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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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Question about the lattice of orthogonal projectors of *real* von Neumann algebras

Let $R$ be a real von Neumann algebra of bounded operators on the real Hilbert space $H$ and ${\cal L}(R)$ the lattice of orthogonal projectors in $R$. Is it true that (similarly to what happens in the complex case) ${\cal L}(R)''= R$? The problem…
V. Moretti
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Definition of Modular Automorphism Group

Is there a way to concisely define the modular automorphism group of a von Neumann algebra $M$ for a "lay" person? I have little more than a vague fuzzy overview of Tomita Takesaki Theory and I'm unsure if it would be wise in my situation to get…
roo
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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: $\| ap \| \leq \| bp\|^{1/2}$ In the proof we have the next two inequalities: $$\|…
MathematicalPhysicist
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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, then for every $a\in M$ operator we…
user26565
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The subalgebras of algebras of $\tau$-measurable operators.

Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful normal semifinite trace $\tau$, let $S(\mathcal{M},\tau)$ be the algebra of $\tau$-measurable operators. Q1: If $\mathcal{B}$ is the subalgebra of $S(\mathcal{M},\tau)$. Whether…
Shaoze Pan
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Type transmutation of von Neumann factors

The crossed product of a Type $III_{1}$ von Neumann factor with its modular automorphism group is generically Type $II$. Does there exist a similar construction turning a Type $II$ (or a Type $III_{1}$) into a Type $I$?
QuantumFieldMedalist
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Minimum of Spectrum

Let $M$ be a von Neumann algebra and $e$ a projector. Is it true that $min Spec_M(x)\leq min Spec_{eMe}(exe)$ for $x$ positive in $M$? Thank you.
Iyari Rojas
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Spectrum of reduced von Neumann algebra

Let $M$ be a von Neumann algebra and $e$ a projector. Do you know if $Spec_{eMe}(exe)\subset Spec_M(x)$ for $x\in M$? Thank you
Iyari Rojas
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Definition of properly infinite projection in a von Neumann algebra

Let $E$ be a projection in a von Neumann algebra $R$. According to Kadison and Ringrose, $E$ is $\textit{properly infinite}$ if $E$ is infinite and for each central projection $P\in R$ either $PE=0$ or $PE$ is infinite. But I've seen other…
SihOASHoihd
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Conditional expectation on Cross product von Neumann algebra

Let $M$ be a von Neumann algebra on a Hilbert space $\mathcal{H}$. Let $(M,G,\alpha)$ be a $W^*$-dynamical system. Let $G$ be discrete. Then the cross product von Neumann algebra $M\rtimes_\alpha G$ is defined in the Hilbert space…
budi
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A question on von Neumann algebra

Let $M$ be a von Neumann algebra and let $\pi: M \rightarrow B(H)$ be a representation. Suppose that $N$ is a von Neumann subalgebra of $M$ such that $E$ is a faithful conditional expectation from $M$ to $N$. Is it true that if $\pi$ is faithful on…
budi
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Von Neumann Correspondence

In Popa's preprint https://www.math.ucla.edu/~popa/popa-correspondences.pdf his initial definition of an $N-M$ correspondence between two von Neumann algebras is a Hilbert space $\mathcal{H}$ and a seperately weakly continuous commuting action of…
sirjoe
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Projection operators and Analytic Elements

Consider the following element: $P_{n}=\sqrt{\frac{n}{π}}\int_{-\infty}^{+\infty}e^{-nt^{2}}τ_{t}(P)dt.$ where $P$ is a projection and $τ_{t}(P)=U(t)PU(t)^*$ for some group of unitaries in a von Neumann algebra. Is $P_{n}$ a projection? How can we…
val 72
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Quantifying the difference between two commutative von Neumann algebras

Given a self-adjoint operator $A$ on a separable Hilbert space, let $A''$ denote the commutative von Neumann algebra generated by $A$. According to [1] and [2], every commutative von Neumann algebra on a separable Hilbert space can be expressed this…
Chiral Anomaly
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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?
mathlover
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