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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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When is the closable composition of operators affiliated to a von Neumann algebra also an affiliated operator?

Let $S$ and $T$ be closed (unbounded) operators affiliated to a von Neumann algebra $M$. Can we say anything about when the closure of their composition $T \circ S$ is also affiliated to $M$ (assuming ...
szantag's user avatar
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Unitarily equivalent von Neumann algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be von Neumann algebras acting on the separable Hilbert spaces $H$ and $K$ respectively. Fact 1: If $\mathcal{A}$ and $\mathcal{B}$ are unitarily equivalent then $\...
E.Papapetros's user avatar
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Convergence of a Sequence in the GNS Space of a von Neumann Algebra with Semi-finite Trace under the $\sigma$-Weak Topology

Let $ M $ be a von Neumann algebra. A Semi-finite trace on $ M^+ $ is a function $\phi$ on $ M^+ $, taking non-negative, possibly infinite, real values, possessing the following properties: Linearity:...
abcdmath's user avatar
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Question about hyperstandard von Neumann algebras and selfpolar cones

Consider the following fragment from the book "Lectures on von Neumann algebras" by Stratila and Zsido, second edition: I have two questions: (1) Why is $\mathscr{M}_q= q\mathscr{M}q \...
Andromeda's user avatar
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Center of reduced von Neumann algebra

Consider the following fragment from the book "Lectures on von Neumann algebras" by Stratila and Zsido: Later, there is the following claim: Here, $z(e)$ is the central support of $e$, i.e....
Andromeda's user avatar
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Direct sum and isomorphism

Let $\mathcal{A}\subseteq \mathcal{B}(H)$ be a von Neumann algebra, $\,H$ be a separable Hilbert space and let $\xi\in H,\,\xi\neq 0$ such that $\overline{\mathcal{A}\xi}=H.$ We consider the ...
E.Papapetros's user avatar
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Let $B$ be a von Neumann subalgebra of $A$. Is it true that $L^2(B)\subseteq L^2(A)$?

Let $A\subseteq B$ be a unital normal inclusion of von Neumann algebras. Is it true that this induces an isometry of the standard Hilbert spaces $$L^2(A)\to L^2(B)?$$ I'm not sure if I can expect this ...
Andromeda's user avatar
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Question about the support of a normal weight on a von Neumann algebra

Consider the following fragment from Stratila's book "Modular theory in operator algebras": I understand everything in this fragment, except that equation $(3)$ holds. How can one show this ...
Andromeda's user avatar
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$s(e+f) = e\lor f$ for projections $p,q\in B(H)$

Let $H$ be a Hilbert space and $p,q\in B(H)$ orthogonal projections, i.e. $$p=p^*= p^2, \quad q=q^* = q^2.$$ I want to show that $s(p+q) = p\lor q$ where $s(p+q)$ is the support projection of $p+q$ ...
Andromeda's user avatar
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Making sense of certain vector-valued integrals in von Neumann algebra theory.

Let $M\subseteq B(H)$ be a von Neumann algebra and $\varphi$ a weight on $M$ with modular automorphism group $\{\sigma_t\}_{t\in \mathbb{R}}$. We define $M_\infty$ to be the set of all elements $m\in ...
Andromeda's user avatar
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Maximal abelian algebras and type I factors [closed]

Let $\mathcal{A}$ be a maximal abelian von Neumann algebra of a type I factor $\mathcal{N\subseteq B}(\mathcal{H})$ where $\mathcal{H}$ is a separable Hilbert space. We assume that $\mathcal{N}$ has ...
val 72's user avatar
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Normality of the Lebesgue integral with respect to a Haar measure

Let $G$ be a locally compact second countable topological group and let $\mu$ be a Haar measure on $G$. I want to show that the weight $\varphi : {L^\infty(G)}^+ \to \overline{\mathbb R_+}$, $f \...
Valentin Massicot's user avatar
3 votes
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Tensor product and extension of $\sigma$-weakly continuous linear map.

Let $M$ be a Von Neumann algebra and let $\Delta$ be a $\sigma$-weakly continuous unital $*$-morphism. We say that $\Delta$ is a comultiplication if $\Delta$ satisfies $(\Delta \otimes \iota)\Delta = (...
Valentin Massicot's user avatar
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Extension of derivation

Let $\mathcal{A}$ be a von Neumann algebra acting on the Hilbert space $H.$ Is it true that every derivation $\delta\colon \mathcal{A}^\prime \to \mathcal{B}(H)$ can be extended to a derivation $\...
E.Papapetros's user avatar
5 votes
2 answers
114 views

Normal character on a group von Neumann algebra

For a locally compact group $G$, I will denote by $L(G)$ its group von Neumann algebra, which is the von Neumann algebra acting on $L^2(G)$, generated by the image of left regular representation $\...
Mogget's user avatar
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Compression and hyperreflexivity

Let $\mathcal{A}$ be a von Neumann algebra acting on the separable Hilbert space $H.$ Is there some projection $S$ belonging to the center $Z(\mathcal{A})$ of $\mathcal{A}$ such that the algebra $S \...
E.Papapetros's user avatar
1 vote
1 answer
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Doubt on exercise on Von Neumann factor

I am trying to solve the last point of the following exercise, but I don't know how to approach it.The exercise says: Let $M \subseteq B(\mathbb{H})$ be a type $\mathbb{II}_1$ factor Von Neumann ...
MBlrd's user avatar
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Doubt on exercise on finite Von Neumann Algebras

I am doing the following exercise: Let $M \subseteq B(\mathbb{H})$ be a von Neumann algebra, $\tau$ be a faithful, tracial, normal state $\tau: M \to \mathbb{C}$. Prove that M is finite. What happens ...
MBlrd's user avatar
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double commutant of unitary operators of a Von Neumann algebra.

Let $M$ be a Von Neumann algebra. I am having trouble showing that the double commutant of unitary operators for $M$ is equal to $M$. I have shown it with projections using borel functioncal calculus. ...
3j iwiojr3's user avatar
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The GNS represenation for a Von neumann algebra is a Von neumann algebra.

Let $M$ be a Von Neumann algebra and $\omega$ be a faithful $\sigma$-weakly continuous positive tracial state on $M$. I know that we have an inner product space on $M$ because $\omega$ is faithful. ...
3j iwiojr3's user avatar
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Multiplication of two unbounded operators and functional calculus

Let $A$ be a positive, self-adjoint unbounded operator defined on a Hilbert space $H$. Let $f,g: [0, \infty]\to \mathbb{R}$ be Borel measurable functions that are bounded on compact subsets. We can ...
Andromeda's user avatar
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3 votes
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Operator on reduced group $C^*$-algebra induces operator on von Neumann algebra

Let $\Gamma$ be a discrete group. Consider its reduced group $C^* $-algebra $C_\lambda^* (\Gamma)$ and von Neumann algebra $L(\Gamma) = C_\lambda^* (\Gamma)'' \subseteq B(\ell^2(\Gamma))$. Let $T:C_\...
Tomás Pacheco's user avatar
1 vote
0 answers
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Disjointification of a sequence of projections in a von Neumann algebra

Let $M$ be a Von Neumann algebra on a Hilbert space $\mathcal H$. Let $\{p_n\}_{n\in \mathbb N}$ be a sequence of non-zero projections in $M$ and $p=\displaystyle \bigvee_{n=1}^\infty p_n$, that is, $...
DenOfZero's user avatar
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Bounded homomorphism

Let $H,K$ be Hilbert spaces and $X,Y$ be subsets of $\mathcal{B}(H,K)$ and $\mathcal{B}(K,H),$ respectively. By $[Y X]$ we denote the w*-closure of the linear span of the set consisting of operators $...
E.Papapetros's user avatar
1 vote
1 answer
60 views

Characterization of the Predual of a von Neumann Algebra

I was reading J. Renault's paper "The Fourier Algebra of a Measured Groupoid" and I am confused about his approach to the predual of a von Neumann Algebra. Let $M:= VN(\mathcal{G})$ be the ...
Tomás Pacheco's user avatar
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centralizer of the tensor product of von Neumann algebra

Let $M$ and $N$ be two von Neumann algebras. Suppose $\omega_1$ and $\omega_2$ are two normal states of $M$ and $N$ respectively. We consider the tensor product von Neumann algebra $M\otimes N$. If we ...
mathbeginner's user avatar
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a von Neumann algebra is isomorphic to the tensor product of von Neumann algebras

The slice maps are defined in the above sceenshot. If we take normal state $\psi$ on $N$, then the slice map $L_\psi$ defined by $L_\psi(\sum x_i\otimes y_i)=\sum \psi(y_i)x_i$ extends to a normal ...
mathbeginner's user avatar
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3 votes
1 answer
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Basic lemma on normality of map between von Neumann algebras

Let $f: M\to N$ be a bounded linear map between von Neumann algebras. Assume that for every net $0 \le x_i\nearrow x$, we have that $f(x_i)\to f(x)$ $\sigma$-strongly. Is it true that $f$ is normal, i....
Andromeda's user avatar
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1 answer
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von Neumann subalgebras of the prime von Neumann algebras

A von Neumann $M$ is called prime if $M=M_1\bar{\otimes} M_2$ implies that $M_1$ or $M_2$ is a type $I$ von Neumann algebra. A factor is prime if and inly if the factor cannot be factorized as the ...
mathbeginner's user avatar
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1 answer
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the type of the intermediate von Neumann subalgebra

Let $M$ be a type III$_1$ factor and $N$ be a type III$_1$ subfactor of $M$. If we know that $P$ contains $N$ amd also is a subalgebra of $M$. Can we determine the type of $P$. Is it of type III?
mathbeginner's user avatar
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-1 votes
1 answer
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strictly semifinite weights restrict to a von Neumann subalgebra

Suppose $\omega$ is a strictly semi-finite weight on a von Neumann algebra $M$ and $N$ is the von Neumann subalgebra of $M$. Is $\omega|_N$ a strictly semifinite weight on $N$?
mathbeginner's user avatar
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index of the subfactor

If $N\subset M$ is an inclusion of type II$_1$ factors, Jones define the index $[M:N]$. If $N$ is the type II subfactor of the type III factor $M$. How to define $[M:N]$?
mathbeginner's user avatar
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2 votes
1 answer
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von Neumann algebra contains the hyperfinite factor $R$

We know the fact that any type II$_1$ factor contains the hyperfinite II$_1$ factor $R$. My question: If we let $M$ be an arbitrary type II$_1$ von Neumann algebra. can we have $R\subseteq M$?
mathbeginner's user avatar
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von Neumann subalgebra which is isomorphic to $R$

Let $M$ be a von Neumann algebra and $N$ be the von Neumann subalgebra of $M$. If $N$ is $*$-isomorphic to the hyperfinite type II$_1$ factor $R$. Can we conclude that $N$ is also a hyperfinite type ...
mathbeginner's user avatar
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2 votes
1 answer
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$\phi$ uwo continuous faithful tracial state on a VNA with GNS.

Let $M \subset B(\mathcal{H})$ be a von neumann algebra and $\phi$ be a uwo continuous faithful tracial state. I understand that we obtain the GNS $(\mathcal{H}_\phi, \pi_\phi, \xi_\phi)$ with $\xi_\...
The Unique Operator's user avatar
1 vote
0 answers
73 views

relative commutant of von Neumann algebra

Let $M$ be a von Neumann algebra. If $N_1$ and $N_2$ are two subalgebras of $M$ such that $N_1$ is $*$-isomorphic to $N_2$. What is the relationship between $A_1:=N_1'\cap M$ and $A_2:=N_2'\cap M$? ...
mathbeginner's user avatar
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1 vote
1 answer
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Showing the double commutant of the image of GNS representation is a factor provided uniqueness of a tracial state.

Let $A$ be a $C^*$-algebra with an identity. If $A$ has a unique tracial state $\varphi:A \rightarrow \mathbb{C}$ i.e. $\varphi(ab) = \varphi(ba)$, $\varphi(x^*x) \geq 0$, and $\varphi(1) = 1$. I am ...
The Unique Operator's user avatar
1 vote
1 answer
46 views

Properties of converging succession of functionals

Given a sequence of functionals $(\omega_n)_{n \in \mathcal{N}}$ on a Von Neumann algebra $\mathcal{W}$ converging to $\omega$, I have the following doubts: Is it true that $\omega_n$ is pure $\...
MBlrd's user avatar
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2 votes
1 answer
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$\sigma$-weakly closed subalgebra of direct product of matrix algebras is again a direct product of matrix algebras

Let $A$ be a $\sigma$-weakly closed $*$-subalgebra of the $W^*$-algebra $\prod_{i\in I}^{\ell^\infty} M_{n_i}(\mathbb{C})$. I believe that we must have $A\cong \prod_{j\in J}^{\ell^\infty} M_{m_j}(\...
Andromeda's user avatar
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-2 votes
1 answer
67 views

types of von Neumann subalgebras [closed]

Let $N$ be a von Neumann subalgebra of the von Neumann algebra $M$. Is the type of $N$ at most the type of $M$?
mathbeginner's user avatar
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1 vote
1 answer
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Showing that a von neumann algebra in the bounded operators of $l^2(S_\infty)$ is a factor

Consider $S_\infty$ i.e. the group of permutations with functions $\sigma:\mathbb{N} \rightarrow \mathbb{N}$ such that $\sigma(n) = n$ for all but finitely many. The left regular represenation defined ...
The Unique Operator's user avatar
-1 votes
1 answer
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relative commutant of type III1 factor [closed]

Let $M$ be a type III$_1$ factor and $N$ be the non-trivial ($N\neq \Bbb C 1$)semi-finite von Neumann subalgebra of $M$. What is the type of relative commutant $N'\cap M$? Is it semi-finite?
mathbeginner's user avatar
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1 vote
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Free Product of interpolated, free group factors

Let $L(\mathbb F_2)$ be the group von Neumann algebra of the free group on two generators. The interpolated free group factors of Dykema and Radulescu are defined as $L(\mathbb F_r)=L(\mathbb F_2)^{1/\...
Jayakumar Ravindran's user avatar
0 votes
1 answer
112 views

Is centre of a Von Neumann algebra trivial? [closed]

Is the centre of a dense sub algebra $A$ of the Von Neumann algebra $M$ is trivial ($Z_A \subset A$) then shall we conclude $Z_M(M)$ is trivial hence $M$ is factor? Remember that we see $Z_A(A) \...
Jayakumar Ravindran's user avatar
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Why does a projector being equal to its own central carrier imply that it is in the center of the commutant of its algebra?

In a passage in a proof, there is the following sequence of implications which I didn't quite understand: "The projector P in a representation $\pi$ of a Banach $*$-algebra $\mathscr{A}$, is its ...
Felipe Dilho Alves's user avatar
1 vote
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If two representations have the scalars a its commutants, why does it imply that they are either unitarily equivalent or disjoint?

In Kadison's article he says that: "Since the commutant of an irreducible representation consists of scalars, two such are either unitarily equivalent or disjoint." Ref:”https://msp.org/pjm/...
Felipe Dilho Alves's user avatar
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1 answer
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Why is the commutant of a unital $*$-representation a von Neumann algebra?

If $\pi$ is a $*$-representation (i.e. a $*$-homomorphism) of a unital Banach$*$-algebra, then $\pi(\mathscr{A})$, is trivially self-adjoint, but why is $\pi(\mathscr{A})'$ a von Neumann algebra? This ...
Felipe Dilho Alves's user avatar
1 vote
0 answers
55 views

Is $L^\infty(G) \otimes L^\infty(G)$ dense in $L^\infty(G \times G)$? [closed]

Let $G$ be a Haussdorf locally compact second countable topological group. Denote $\mu$ a Haar mesure on $G$. We can embed the algebraic tensor product $L^\infty(G) \otimes L^\infty(G)$ into $L^\infty ...
Valentin Massicot's user avatar
2 votes
1 answer
55 views

help showing a property for a weak operator closed $^*$-subalgebra of the bounded operators of a Hilbert space.

Let $A$ be a weak operator closed $^*$-subalgebra of the bounded operators of a Hilbert space. If $T \in A$, then I am trying to show that $P_{(\ker T)^\perp} \in A$(projection for $(\ker T)^\perp$. ...
The Unique Operator's user avatar
2 votes
2 answers
61 views

Why do elements of the Gel'fand spectrum map self-adjoint elements to points in the usual spectrum of the element?

A. Döring says in the book "Deep Beauty Understanding the Quantum World Through Mathematical Innovation", that, for the Gelfand spectrum defined as: Definition: A Gelfand spectrum $\Sigma_V$...
Felipe Dilho's user avatar

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