# Questions tagged [von-neumann-algebras]

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, non-commutative geometry, and quantum field theory.

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### Projections in a diffuse von Neumann algebra

If a von Neumann algebra $M$ has no minimal projections, we call it a diffuse von Neumann algebra. Suppose $e$ is a projection in a diffuse von Neumann algebra $M$. Since $M$ is diffuse, we can find ...
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### Normality implies lower $w$-semicontinuity

I got stuck with the following problem while following Section $10.14$ of the book 'Lectures on von Neumann algebras' by Stratila and Zsido. Problem: Let $\mathcal{M}$ be a von Neumann algebra. Show ...
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### noncommutative Holder inequality

I am learning the noncommunicative holder inequality. It is based on operator theory. I have two problems when i try to understand the proof of noncommunicative holder inequality Let $M$ be a von ...
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### does $X$ has an $\ell_2$-complemented copy if the dual $X^*$ has an $\ell_2$-complemented copy

Let $X$ be a Banach space. If we know that $X^*$ has a complemented subspace isomorphic to $\ell_2$, then can we say that $X$ has a complemented subspace isomorphic to $\ell_2$? Precisely, I want to ...
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### GNS construction for a normal state on a von Neumann algebra

Let $M$ be a von Neumann algebra. For any faithful normal state $\omega$ on $M$, according to the GNS construction, we have $\omega(x)=\langle x \Omega,\Omega\rangle$, where $\Omega$ is a cyclic ...
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### Determining the normal functionals on the direct sum of von Neumann algebras

Consider a collection $\{M_i: i \in I\}$ of von Neumann algebras and consider their $\ell^\infty$-direct sum $M$. I'm trying to show that a normal functional $\omega$ on $M$ is of the form \omega(m) ...
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### Showing the existence of a right-inverse in a von Neumann algebra

In this paper, specifically Theorem 4.1 on page 10, one of the last steps in the proof involves saying that a particular right-inverse exists. I'll try to restate what I think are the relevant pieces ...
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### Show that a certain net converges in a von Neumann algebra.

Consider the abstract von Neumann algebra $M = \ell^\infty\text{-}\bigoplus_{i \in I} B(H_i)$. Moreover, we assume $\dim H_i< \infty$ for all $i \in I$. Let $x_i$ be the identity on $B(H_i)$ and ...
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### What is an invariant mean on $L^\infty(\mathbb{R})$?

Consider the von Neumann algebra $M:=L^\infty(\mathbb{R})$, which consists of (classes) of essentially bounded measurable functions $\mathbb{R}\to \mathbb{C}$. Here $\mathbb{R}$ has the classical ...
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### Show that $(M_*)^*\cong M$ for a von Neumann algebra $M$.

Let $M \subseteq B(H)$ be a von Neumann algebra. Denote $B(H)_*$ to be the $\sigma$-weakly continuous functionals on $B(H)$ and let $M_*= \{\omega\vert_M: \omega \in B(H)_*\}$. I want to prove that ...
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### What do weak-star limits of normal states on von Neumann algebras look like?

Let $N$ be a von Neumann algebra and let $(\phi_\lambda)$ be a net of normal states on $N$ so that $\phi_\lambda\to\phi$ in the weak-* topology, i.e. $\phi_\lambda(x)\to\phi(x)$ for all $x\in N$. What ...
The Theorem in the screenshit is from Takesaki's book (Vol 2, Chapter VIII, section 3) When reading the proof of (iii)→(i), I met with troubles. How to prove $n_{\varphi}$ is a left ideal according ...