Let $\omega$ be a positive linear functional on $M$ which is a Von Neumann Algebra. Suppose $\omega$ is completely additive (i.e. $\omega$ applied to a strongly convergent sum of mutually orthogonal projections is the same as the convergent complex valued sum of $\omega$ applied to the projections). Then supposedly, $\omega$ is the pointwise convergent sum: […]

Let $B(H)$ be the set of bounded operators on a Hilbert space $H$. I know that $u_{\alpha}\longrightarrow u$ in S.O.T if and only if $u_{\alpha}(x)\longrightarrow u(x)$ for all$x\in H$. I know that $\cdot : B(H)\times B(H) \longrightarrow B(H)$ such that $(u,v)\longmapsto uv$ is separately continuous and jointly continuous on bounded set. […]

Let $H$ be an infinite dimensional separable Hilbert space. Bicommutant theorem : Let $\mathcal{S}$ be $*$-subset of $B(H)$, then $\mathcal{S}”$ is the strong closure $\overline{\langle \mathcal{S} \rangle}$ of the algebra $\langle \mathcal{S} \rangle$ generated by $\mathcal{S}$ (called the von Neumann algebra generated by $\mathcal{S}$). Now if $\mathcal{S}$ is a non-selfadjoint subset of $B(H)$, what can […]

Let $A$ and $B$ are two factor von neumann algebras that act on two infinite dimensional Hilbert spaces H and K respectively. Let $\Phi:A\longrightarrow B$ is an additive bijective map with some other conditions that it is not necessary to state them. my question is that if we know $\mathbb{R}I\subseteq \Phi(\mathbb{R}I)$, can we conclude by […]

Let $\mathcal H, \mathcal K$ be Hilbert spaces and $M \subseteq B(\mathcal H)$ a (concrete) von Neumann algebra (Here, $B(\mathcal H)$ denotes the algebra of bounded operators on $\mathcal H$). Furthermore, let $$\pi \colon M \to B(\mathcal K)$$ be a *-Homomorphism. Note that $\pi$ is automatically a contractive, i.e. especially bounded, operator. I want to […]

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : An operator $T \in B(H)$ is irreducible if $W^{*}(T)=B(H)$. Definition : A $C^{*}$-algebra is exact if it preserves exact sequences under the minimum tensor product. Property : A $C^{*}$-algebra is exact if and only if : it’s […]

The problem I am considering stated formally is this: Show that if a sequence in $\ell_\infty^*$ is weak*-convergent, then it is also weakly convergent. We may reduce this to the case where the sequence is weak$^*$-null, and show that it is weakly null. This is a special case of a result of Grothendieck from the […]

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