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Let $H$ be a Hilbert space and $\mathcal{X}\subseteq \mathcal{B}(H)$ be an operator space. We denote $d(T,\mathcal{X})=\inf_{X\in\mathcal{X}}\|T-X\|,\,\,T\in\mathcal{B}(H)$ and $r_{\mathcal{X}}(T)=\su …

Do we know examples of separably acting von Neumann algebras $\mathcal{M}$ such that $\mathcal{M}$ is unitarily equivalent to $M_{k}(\mathcal{M})$ for some $k\in\mathbb{N}\,?$ Obviously there are exam …

Do we have a counterexample of injective von Neumann algebras $\mathcal{M}$ and $\mathcal{N}$ acting on the Hilbert space $H$ such that $\mathcal{M}\cap \mathcal{N}$ is not injective ?

Let $\mathcal{M}$ be a type $II_1$ factor with tracial state $\tau$ and $P$ be a projection in $\mathcal{M}$ such that $\tau(P)=1/n$ for some natural number $n.$ It is known (Ananatharaman-Popa "An in …

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 $\m …