Articles of von neumann algebras

Positive Linear Functionals on Von Neumann Algebras

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: […]

‎Jointly ‎continuous of product in $B(H)$

‎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. ‎ ‎ […]

How generalize the bicommutant theorem?

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 […]

Is this equality true or it is not necessarily true?

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 […]

*-homomorphism between concrete von Neumann algebras is SOT-SOT continuous iff it is WOT-WOT continuous

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 […]

Does an irreducible operator generate an exact $C^{*}$-algebra?

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 […]

$\ell_\infty$ is a Grothendieck space

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 […]