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, ...

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The Predual of a von Neumann algebra

Let $M$ $\subseteq$ $B(H)$ be a von Neumann algebra. I am wondering how does $M_*$ sit inside $B(H)_*$ upto isometry. Note - $M_*$ denotes the predual of $M$. Thanks for any help.
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Radon- Nikodym Theorem For von Neumann algebras

Sakai's Radon Nikodym Theorem for von Neumann algebra goes as follows: Let $\phi$ and $\psi$ be normal forms on a $von$ $Neumann$ $algebra$ $M$ such that $\phi$ $\leq$ $\psi$. Then $\exists$ $a$ ...
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Polar Decomposition of Forms in a von Neumann algebra

The polar decomposition of forms in a von Neumann algebra goes as follows - Let $\phi$ be a $\sigma$-weakly continuous form on a von Neumann algebra $M$. Then there exist a normal form $|\phi|$ and a ...
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Understanding the bidual of a $C^*$-algebra as a $C^*$-algebra

I have a lot of problems trying to understand the double dual of a $C^*$-algebra. Let $A$ be a $C^*$-algebra, I read that if you endow the bidual Banach space $A^{**}$ of $A$ with the weak-*topology, ...
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Flip automorphism for a $II_1$ factor is not inner

It is known that for a $II_1$ factor $M$, the flip automorphism defined on $M \overline{\otimes} M$ by $a \otimes b \mapsto b \otimes a$ is not inner. A proof can be found on Vol. IV of the books by ...
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Relation between tracial states on von Neumann algebras and their GNS representations

Let $M$ be a von Neumann algebra acting on a Hilbert space $H$, and let $\tau$ be a faithful tracial state on $M$. What is the relation between the GNS representation of $(M,\tau)$ and the original ...
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124 views

Closed unit ball of B(H) is not compact in strong operator topology of B(H).

In operator theory we prove that closed unit ball of B(H) is compact in weak operator topology and is closed in strong operator topology. But a book of operator theory states that closed unit ball of ...
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24 views

In what topology is the unitary group of a unital C*-algebra locally compact?

If a unital $C^*$-algebra $\mathcal{A}$ is finite-dimensional then the unitary group $\mathcal{U}(\mathcal{A})$ is compact with respect to the norm topology. My question is: what if ...
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The 1-Norm on a Quantum Group as a Supremum

To this MO question, Yemon Choi comments that If $\tau$ is a faithful normal trace on a von Neumann algebra $M$, then IIRC $\tau(|x|)$ is equal to the supremum of $|\tau(xy)|$ as $y$ runs over all ...
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Can a sequence of von Neumann algebras determine a maximal directed set of subalgebras?

Can a von Neumann algebra $A$ have an infinite sequence $A_0 \subset A_1 \subset A_2 \subset ...$ of sub-vN-algebras such that every other sub-vN-algebra $B \subseteq A$ satisfies, for some $n \geq 0$ ...
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Unique trace on a type $II_1$ von Neumann Algebra

Let $M \subseteq B(H)$ be a type $II_1$ von Neumann Algebra. Then any two non-zero ultraweakly continious normalised traces $Tr,tr : \rightarrow \mathbb{C}$ are equal. I'm trying to understand this ...
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Sequential Murray-von Neumann equivalence of projections

How general is the following statement about Murray-von Neumann equivalence of projections in a von Neumann algebra? Let $M$ be a von Neumann algebra and let $p,q\in M$ be projections. If there ...
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von Neumann algebra associated to the full group C*-algebra

Lance's theorem asserts a discrete group $G$ is amenable if and only if the reduced and full groups C*-algebras coincide. The group von Neumann algebra is the weak closure of the reduced group ...
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Every conditional expectation is normal?

Let $M$ be a von Neumann algebra and let $N$ be a von Neumann subalgebra and let $E$ be a conditional expectation $M\to N$. Let $i$ be the canonical inclusion of $M$ into $M^{**}$. Claim: $E$ is ...
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33 views

Weak operator limit of projections in $B(H)$

Let $H$ be infinite dimensional and $\cal P$ be the set of all projections in $B(H)$. Show that $\cal P$ is weak operator dense in $(B(H))^+_{\|.\|\leq 1}$, the set of positive operators in the unit ...
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Abstract Von Neumann Algebras

I have just read this question Is a von Neumann algebra just a C*-algebra which is generated by its projections? and am wondering about Robert Israel's answer when he says that a subalgebra of $C(X)$ ...
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169 views

A theorem about a tracial state in von Neumann algebra

I am reading a book about C*-algebra. There is a quotation below. Let $M$ be a von Neumann algebra with a faithful normal tracial state $\tau$ and let $1_{M}\in N\subset M$ be von Neumann subalgebra. ...
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How a trace of an ideal act on an element of the whole algebra? [closed]

Let A be a C* algebra with an ideal I. Suppose $\tau$ is a trace on I. Let $x\in I$. Then how to understand $\tau(range(x))$? i.e. how $\tau$ acts on the range projection of x?
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Closed unit ball of positive bounded operator space and its extreme point

Let $H$ be infinite dimensional Hilbert space. Then the closed unit ball of positive bounded operator space $B(H)^+$ is not the convex hull of the projections of $B(H)$. Please help me. Thanks.
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Separating and cyclic vector

Let $\{\Gamma_i , \mu_i\}_{i\in I}$ be a family of probability measure spaces and suppose $I$ is uncountable. Let $\{\Gamma , \mu\} = \prod_{i\in I} \{\Gamma_i,\mu_i\}$ be the product measure space. ...
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Inclusion of representations: $\pi_1(A)^{''}\subseteq \pi_2(A)^{''} \Rightarrow \pi_1(A)\subseteq \pi_2(A)$?

Let $A$ be a $C^*$-algebra and $\pi_1 , \pi_2$ two *-representations on a same Hilbert space $\mathcal{H}$ so that the inclusions $\pi_1(A)^{''}\subseteq \pi_2(A)^{''}$ of the associated von Neumann ...
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29 views

Irreducible representation and rank one projetion

Let $A$ be a C*-algebra with a nonzero minimal projection $e$. a - Show that if $\{\pi, H\}$ is an irreducible representation of $A$ such that $\pi(e) \neq 0$, then $\pi(e)$ is a projection of rank ...
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Sot closure of the unit ball of a subalgebra of $B(H)$

Let $A$ be a $C^* -$ subalgebra of B(H) and $S$ be the closed unit ball of $A$. 1- $S$ is convex and bounded, so $ S = weak^* -cl ~S$. (Is it correct?) 2- By the Kapalansky density theorem, we have ...
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Is an ultrapower of the hyperfinite factor still hyperfinite?

Let $\mathcal{R}$ be the hyperfinite type $II_{1}$ factor and let $\mathcal{U}$ be a free ultrafilter on $\mathbb{N}$. Is it true that $\mathcal{R}^{\mathcal{U}}$ is never hyperfinite ? How can I see ...
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Questions about the Kapalansky density theorem

I'm studying Takesaki's Theory of operator algebras book by myself. The following is a theorem from that book: I have several questions about this proof: 1- He claims, in the first line of ...
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Arveson spectrum for a unitary representation of a group on a Hilbert space

Let $G = \mathbb{R}$. By Stone's theorem, $U(t)\in\mathcal{B}(\mathcal{H})$ is generated by a self-adjoint operator $H$ (for which there is a resolution of the identity P(p), by the spectral theorem) ...
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Why does one only consider one-parameter groups in Borchers-Arveson theorem?

The theorem (Operator algebras and Quantum statistical mechanics vol. 1, Bratteli, Robinson, Thm. 3.2.46 p.261) roughly says that if one has a one parameter automorphism group $t \rightarrow\alpha_t$ ...
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Where does the double commutant theorem fails for $AW^*$-algebras?

Commutative $AW^*$-algebra are characterized as those $C^*$-algebras such that their space of projections is a complete boolean algebra (see http://en.wikipedia.org/wiki/AW*-algebra). Von Neumann ...
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$\sigma$-weak topology versus weak operator topology

The reference text for this question is: Pedersen, Analysis Now, GTM 118. The $\sigma$-weak topology on $B(H)$ (the bounded linear operators on a Hilbert space $H$) is the weak$^*$-topology on ...
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Subgroups of an infinite product

$G := \prod\limits_{n=1}^{\infty} (\mathbb{Z}/3^{n} \rtimes \mathbb{Z}/2 )$ $g_m = ((m,0),(m,0), (m,0),\dots) \in G$ $H = \prod\limits_{n=1}^{\infty} (\{0\},\mathbb{Z}/2 ) \subset G$ Questions: ...
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A question on a lemma in von Neumann algebra.

Let $\mathfrak{U}$ be a von Neumann algebra, the lemma says that: If $p\in \mathfrak{U}$ is a projection and $a,b \in \mathfrak{U}$ s.t $0\leq a \leq b \leq 1$, then: $\| ap \| \leq \| bp\|^{1/2}$ ...
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Affiliated algebras

I am searching for articles on affiliated $C^*$-algebras and affiliated von Neumann algebras. I know that Wonorowicz wrote articles about this topic and also Pedersen wrote a section in his book ...
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when a crossed product group is inner amenable

Denote $K, H$ to be countable discrete groups, then I am interested whether the crossed product group $G=H\rtimes_{\alpha} K$ is inner amenable or not. For example, when $\alpha$ is trivial, ...
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What is the definition of hyperstonean space?

I've seen several questions and answers on the Gelfand transform for commutative $C^*$-algebras leading to a characterization of commutative Von Neumann algebras as those whose spectrum is ...
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Polar decomposition

Every $x\in B(H)$ has a representation such as $x= u|x|$ (by polar decomposition) where $u$ is a partial isometry. In a paper, the author claims that$$u = {\rm strong} - \lim_{\epsilon\to 0} ...
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Von Neumann algebra generated by a subalgebra

Let A be a C*-algebra of operators on a Hilbert space H. Show that if $A\subset K(H)$, then $\{A'\cap K(H)\}'\cap K(H) = A$ I do not have any idea about it. Please give me a hint. Thanks.
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Image of a commutative C*-algebra

Let $A$ be an unital commutative C*-subalgebra of $B(H)$, and $\Omega$ be its character space. By spectral theorem $$\phi: B_\infty(\Omega)\to B(H);~~~~~f\to \int f \, dP$$ is a $*-$ homomorphism ...
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about ultrapowers

Let $\mathcal{R}$ denote the hyperfinite type $II_{1}$ factor, with $\mathcal{R}^{\omega}$ the ultrapower of $\mathcal{R}$, in respect to some ultrafilter $\omega$ on $\mathbb{N}$. I'm reading a paper ...
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Prove a factor is III$_{\lambda}$ type

This question is from the Sunder's book An Invitation to von Neumann Algebras Ex 4.2.13 Let M be a semifinite factor with fns trace ${\tau}$. Let ${\theta}$ be an automorphism of M such that ...
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$C^*$-algebras, von Neumann algebras, unbounded operators and quantum mechanics in connection

I am studying the theory of $C^*$-algebras, von Neumann algebras and unbounded operators in courses on Functional Analysis and Opertor Algebras. Now I want to apply this knowledge to (algebraic) ...
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embedding of $\prod_{n\in\mathbb{N}}M_{n}(\mathbb{C})$ in a type $II_{1}$ factor

Suppose $M$ is a type $II_{1}$ factor with trace $\tau$. Let $\lbrace p_{n}\rbrace_{n\in\mathbb{N}}$ be an increasing sequence of projections such that $\tau(p_{n})\rightarrow 1$. Now, let's consider ...
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Types of von neumann algebras

A von Neumann algebra M is said to be finite, infinite, properly infinite, or purely infinite according to the property of the identity projection 1. I think that this classification of types are ...
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an additive bijective map

Let H be a Hilbert space and $\Phi:B(H)\longrightarrow B(H)$ is an additive bijective map. If $\mathbb{R}IāŠ†\Phi(\mathbb{R}I)$, can we conclude by the bijectivity of $\Phi$ that ? ...
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Is this equality true or it is not necessarily true?

Let $A$ and $B$ are two factor von neumann algebras that act on two infinite dimensional Hilbert spaces H and K respectively. Let $\Phi:A\longrightarrow B$ is an additive bijective map with some other ...
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Ultraweak closed left ideal of a von Neumann algebra

The following is a proposition of Takesaki's Operator Theory: My questions are: 1- He claims for two sided ideal $\cal m$, $e \in M\cap M'$. While I think for $\sigma -$ weakly closed two sided ...
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Is the image of a von Neumann algebra under a C*-homomorphism a von Neumann algebra as well?

If $\varphi: A\to B$ is a (norm-continuous, unital, involutive) homomorphism of $C^*$-algebras, then the image $\varphi(A)$ is closed in $B$ and therefore is a $C^*$-algebra with the $C^*$-norm ...
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Show that $\pi(M)'' = \pi(M'')$

Let $M$ is a $*-$ subalgebra of $B(H)$. Let $\bar H$ denote the direct sum $\sum H_i$ where $\{H_i\}$ is a family of replicas of $H$. Define $$\pi :x\in B(H) \to \bar x \in B(\bar ...
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Noncommutative version of Littlewood's First Principle

There are definitely noncommutative analogues for Lusin's theorem and Egoroff's theorem (found in Blackadar for example). I'm curious if there is a version of the first principle: Every Lebesgue ...
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von Neuman algebras with trace

I started to learn von Neumann algebras and I wonder about the proof of the following. Can anyone outline it? If $\tau$ is a trace on a von Neumann algebra $M$, in other words $M$ is $II_1$-factor, ...
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QD C*-algebra's representation theorem

Here is a question from the proof of the "QD C*-algebra's representation theorem" in P245 of book "C*-algebras and Finite-Dimensional Approximations" by Nate and Taka. For a separable unital ...