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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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Free probability and projections

Let $(\mathcal{A},\varphi)$ be a free probability space, where $\mathcal{A}$ is a von Neumann algebra and $\varphi$ a finite and faithful trace. Let furthermore $p\in\mathcal{A}$ be a projection. ...
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Reference request unital normal *-homomorphisms $B(H) \to B(K)$

It is known that if $\varphi \colon B(H) \to B(K)$ is a unital non-zero normal $*$-homomorphism (for Hilbert spaces $H$ and $K$), then there exists a Hilbert space $K'$ and a unitary $U' \colon K \to ...
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Regarding cyclic vector of $L^{\infty}$ in $L^{2}$

What are cyclic vectors for $L^{\infty}([0,1],\mu)\otimes L^{\infty}([0,1],\mu)$ in $L^{2}([0,1],\mu)\otimes L^{2}([0,1],\mu)$ other than $1\otimes 1$? I want nonconstant cyclic vector in $f\otimes g$ ...
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connection between unital $C^*$ algebra and finite von neumann algebra

Let $A$ be a unital $C^*$-algebra with a tracial state $\tau$, $L^2(A,\tau)$ is the Hilbert space induced by the GNS constructtion.Suppose $\lambda$ is the left action of $A$ on $L^2(A,\tau)$ ,does ...
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1answer
34 views

On ergodic theory

Suppose there exist an action of group $G$ on $L^{\infty}(X,\mu)$ via measure preserving transformation ( the left translation Koopmans action). $\mu$ is probability measure. Suppose the action is ...
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On computing multiplicity function for self adjoint operator with nonatomic spectral measure

Suppose $T$ is a self-adjoint operator in $B(\mathcal{H})$ with $\sigma(T)$ is spectrum of $T$. $\mu$ is a spectral measure. For the operators having general continuous spectrum how to calculate the ...
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Density of $\mathcal{L}^2(\mathbb{R})^+\cap\mathcal{L}^{\infty}(\mathbb{R})^+$ in $\mathcal{L}^{\infty}(\mathbb{R})^+$

I am getting in a world of confusion here. One of my problems is the nomenclature for topologies on $\mathcal{L}^{\infty}(\mathbb{R})$ (or rather $B(H)$ for $H$ a Hilbert space) so straight away I ...
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1answer
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On spectral multiplicity of left shift operators

Let $U$ be an operator defined on $l^{2}(\mathbb{Z})$ by $U(e_{n})=e_{n-1}$, where $e_{n}$ is an orthonormal basis of $l^{2}(\mathbb{Z})$. $U$ is a left shift operator. Since $U$ is unitary operator ...
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26 views

On spectral multiplicity

How direct integral decomposition of self-adjoint operator in $B(\mathcal{H})$ is connected with multiplicity? If we take $M_{f}$ on $L^{2}(X,\mu)$ what is it spectral multiplicity? How is direct ...
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Dual of compact operator and dimension of dual space

The trace class operators are the dual of the compact operators It is proved that dual of space of compact operator $K(H)$ is space of trace class operator $L_1(H)$. Now, $L_1(H) \subseteq K(H)$, ...
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Left shift operators

Let $T$ be the left shift operator on $B(l^{2}(\mathbb{N}))$. How to see that von Neumann algebra generated by $T$ is $B(l^{2}(\mathbb{N}))$?
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Regarding group action on vN algebras

$G$ be a discrete countable group acting on $M$ via automorphisms of $M$. Does there exist a faithful normal state on $M$ which preserves the action means $\varphi(\sigma_{g}(x))=\varphi(x)$, $\sigma:...
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1answer
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Inverse of compression (von Neumann algebra)

I am stuck with this seemingly easy problem but I am having trouble showing this: Let $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ be a von Neumann algebra realized inside a subalgebra of the ...
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1answer
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Traciality of compressions of von Neumann algebras

Let $\phi_1$ be a linear functional on a von Neumann algebra $\mathcal{A}.$ (I need the result in particular for $\Pi_1$-factors), satisfying traciality. With "traciality" I mean the following: For $...
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Calculation of inner product I guess! [closed]

Suppose ${\alpha_{t}}$ is a $\sigma-$weakly continuous one parameter group of automorphism in a Von neumann algebra and u is a unitary. Is it true that $\langle \alpha_{t}(u)\xi,u\xi \rangle= \langle \...
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1answer
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Properly infinite projection/von Neumann algebra

The definitions I am using are: Def: a projection $e \in M$, von Neumann algebra, is said to be purely infinite, if given any projection $p \in Z(M) \subset M$, then $pe$ is finite if and only if $pe=...
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Existence of conditional expectations onto masas.

Given an inclusion $N\subset M$ of von Neumann algebras, a conditional expectation is a map $E:M\to N$ that is a projection ($E^2=E$) and it has $\|E\|=1$. This automatically implies that $E$ is ...
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1answer
106 views

Spectral integral form of tensor products

$T_{1}\otimes T_{2}$ is the operator acting on $\mathcal{H}\otimes \mathcal{H}$ then where $T_{1},T_{2}$ are self adjoint operators. What is the spectral resolution of identity for $T_{1}\otimes T_{2}$...
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0answers
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von Neumann sub algebras of $L^{\infty}$

I know that $L^{\infty}([0,1],\mu)$ acting on $L^{2}([0,1],\mu)$ is a von Neumann algebra. How will be the von Neumann sub algebras $L^{\infty}([0,1],\mu)$ look like?
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7 doubts about the von Neumann algebra [closed]

A von Neumann algebra, or $W^*$-algebra, is a $*$-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type ...
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1answer
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Normal u.c.p extension of Schur-multiplier

I'm struggeling with the proof of a theorem in [BO08]. The first part before the line is what I think I understood. The part after that I don't understand at all. Let $\Gamma$ be a discrete group and ...
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0answers
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How to identify infinite tensor product of c-star algebras

We know the definition of the infinite tensor product of $C^{*}$-algebras as an inductive limit. What is the infinite tensor product of $\mathbb{C}$ is isomorphic to? Similar for abelian $C^{*}$-...
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1answer
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Does the von Neumann algebra generated by a normal operator contain all commuting projections?

Let $H$ be a Hilbert space and $T\in B(H)$ a bounded normal operator. Let $\mathscr{A}$ be the von Neumann algebra generated by $T$. Is it true that $\mathscr{A}$ contains every orthogonal projection ...
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1answer
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Importance of pure states in $C^{*}$ algebras

In a tensor product of $C^{*}$-algebras I have seen some proofs are used by the argument of norms by pure states. Why pure states are essential to study related to vN algebras and $C^{*}$-algebras?
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On c-star tensor norms and von Neumann tensor norm

Does on finite tensor products of matrix algebras, all the $C^*$-norms and von Neumann tensor product norm coincide, or rather they are algebraically isomorphic? What about abelian von Neumann ...
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1answer
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Weakly dense C*-algebra in a commutative von Neumann algebra and order convergence

Let $H$ be a Hilbert space and $\mathscr{A}$ a commutative norm-closed unital $*$-subalgebra of $\mathcal{B}(H)$. Let $\mathscr{M}$ be the weak operator closure of $\mathscr{A}$. Question: For given ...
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1answer
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What is the definition of identity bimodule in the context of von Neumann or $\ast$-algebras?

I will just repeat the question in the title: What is the definition of identity bimodule in the context of von Neumann or $\ast$-algebras? I know what a bimodule is but I never heard up to ...
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1answer
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About “names” of von Neuman algebra morphisms

I have actually a basic quastion about maps between von Neumann algebras. If I have a map $f:N \to M$ being $N$ and $M$ von Neumann algebras. when this map is considered: completely positive, normal ...
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Is strong topology on a W*-algebra a dual topology?

I've come up with this question when reading Shoichiro Sakai's book $C^{*}$-Algebras and $W^{*}$-algebras Chapter 1.8. Let $A$ be a $W^{*}$-algebra (a $C^{*}$-algebra with a Banach space predual) and ...
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Proving difficulty in MASA [closed]

Prove that $ L^\infty{(\mathbb{R},\mu)}$ is a masa in $B(L^2(\mathbb{R},\mu))$, $\mu$ is sigma finite measure in particular Lebesgue, and what is the cyclic vector of it?
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What does the enveloping von Neumann algebra functor do to locally compact Hausdorff spaces?

Given a $C^*$-algebra $A$, we have an enveloping von Neumann algebra $A^{**}$ which is adjoint to the forgetful functor from the category of $W^*$-algebras to $C^*$-algebras. Because commutativity is ...
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1answer
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Bell type inequalities - von Neumann Algebras

I have a problem with proving part (4) of the following Lemma 2.1. It concerns Bell-type Inequalities. I attach the relevant definitions the lemma and the suggested proof (It begins with "Since..."). ...
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Existence of infinite family of orthogonal projections in finite vN algebra

Can we always have an infinite family of orthogonal projections in type $II_{1}$ vN algebra such each of which has full central support?
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1answer
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Type III Von Neumann algebras and spectra of the modular operators.

I´m studying the paper of Fredenhagen. There he said that he would prove that the algebra of local observables under certain conditions is of type III, by showing that all modular operators satisfy $...
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1answer
22 views

dimension of a von Neumann algebra

Is there any dimension on a von Neumann algebra? Is there any relationship between finite von Neumann algebras and finite dimensional von Neumann algebras?
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Example of norm separable c-star algebras [closed]

I want to know enough examples of norm separable $C^{*}$-algebras which are neither finite dimensional nor commutative.
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examples of non-unital commutative $C^*$-algebras

I know that all the non-unital commutative $C^*$ algebras are isomorphic to $C_0(\Omega)$,where $\Omega$ is a locally compact space. Can anyone show me some common non-unital commutative examples.I ...
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1answer
37 views

non isomorphic finite dimensional $C*$ algebras

How many non isomorphic finite dimensional $C^*$ algebras if the dimensions without a bound? Is it countable or uncountable?
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1answer
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spectrum of $C^*$ algebras

When $A=\bigoplus B(\Bbb C^n)$ ($c_0$ direct sum),how to compute the spectrum of $A$ ?What about the conclusion If we replace the $\ell ^\infty $ direct sum with $c_0$ direct sum?
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1answer
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arbitrary $n$-dimensional matrix algebras in II$_1$ factor

I'm struggling to show that in a type II$_1$ and any $n$ $\exists$ a subfactor $M$ such that $M \cong M_n$ . I suppose it should follow from the isomorphism between the equivalence classes of ...
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1answer
28 views

approximate identity element

Let $I\subset \prod_n B(H_{m_n})$ be a separable $C^*$ algebra,where $\prod B(H_{m_n})$ denotes the $\ell ^{\infty}$direct sum of $B(H_n)$ and $dim(H_{m_n})<\infty$. We suppose that there exits a ...
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1answer
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equivalent projections in finite factors are unitarily equivalent

Why are two Murray von Neuman equivalent projections $p$ and $q$ in a finite factor unitarily equivalent?
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1answer
27 views

image of some ideal under the quotient map

! I am still confused about the range of $\sigma(I_{\omega})$.Since $\|x_n\|_2\to 0,$we have $\|x_n\|\to 0$,$\sigma(I_{\omega})=0,$then $J=0$,it is trivial. Is my understanding correct?
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1answer
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Equality of tracial states on dense $C^*$-subalgebra implies equality on generated von Neumann algebra?

Maybe this is a simple question, but I'm not sure about the following: Let $\cal M$, $\cal N$ be von Neumann algebras and $X\subseteq \cal M$ a weakly dense (possibly separable) $C^*$-subalgebra. Let ...
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1answer
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identify GNS construction as asubalgebra of $R^{\omega}$

I have two questions In lemma6.5.5. 1.why can we identify the GNS representation of $\prod M_{k(n)}(\Bbb C)/\bigoplus M_{k(n)}(\Bbb C)$ with respect to $\tau _{\omega}$ with a subalgebra of $R^{\...
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1answer
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Some clarifications required

Is the inductive limit of tensor product $L^{\infty}(X,\mu)^{\otimes \infty}$ is isomorphic to $L^{\infty}(X,\mu)$?
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1answer
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functional calculus

Suppose $A$ is a non-unital $C^*$ algebra,$B$ is another $C^*$ algebra.Suppose $\phi:A\rightarrow B$ is a non-zero $*$ homomorphism and $x_0$ is a normal elememt in $A$,by continuous functional ...
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1answer
38 views

strictly positive element

If $A$ is a non-unital separable $C^*$ algebra,does there exist a strictly positive idempotent element in $A$ ?
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Do all *-isomorphisms between von Neumann algebras preserve strong operator topology?

Do all $*$-isomorphisms between von Neumann algebras preserve the strong operator topology? Seems clearly true for $*$-isomorphisms with a unitary implementation, but I don't see the answer for other ...
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On support of a spectral measure

Suppose $A$ is the self-adjoint unbounded operator in Hilbert space $\mathcal{H}$, if spectral measure $E$ of $A$ is supported on $[0,a]$, then prove that the resolution of identity $E_{a}=1_{(-\infty,...