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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.

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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Kaplansky for Projections

Let $H$ be a Hilbert space, and $A$ a $C^*$-subalgebra of $B(H)$ (the bounded operators on $H$). Let $B$ be the strong-operator closure of $A$, so that in particular, $B$ is a von-Neumann-algebra. According to the Kaplansky-Theorem: The … in the…
RS8
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Generators of von Neumann algebras

Let us suppose that M is a von Neumann algebra on some hilbert space $H$ such that $M = A''$ for some $C^*$-algebra $A\subseteq B(H)$. I am wondering when $A$ will be dense in $M$ and by which topology. I also want to know that if I have two…
unknown is my last name
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Generator of an abelian von Neumann Algebra

I want to show that $L^{\infty}(S^1)$(where $S^1$ is equipped with its Haar Measure) as a von Neumann algebra is generated by the multiplication operators $M_{e^{in\theta}}$ where $n \in Z$ and $\theta\in R$. Note - 1. For any $f\in…
Ester
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Computing complex or irrational powers of the modular operator

Let $M\subset B(H)$ be a von Neumann algebra, with cyclic separating vector $\xi$. Then the modular conjugation operator $S$ is defined to be the closure of the operator $$S_{0}:M\xi\to M\xi\text{ defined by }S_{0}(x\xi) = x^{*}\xi$$ Then the…
roo
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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.
Ester
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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 partial isometry $v\in M$, such that $\phi(x) =…
Ester
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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 exists a sequence $\{x_n\}_{n\in\mathbb N}\subset M$…
Phoenix87
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Trace preserving isomorphism on von Neumann algebras

Is the condition on an isomorphism between von neumann algebras which says that the trace is conserved the same thing as the notion of spatial isomorphism?
Iyari Rojas
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Subfactor of hyperfinite one

Is there a strict subalgebra of the hyperfinite $II_1$ factor that is separable and type $II_1$ factor?
Iyari Rojas
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von Neumann algebra generated by a $C^*$-algebra

Let $A\subseteq B(H) $ be a $C^*$ algebra. I know that if $A$ is unital, then by von Neumann density theorem SOT closure$(A)=A''$ and $A''$ is the von Neumann algebra generated by $A$ in $B(H)$. Is it true that if $A$ is not unital, still SOT…
budi
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Restriction of a faithful representation of a von Neumann algebra

Suppose that $M$ is a von Neumann algebra represented on a Hilbert space $H$. Suppose $H_0$ is a subspace of $H$ invariant under $M$ so that we get a representation $\pi: M \rightarrow B(H_0) $. Will $\pi$ be faithful? I do not much about von…
budi
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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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