A von Neumann algebra is a unital *-subalgebra of the algebra of bounded operators on a Hilbert space, closed in the weak operator topology. Also called a $W^*$-algebra, and may be regarded as a non-commutative generalization of $L^\infty$ space. These algebras are extensively used in knot theory, ...

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679 views

$\ell_\infty$ 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 ...
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327 views

Complementability of von Neumann algebras

Is every von Neumann algebra complemented in its bidual? It is certainly true for commutative von Neumann algebras as their spectrum is hyperstonian. Is it 1-complemented?
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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^{*}...
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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 ...
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270 views

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{...
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Do we have Maximal Abelian Algebras (MAAs)?

Let $\mathcal{H}$ be a Hilbert space and $B(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$. A MASA $\mathcal{M}$ is a subalgebra of $B(\mathcal{H})$ that is abelian and self-...
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47 views

‎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 ...
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218 views

Closed unit ball of B(H) is not compact in strong operator topology of B(H).

In operator theory we prove that closed unit ball of B(H) is compact in weak operator topology and is closed in strong operator topology. But a book of operator theory states that closed unit ball of ...
4
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217 views

Complemented ideals in von Neumann algebras

Let $I$ be an ultraweakly closed ideal in a von Neumann algebra $M$. For example, this can be the kernel of an ultraweakly continuous homomorphism. Is it true that there is another ideal $J\subset M$ ...
3
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67 views

Arveson spectrum for a unitary representation of a group on a Hilbert space

Let $G = \mathbb{R}$. By Stone's theorem, $U(t)\in\mathcal{B}(\mathcal{H})$ is generated by a self-adjoint operator $H$ (for which there is a resolution of the identity P(p), by the spectral theorem) ...
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43 views

If $\|p-q\|<1$ then they are Murray–von Neumann equivalent- proof

$\newcommand{\ran}{\operatorname{ran}}$ Let $p$, $q$ be projections in a $C^*$-algebra $A$. If $\|p−q\|<1$ then $p$ and $q$ are MvN equivalent. I'm tryind to understand the following proof: ...
2
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1answer
39 views

projection in a factor von Neumann algebra.

We know that center of a factor von Neumann algebra $\mathcal{A} $ is trivial. Let $P_1$ be a projection in $\mathcal{A} $ such that $P_1\neq I,0$ . undoubtedly there exist another projection like $...
2
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50 views

Zhou operator theory book, Kaplanskys formula

In Zhou's operator theory book, Kaplanskys formula has stated that if $P$ and $Q$ are projection in a von neumann algebra $A$ acting on $H$, then $P\vee Q-Q\sim P-P\wedge Q$. In the proof, it says ...
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178 views

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 ...
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0answers
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Why does one only consider one-parameter groups in Borchers-Arveson theorem?

The theorem (Operator algebras and Quantum statistical mechanics vol. 1, Bratteli, Robinson, Thm. 3.2.46 p.261) roughly says that if one has a one parameter automorphism group $t \rightarrow\alpha_t$ ...
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1answer
51 views

The involution of tensor product

Proposition 3.1.8 (Linear independence). If $\{x_{1},...,x_{n}\}\subset X$ are linearly independent, $\{y_{1},...,y_{n}\}\subset Y$ are arbitrary and $$0=\sum\limits_{i=1}^{n}x_{i}\otimes y_{i}\in X\...
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37 views

Sequences of operators in $B(H)$ which converge in the weak operator topology

Let $H$ be a Hilbert space. Suppose that $\{u_n\} \subseteq B(H)$ is a sequence which converges to some operator $u$ in the weak operator topology, which means that for all $x,y\in H$ one has ‎$$‎‎‎‎\...
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40 views

an additive bijective map

Let H be a Hilbert space and $\Phi:B(H)\longrightarrow B(H)$ is an additive bijective map. If $\mathbb{R}I⊆\Phi(\mathbb{R}I)$, can we conclude by the bijectivity of $\Phi$ that ? $\mathbb{R}I=\Phi(\...