on locally compact abelian groups equipped with their respective Haar measure. In particular, let $n \geq 2$ and suppose $G_1$, $G_2$, $\ldots$, $G_n$ are LCA groups. We ...
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For a family of projections how to prove $\vee(I-E_a) \ge I-\wedge E_a$?

In the book "Fundamentals of the Theory of Operator Algebras" by Richard V. Kadison and John R. Ringrose at page 111 (a screenshot is attached below) the authors claims: Since the map $E \...
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Must a sequence of projections strongly converge to 0 if their values given by a faithful normal state converge to 0?

If $\omega$ is a faithful normal state on a von Neumann algebra $R$, and $P_1, P_2, \ldots$ are projection operators in $R$ such that $\omega(P_n) \rightarrow 0$ as $n \rightarrow \infty$, must the $...
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Proof of Kaplansky density theorem

I am learning Kaplansky density theorem, there is a problem as following: If $C$ is a unital $C^*$-subalgebra of $B(H)$ and $A$ is its SOT clousure, $ReA$ and $ReC$ is set of self adjoint elements in $...
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Commutant of a type $II_{\infty}$ von Neumann algebra

If $M$ is a type $II_{\infty}$ von Neumann algebra, can we conclude that the commutant $M'$ of $M$ is a type $II_{\infty}$ von Neumann algebra?
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Question about polar decomposition inside a von Neumann algebra.

Let $M \subseteq B(H)$ be a von Neumann algebra. Let $x = v|x|$ be the polar decomposition of $x$. It is well-known that $v,|x| \in M$. Is it true that the element $v^*v$ is an element of the von ...
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If $v$ is in the von Neumann algebra generated by $x$, then $v^*v$ is in the von Neumann algebra generated by $x^*x$.

Let $M \subseteq B(H)$ be a von Neumann algebra. Given $S \subseteq M$, let $VNA(S)$ be the von Neumann algebra generated by $S$. Note that I require von Neumann algebras to be unital, so $VNA(S) = S''...
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Deforming an approximate algebra into an exact algebra

Consider a linear subspace of matrices, $M \subset \textrm{Mat}_n(\mathbb{C})$, which is $\epsilon$-approximately closed under multiplication, i.e. for all $x,y \in M$, there exists $z \in M$ such ...
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Kaplansky's formula

I want to prove the Kaplansky's formula For two projections $p, q$, $$(p\lor q)-p\sim q-(p\land q).$$ A step where I'm stuck is in trying to prove that the projection onto the kernel of $(1-p)q$ is $...
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Group Von Neumann Algebra Application

This is a follow up question on Von Neumann Algebra of an Infinite Group. Let $\mathbb{Z}$ be the additive group of the integers. What is the Von Neumann algebra of $\mathbb{Z}$? Let $M$ be the Von ...
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Doubt in Murray-von Neumann equivalence

I'm trying to understand the following proof: I understand the proof entirely except the displayed formula after "Hence we have". Basically I don't see why $$e_{2n}-e_{2n+1} \sim f_{2n+1}-...
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Supremum of projections in von Neumann algebra

Let $\{P_i:i\in I\}$ be a set of projections in a von Neumann algebra $M$ and $J\subseteq I$ be finite. Then $P_J:= \bigvee_{i\in J} P_i$ is the smallest projection $P$ such that $\left(\sum_{i\in J}...
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When is a “block” of a von Neumann algebra itself a von Neumann algebra?

Consider a finite-dimensional Hilbert space $H$ and a von Neumann algebra $M$ on $H$, i.e., an algebra of operators on $H$ which is closed under Hermitian conjugation and contains the identity. ...
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Prove a von Neumann algebra is diffuse

In Lemma 2.2, The authors want to prove $M_{\varphi}$ is diffuse. How to check that $eMe$ falls in $A$?
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Properties of a state pulled back from quotient $\mathrm{C}^*$-algebra

Let $\mathcal{A},\,\mathcal{B}$ be $\mathrm{C}^*$-algebras and $\pi:\mathcal{A}\rightarrow \mathcal{B}$ a surjective unital $*$-homomorphism. Suppose self-adjoint $f\in \mathcal{A}$ has connected ...
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Von Neumann Algebra of an Infinite Group

Let $\mathbb{Z}$ be the additive group of the integers. What is the Von Neumann algebra of $\mathbb{Z}$? Let $M$ be the Von Neumann algebra of $\mathbb{Z}$. I know that we have $M \subset B(\ell^2(\...
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does $X$ has an $\ell_2$-complemented copy if the dual $X^*$ has an $\ell_2$-complemented copy

Let $X$ be a Banach space. If we know that $X^*$ has a complemented subspace isomorphic to $\ell_2$, then can we say that $X$ has a complemented subspace isomorphic to $\ell_2$? Precisely, I want to ...
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Von Neumann Algebra built from the bilateral shift

Let $T: \ell^2(\mathbb{Z}) \rightarrow \ell^2(\mathbb{Z})$ be the bilateral shift, and let $S= \{ I, T, T^* \}$. Why do we have $S'= S'' = \{ W\in B(\ell^2(\mathbb{Z})) : \langle e_n, We_k \rangle = \...
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Commutant built from the Unilateral Shift

Let $T: \ell^2 \rightarrow \ell^2$ be the unilateral shift, and let $\mathcal{T}= \{ I, T, T^* \}$. Show that $\mathcal{T}' = \{\lambda I_{\ell^2} : \lambda \in \mathbb{C} \}$ and $\mathcal{T}'' = B(\...
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Commutant of a set of operators in $B(H)$

Let $\mathcal{T} \subset B(T)$ be a set of operators. Then we define the commutatnt $\mathcal{T}' = \{ S \in B(H) : ST =TS, \forall T \in \mathcal{T}\}$. I'm trying to show $\mathcal{T}' = \mathcal{T}'...
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When can we make $H$ separable in $A\subset B(H)$?

I came across the fundamental result in operator algebras that any $C^*$-algebra can be embedded in some $B(H)$. Is there some characterization of a $C^*$-algebra (or von Neumann algebra) $A$ such ...
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Increasing net of projections in von Neumann algebra

In $B(H)$, there exists an increasing sequence of finite rank projections convergent to the identity in SOT. Is there an analogous statement for a von Neumann algebra $M\subseteq B(H)$?
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Spatial isomorphism between type I factors

I'm reading a paper from Christopher J. Fewster about split property, and I'm trying to catch one of his assertions. He asserts that every type I factor $M$ (a concrete von Neumann algebra over $\...
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commutant of the corner of a von Neumann algebra

Let $M$ be a von Neumann algebra and $p$ is a projection in $M$. Does there exist relationship between the commutant $(pMp)'$ of $pMp$ and $pM'p$. I know the fact that if $p\in M'$, we have $(pMp)'=pM'...
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Trace on $M_n(A)$ where $A$ is a tracial von Neumann algebra

Let $A$ be a tracial von Neumann algebra with trace $\tau$. If I'm not mistaken, this trace is unique. For $X=[x_{ij}]\in M_n(A)$, define $$\tau_n(X):=\frac{1}{n}\sum_{i=1}^n\tau(x_{ii}).$$ I find ...
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intersection of two von Nuemann algebras

Let $M_1$ and $M_2$ be two von Neumann algebras. $p=p^*=p^2$ and $p\in M_1\cap M_2$. Can we conclude that $pM_1p\cap pM_2p=p(M_1\cap M_2)p$? It is trivial that $p(M_1\cap M_2)p\subset pM_1p\cap pM_2p$....
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contruced a $*$-isomorphism between two von neumann algebras

Let $M$ be a von Neumann algebra and $p$ is a projection in $M$. Can we contruct a $*$-isomorphism between $M$ and $pMp$?
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Spectral Projection and GNS in Commutative Case

For $N>4$, let $A=C([0,N])$ and $1_A=\mathbf{1}_{[0,N]}$ (the identity on $A$). Consider $\operatorname{id}:=\operatorname{id}_{[0,N]}=(t\mapsto t)$ (not the identity on $A$ but $f(t)=t$) and the ...
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Norm bounded sets in $C^*$-algebras/von Neumann algebras

I was reading about the weak operator topology in wiki, and I saw that: Norm-bounded sets in $B(H)$ are pre-compact in WOT. I was wondering: Is there an analogous statement for von Neumann algebras? $...
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GNS construction for a normal state on a von Neumann algebra

Let $M$ be a von Neumann algebra. For any faithful normal state $\omega$ on $M$, according to the GNS construction, we have $\omega(x)=\langle x \Omega,\Omega\rangle$, where $\Omega$ is a cyclic ...
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Determining the normal functionals on the direct sum of von Neumann algebras

Consider a collection $\{M_i: i \in I\}$ of von Neumann algebras and consider their $\ell^\infty$-direct sum $M$. I'm trying to show that a normal functional $\omega$ on $M$ is of the form $$\omega(m) ...
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Showing the existence of a right-inverse in a von Neumann algebra

In this paper, specifically Theorem 4.1 on page 10, one of the last steps in the proof involves saying that a particular right-inverse exists. I'll try to restate what I think are the relevant pieces ...
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Show two topologies coincide on the unit ball.

Consider the following lemma from "Lectures on von Neumann algebras": I understand the proof of $(i)$ and $(ii)$. However, the proof says that $(iii)$ and $(iv)$ follow immediately from $(i)...
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Question about initial topology and dual vector space

Consider the following fragment from the book "Lectures on von Neumann algebras". Why is the line $\varphi$ is $\sigma(\mathcal{E}, \mathcal{F})$-continuous $\implies$ there exist $\psi_1, ...
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59 views

Show that a certain net converges in a von Neumann algebra.

Consider the abstract von Neumann algebra $M = \ell^\infty\text{-}\bigoplus_{i \in I} B(H_i)$. Moreover, we assume $\dim H_i< \infty$ for all $i \in I$. Let $x_i$ be the identity on $B(H_i)$ and ...
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What is an invariant mean on $L^\infty(\mathbb{R})$?

Consider the von Neumann algebra $M:=L^\infty(\mathbb{R})$, which consists of (classes) of essentially bounded measurable functions $\mathbb{R}\to \mathbb{C}$. Here $\mathbb{R}$ has the classical ...
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Show that $(M_*)^*\cong M$ for a von Neumann algebra $M$.

Let $M \subseteq B(H)$ be a von Neumann algebra. Denote $B(H)_*$ to be the $\sigma$-weakly continuous functionals on $B(H)$ and let $M_*= \{\omega\vert_M: \omega \in B(H)_*\}$. I want to prove that ...
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What do weak-star limits of normal states on von Neumann algebras look like?

Let $N$ be a von Neumann algebra and let $(\phi_\lambda)$ be a net of normal states on $N$ so that $\phi_\lambda\to\phi$ in the weak-* topology, i.e. $\phi_\lambda(x)\to\phi(x)$ for all $x\in N$. What ...
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Characterization of semi-finiteness in terms of the modular automorphism group

The Theorem in the screenshit is from Takesaki's book (Vol 2, Chapter VIII, section 3) When reading the proof of (iii)→(i), I met with troubles. How to prove $n_{\varphi}$ is a left ideal according ...
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27 views

Check a set is a von Neumann subalgebra

Let $M$ be a von Neumann algebra and $\varphi$ is a fithful normal positive linear functional on $M$. Define $S:=\{x\in M: x\varphi=\varphi x\}$, where $x\varphi(y):=\varphi(yx),\varphi x(y):=\varphi(...
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Faithful semi-finite normal weights

The above screenshot is from Takesaki's book(Vol 2, Chapter VIII) I wonder how to prove the implication (ii)→(i)? By Theorem 2.11, we have $\sigma_t^{\psi}(x)=h^{-it}\sigma_t^{\varphi}(x)h^{it}$ for ...

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