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        Questions tagged [von-neumann-algebras]

        Subtag of [tag:oa.operator-algebras] for questions about von Neumann algebras, that is, weak operator topology closed, unital, *-subalgebras of bounded operators on a Hilbert space.

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        97 views

        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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        1answer
        138 views

        On diagonal part of tensor product of $C^*$-algebras

        Suppose we have a $C^*$-algebra $\mathcal{U}$, Consider the $C^*$-subalgebra generated by elements of the form $a\otimes a$, what is it isomorphic to? Is it isomorphic to $\mathcal{U}$ itself?
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        votes
        1answer
        106 views

        Ultraproduct of non-commuative $L^p$-spaces

        Let $1<p<\infty.$ Let $I$ be a non-empty set and $\mathcal{U}$ be an ultrafilter over $I.$ Let $M_i$ be von Neumann algebras equipped with normal faithful semifinite traces $\tau_i,$ $i\in I.$ ...
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        2answers
        187 views

        Actions of locally compact groups on the hyperfinite $II_1$ factor

        Let $R$ be the hyperfinite $II_1$ factor, and let $G$ be a locally compact group. (1) Does there always exist a continuous, (faithful) outer action of $G$ on $R$? (2) If so, how does one ...
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        2answers
        86 views

        Polar decomposition of tensor product of operators in von Neumann algebra

        If $T=V|T|\text { and } S=W|S|$ is the polar decomposition of $T$. Is it true that the polar decomposition of $T\otimes S$ is $T\otimes S=(V\otimes W)(|T| \otimes |S|)$. If $T$ and $S$ are self-...
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        votes
        2answers
        135 views

        Ultraweak topology in abelian von Neumann algebras

        Let $A$ be an abelian von Neumann algebra acting on the (not necessarily separable) Hilbert space $\mathcal{H}$ (with identity $I$). From the Gelfand-Neumark theorem, there is a compact Hausdorff ...
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        1answer
        90 views

        Computing multiplicity function for self adjoint operator with nonatomic spectral measure

        Suppose $T$ is a self-adjoint operator in $B(H)$ with $\sigma(T)$ a spectrum of $T$. $\mu$ is a spectral measure. For the operators having a generally continuous spectrum how to calculate the ...
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        55 views

        Sequence of unitaries in type III von Neumann algebra

        Consider a type III von Neumann algebra $\mathcal{M}$ and an isometry $w$. How does one show that there exists a sequence of unitaries $u_n\in\mathcal{M}$ that converge strongly to $w$? For instance,...
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        67 views

        Morita equivalence for graded von Neumann algebras

        I am interested in understanding Morita equivalence of $Z_2$-graded von Neumann algebras. In the ungraded case, Rieffel showed that all Type I factors are Morita-equivalent, while for Type III factors ...
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        1answer
        80 views

        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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        votes
        1answer
        180 views

        Property $\Gamma$ in terms of Correspondences

        A type $II_{1}$ factor $M$ with trace $\tau$ has Property $\Gamma$ if for every finite subset $\{ x_{1}, x_{2},..., x_{n} \} \subseteq M$ and each $\epsilon >0$, there is a unitary element $u$ in $...
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        0answers
        86 views

        Pimsner-Popa basis dealing with higher relative commutants

        Let $(N \subseteq M)$ be a finite index inclusion of ${\rm II}_1$ factors. Let $e_1$ be the Jones' projection. A finite subset $\{\lambda_i, i \in I\} \subset M $ is called a (right) Pimsner-Popa ...
        2
        votes
        1answer
        75 views

        Real rank 0 implies stable rank 1 on $C^\ast$-algebras?

        A $C^\ast$ algebra has defined stable rank (https://www.univie.ac.at/nuhag-php/bibtex/open_files/2079_Rieffel-StableRank.pdf) and real rank (https://core.ac.uk/download/pdf/82123484.pdf), which are ...
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        votes
        1answer
        138 views

        What is the story behind this Hilbert space in the definition of Hilbert Modules

        Here is Deflnition 1.5 of Hilbert module in "L^2-invariants: theory and applications to geometry and K-theory", Springer-Verlag, 2002, by W. Lück: A Hilbert $\mathcal N(G)$-module $V$ is a Hilbert ...
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        1answer
        128 views

        Projections in the tensor product of von Neumann algebras

        This question seems elementary, but I have already asked an expert who does not know the answer, so I would like to post here. Let $M$ and $N$ be von Neumann algebras, and let $M\bar{\otimes}N$ be ...

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