All Questions
Tagged with oa.operator-algebras von-neumann-algebras
172 questions with no upvoted or accepted answers
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Integral decomposition
Let $\mathcal{A}$ be a separably acting von Neumann algebra and let $$\mathcal{A}\cong \int_{\Gamma}^{\oplus} \mathcal{A}_{\gamma}\,d\mu(\gamma)$$ be its direct integral decomposition into factors $\...
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Intersection of type-I von-Neumann algebra factors
Is the intersection of a (possibly infinite) family $\{\mathcal M_i\}$ of type-I von-Neumann algebra factors (over the same Hilbert space $\mathcal H$) again a type-I von-Neumann algebra factor?
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Type III von Neumann algebra
Let $\mathcal M$ be a type $\mathrm{III}$ von Neumann algebra. Is it true that for all $n\geq 1,$ $\mathcal M$ contains a copy of $M_n$ as a von Neumann subalgebra? By Theorem 9.24 of these lecture ...
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Monotone convergence theorem for increasing net of positive functions
Suppose that we have $(\Omega,\mu)$ a $\sigma$-finite measure space. I have the following question.
(Assume that $(f_i)_{i\in I}$ be an increasing net of positive measurable functions such that $f_i\...
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Dual operator space
Suppose $E$ is an operator space and $E^*$ is the dual operator space. It is well known that the matricial norm structure on $E^*$ is given by the formula $\|[f_{ij}]_{i,j=1}^n\|_{M_n(E^*)}:=\sup\{\|...
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Monotone series of projections converging to 1 in von Neumann algebra
The following statement is being used a lot in the literature, and I wonder how to prove it.
Let $M$ be an infinite-dimensional von Neumann algebra (with unit element), show that there is an ...
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Limit of spectral projection of increasing sequence of positive operators
Let $\mathcal M$ be a von Neumann algebra. Suppose $(x_\alpha)\subset \mathcal M$ is bounded increasing net of positive operators converging to a positive operator $x\in\mathcal M.$ Is it true that $\...
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On mixing and weak mixing subalgebras of finite von Neumann algebras
Let $M$ be a full $\mathrm{II}_1$ factor. Consider mixing and weak mixing subfactors $B$ and $C$ of $M$. Are $B$ and $C$ full?
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On prime factors
Let $M$ be a prime $\mathrm{II}_1$ factor. Let $N$ be a non hyperfinite finite index subfactor $N$, is $N$ prime factor?
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On an application dominated convergence theorem in vN algebras
$M$ be a $\mathrm{II}_{1}$ factor equipped with the faithful normal trace $\tau$ in the standard form. Let $\tau(Jx'J\eta)=0,\forall x' \in M'$ and fixed $\eta$ in $L^{1}(M,\tau)$. Is it true $\eta=0$?...
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On invertibility of ergodic averages
Let $x$ be invertible unbounded operator affiliated operator to the $\mathrm{II_{1}}$ factor $(M,\tau)$. Under which condition on $x$, the iterates also $1+\sigma(x)+\cdots+\sigma^{n}(x)$ are ...
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On examples of action of C-star simple group on von Neumann algebra
Can there exist a faithful action of a $C^{*}$-simple group $G$ on a von Neumann algebra $(M,\varphi)$ equipped with faithful normal state $\varphi$ such that action preserves the state $\varphi$ and ...
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Characterzing compact actions on von Neumann algebra
Suppose $G$ is a countable discrete group acting on vN algebra $M$, the action is compact. Can we have a topology on Aut$(M)$ such that $\{\sigma_{g}\in \text{Aut}(M):g \in G\}$ form a compact subset ...
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On cyclicity of a module
Let $A$ be a $\text{ von Neumann algebra }$, $\mathcal{H}$ is a cyclic $A$ module, $G$ be a finite group acting on $A$, is $\mathcal{H}$ cyclic module over fixed point subalgebra of the action? ...
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On existence of sequence of unitaries in $II_{1}$ factor $M$
Let $M$ be a $\mathrm{II}_{1}$ factor acting on $L^{2}(M, \tau)$ in standard form, let $\{e_{n}:n \in \mathbb{N}\}$ be fixed orthonormal basis of $L^{2}(M, \tau)$, does there exist sequence of ...
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On cyclicity of fixed point algebra of flip automorphism
Let $M$ be a von Neumann algebra having a cyclic vector in $\mathcal{H}$, is the fixed point subalgebra under the flip automorphism on $M\otimes M$ has a cyclic vector in $\mathcal{H}\otimes \mathcal{...
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On $s$-numbers in finite von Neumann algebra
$T$ is an operator in $M$, $M$ is finite von Neumann algebra. There is a notion of singular value function that is ($s$-numbers). My question is: what is $s$-number for tensor product of two operators ...
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Injectivity of Fermion algebras
Is the Fermion algebra or $-1$-Fock space as defined in https://arxiv.org/pdf/math/0303045.pdf hyperfinite? Any references?
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semifinite projection
Let $M$ be von Neumann algebra, $p$ be semiefinite projection and $q$ be projection in $M$ such that $Z(q)=Z(p)$.
( $p$ is semifinite projection if every nonzero subprojection of $p$ contains a ...
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The completely reducible bimodules coming from subfactors
This post is a sequel of: Are all the R-R-bimodules completely reducible?
Question: For which (as general as possible) class of subfactors $(N \subset M)$, the bimodule $_NM_M$ is known completely ...
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Weights on Von Neuman factors
Let A be a type $I$ factor on a Hilbert space H. Let $\varphi$ be a semi-finite normal weight on $A^{+}$ is it possible to say that then there exist Hilbert spaces $H_2 \subset H_1$ and an isomorphism ...
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What is a type $\text{II}_1$ factor von Neumann algebra?
After finding formal definitions in various texts (see, eg, Witten, Notes On Some Entanglement Properties Of Quantum Field Theory, Rev. Mod. Phys. 90, 45003 (2018), doi:10.1103/RevModPhys.90.045003 ...
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