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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 multipli …

Consider the von Neumann subalgebra of $M\otimes M$ by $ B= \mathrm{vN} \{T\otimes T: T\in M\}$. What is the commutant of B?

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} …

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 $M$?

$M$ 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 $\eta=0$ …

Let the symmetric group $G=S_{k}$ act on the von Neumann algebra of the free group $L(F_k)$ via permuting its generators. Is the fixed point algebra under the action isomorphic to the whole algebra, i …

Let $M$ be a hyperfinite type $\mathrm{III}_\lambda$ factor. $N$ be a finite index type $\mathrm{III}$ subfactor, is it true $N$ is hyperfinite type $\mathrm{III}_{\lambda}$?

Let $M$ be a type $II_{1}$ factor, Let $B$ is an infinite dimensional nonabelian subalgebra. Is it true that $B$ always type $II_{1}$ ?

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{ …

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 o …

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 unitari …

$T$ 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 …

If $M$ is in standard form, consider the action of a finite group on $M$, does the fixed point subalgebra under the action is in standard form? What we can say if $M$ is hyperfinite $\mathrm{II}_{1}$ …

There is a definition in Haagerup's paper on $L^{P}$ spaces for weights, my question is after putting the norm is it become semifinite $L^{P}$ space on the crossed product? I am not clear please help. …

Let $M$ be a vN algebra (represented GNS space with respect to state) in standard form. Under which condition we can say a subalgebra $B$ of $M$ is also in standard form? If there exist $\varphi$ pres …