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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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        74 views

        Could we assume without loss of generality that all coefficients are positive?

        Let $\alpha$ be an element in the group algebra $\mathbb CG$ of a torsion-free group $G$. Assume that, as an operator acting on $\ell^2(G)$, $\alpha$ is positive. Does there exist $\beta\in\mathbb CG$ ...
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        49 views

        Being additive map using spectral decomposition theorem

        Let $(\mathcal{A},\tau_\mathcal{A})$ and $(\mathcal{B},\tau_\mathcal{B})$ be semifinite von Neumann algebras with normal semifinite faithful traces $\tau_\mathcal{A}$ and $\tau_\mathcal{B}$. I defined ...
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        64 views

        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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        52 views

        Strong operator convergence of support in non-commutative $L^p$ spaces

        Let $M$ be a von Neumann algebras with normal faithful semifinite trace $\tau.$ Let $L^p(M)$be the associated non-commutative $L^p$-space. Suppose, $x\geq 0$ be an element of $L^p(M)$ and $0\leq x_n\...
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        159 views

        Hausdorff dimension and von Neumann dimension

        There are two subjects in which non-integral dimensions appear: fractal geometry: consider the well-known Hausdorff dimension of fractals. von Neumann algebra: consider a type ${\rm II_1}$ ...
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        77 views

        Enveloping von Neumann algebra of Clifford algebra

        As explained in the book "Spinors in Hilbert Space" by Plymen and Robinson, if $V$ is a complex (separable) Hilbert space with a real structure, and $\mathrm{Cl}(V)$ the corresponding Clifford algebra,...
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        46 views

        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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        55 views

        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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        92 views

        Some doubt on crossed product von Neumann algebras

        There are two definitions in different books. Let $G \curvearrowright M$, then there is the definition of forming Group ring $M[G]$, define product and addition then make its algebra. represent the ...
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        1answer
        64 views

        On existence of fixed point operator

        Let $M$ be an infinite dimensional non-type $I$ factor, given $\xi$ in $\mathcal{H}$, does there exist a not identify operator $x$ in $M$ such that $x\xi=\xi$, I have tried with taking projection $P_{\...
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        2answers
        143 views

        Invariance of spectrum under conjugation

        Let $T$ be a self-adjoint invertible operator on $\mathcal{H}$ with a continuous spectrum, means the spectral measure is nonatomic. For which class of invertible operators $V$( with continuous ...
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        1answer
        87 views

        On commutant of $II_{1}$ factors

        Suppose $M$ is $II_{1}$ factor but need not be in standard form. Under what condition (on $M$ or Hilbert space) is the commutant $M'$ of $M$ again $II_{1}$ factor on the Hilbert space acted by $M$?
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        41 views

        Projections in tensor product of vN algebras

        Can we write any projection in the tensor product vN algebra $M\otimes N$ in terms of limits of projections $p\otimes q$, where $p$ and $q$ are projections in M, N or somewhat relate the projections ...
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        1answer
        84 views

        On boundedness of sequence of operators in vN algebra

        Let $x_{n}$ be a sequence of operators in vN algebra $M$, $\Omega$ is a cyclic vector for $M$, if $x_{n}\Omega$ converges in $\mathcal{H}$, can we say there exist a subsequence $\{y_{n}\}$ of $\{x_{n}\...
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        0answers
        47 views

        Separating vector on dense subalgebra

        Suppose $M$ be a vN algebra and $U$ be a S.O.T dense self-adjoint subalgebra of $M$ has separating vector, does $M$ have? If not give a counterexample. Or there is a condition on M like type II_{1} or ...

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