Questions tagged [von-neumann-algebras]
Subtag of the [oa.operator-algebras] tag for questions about von Neumann algebras, that is, weak operator topology closed, unital, *-subalgebras of bounded operators on a Hilbert space.
647 questions
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Trying to show a map between two Von Neumann algebras is normal
In a paper by Izumi, he shows that if $M$ is a Von-Neumann algebra, and we have a normal u.c.p map $\varphi:M\to M$, then the fixed points of $\varphi$ have the structure of a Von Neumann algebra (and ...
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Tracial comparison and orthogonality for sequences of projections in semifinite von Neumann algebras
Let $M$ be a von Neumann algebra with no minimal projections, and let $\phi$ be a faithful normal semifinite trace on $M$. Suppose $(p_n)_{n\ge 1}$ and $(q_n)_{n\ge 1}$ are sequences of non-zero ...
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Understanding the key ideas in the proof of Connes' characterization of sigma-finite von Neumann algebras
In the 1974 paper by Connes "Caractérisation des espaces vectoriels ordonnés sous-jacents aux algèbres de von Neumann", a correspondence is shown between self-dual, facially homogeneous, and ...
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Projections in the bidual of a C*-algebra that are not sequentially accessible
Let $A$ be a $C^*$-algebra with enveloping von Neumann algebra $A^{**}$.
Call a bounded sequence $(a_n)\subset A$ weakly Cauchy in $A$ if $\phi(a_n)$ is a Cauchy sequence in $\mathbb C$ for every $\...
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Compact quantum groups in the locally compact framework
Background: I want to understand compact quantum groups and the dual groups in the framework of "Locally Compact Quantum Groups" in the sense of Vaes and Kustermans. I'm not familiar with ...
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Mutual commutants of commuting isomorphic type $\mathrm{III}_1$ factors
Let $\mathcal{A}$ and $\mathcal{B}$ be type $\mathrm{III}_1$ factors acting on a Hilbert space $H$ such that
$$\mathcal{A} \cong \mathcal{B} \quad \text{and} \quad [\mathcal{A}, \mathcal{B}] = 0.$$
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Existence of two-sided Pimsner-Popa basis and subfactor planar algebra
Given an extremal subfactor $N\subset M$ with $[M:N]<\infty$, Vaughan Jones asked whether there exists a two-sided Pimsner-Popa basis- that is, whether there exists a set $\{\lambda_i: 1\leq i\leq ...
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Central extension of a hyperlinear group is hyperlinear
$\DeclareMathOperator{\vN}{vN}$ I have a very vague question regarding the quotient weak expectation property (QWEP) for twisted group von Neumann algebras. Recall that a discrete group $G$ is ...
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Measurable fields of Hilbert spaces with Serre-Swan duality
The ncatlab article measurable field of Hilbert spaces uses Takesaki's definition of a measurable field of Hilbert spaces (MFoHS). Then it claims that MFoHSs has a Serre-Swan Duality. That is, the ...
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Preprint by Petz on noncommutative ergodic theory
In the collection "Aspects of Positivity in Functional Analysis. Proceedings of the Conference held on the Occasion of H.H. Schaefer's 60th Birthday", there is a contribution by Denes Petz ...
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Passage of amenability, and the Følner condition to corners of von Neumann Algebras
Let $(M,\tau)$ be a tracial von Neumann algebra, and $q \in M$ be a (nonzero) projection, and let $\mathcal H = L^2(M)$ be the standard representation.
Using the definition of injectivity ($M \subset ...
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TRO equivalence
Kadison and Ordower in a joint work posed the next question (which remains open): "Is every hyperreflexive subspace completely hyperreflexive?"
Recall that a reflexive subspace $\mathcal S\...
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A question on freeness
Let $\{\mathcal{A}_i\}_{i=1}^3$ be a collection of unital free sub algebra in a noncommuttaive probability space $(\mathcal A,\varphi)$ (see https://arxiv.org/pdf/0911.0087 for definitions). Is it ...
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Proving the existence of projections satisfying specific conditions for a tracial weight on a semifinite von Neumann algebra
Let $M$ be a von Neumann algebra that contains no minimal projections, and let $\varphi$ be a faithful, normal, semifinite tracial weight on $M$. Consider a $*$-automorphism $\theta: M \to M$ ...
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Reference request: uncountable Connes' T-invariance
This question is in the context of Tomita–Takesaki theory. Its brief introduction can be found in wiki and I will borrow terminologies used there. Given $\mathcal{M}$ a $\sigma$-finite factor and $\...
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