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6 votes
1 answer
175 views

Lifting a semicommutative von Neumann algebra into an algebra of operator-valued functions

Given a complete probability space $\Omega$ and a von Neumann algebra $\mathcal M$, it is well-known that the tensor product von Neumann algebra $\mathrm L^\infty(\Omega)\overline\otimes\mathcal M$, ...
P. P. Tuong's user avatar
0 votes
0 answers
390 views

Monotone convergence theorem for increasing net of positive functions

Suppose that we have $(\Omega,\mu)$ a $\sigma$-finite measure space. I have the following question. (Assume that $(f_i)_{i\in I}$ be an increasing net of positive measurable functions such that $f_i\...
A beginner mathmatician's user avatar
3 votes
0 answers
227 views

Is there a noncommutative version of von Neumann's ergodic theorem? [closed]

The two most celebrated ergodic theorems are Birkhoff's ergodic theorem and von Neumann's ergodic theorem. E. C. Lance in his remarkable work (Ergodic Theorems for Convex Sets and Operator Algebras) ...
Neil hawking's user avatar
6 votes
2 answers
248 views

Extension of a von Neumann algebra by a von Neumann algebra

I asked this question at MSE now I repeat it at MO: Let $A,B,C$ be $3$ unital $C^*$ algebras. Assume that we have the following short exact sequence of $C^*$-algebras: $$0\to A\to C\to B\...
Ali Taghavi's user avatar
2 votes
1 answer
324 views

Direct proof a property of hyperstonean spaces

First, let me state some basic facts and definitions for my question. I believe these are well-known among experts working on von Neumann algebras, but let me state them anyway since my question is ...
Rick Sternbach's user avatar
2 votes
0 answers
164 views

An operator valued Egoroff's theorem

The following statements suggests $B(H)$-valued Egoroff's theorem when $H$ is a separable Hilbert space. Probably it will be hold even if a von Neumann algebra $M$ whose predual is separable is ...
ABB's user avatar
  • 4,058
6 votes
2 answers
1k views

Commutative von Neumann algebras and localizable measure spaces

This is not my subject so I apologize if my question is too obvious or understood from other pages. I read some pages such as Reference for the Gelfand-Neumark theorem for commutative von Neumann ...
Ilan Barnea's user avatar
  • 1,344
17 votes
3 answers
3k views

Which sigma-ideals in a sigma-algebra are ideals of null sets?

My question is motivated, to be somewhat vague, by an attempt to see how much a measure space is defined by the set of null sets. In other words, assume we are not given a concrete measure on a space ...
Super-Measurable Analyst's user avatar
3 votes
1 answer
565 views

When does a $W^*$-algebra have a standard Borel spectrum?

EDIT: André Henriques has commented below that the correct separability condition is not weak-* separability as I have written below, but separability of the predual. This post came out a bit long, ...
Super-Measurable Analyst's user avatar
7 votes
2 answers
518 views

Is there an inclusion of $L_\infty(G)$ into $C_0(G)^{**}$?

This is probably a silly question with which I am stuck however. Let $G$ be a locally compact group. It seems to me that there is a canonical map of $L_\infty(G)$ into $M=C_0(G)^{**}$ (the latter is ...
Yulia Kuznetsova's user avatar
11 votes
4 answers
2k views

What kind of completion is this?

Let $X$ be a compact Hausdorff space, and $C(X)$ the unital commutative C*-algebra of continuous functions on it. The double Banach dual $C(X)^{**}$ is a commutative von Neumann algebra and hence has ...
Chris Heunen's user avatar
  • 3,937
2 votes
2 answers
865 views

Decomposition of an abelian von Neumann algebra

Hi, I came across the statement below and I couldn't figure out why it is true. I was hoping someone could explain it or give me a good reference. Thank you in advance. "Let $\pi$ be a non-degenerate ...
Wishiwere Smarter's user avatar
37 votes
5 answers
4k views

Reference for the Gelfand duality theorem for commutative von Neumann algebras

The Gelfand duality theorem for commutative von Neumann algebras states that the following three categories are equivalent: (1) The opposite category of the category of commutative von Neumann ...
Dmitri Pavlov's user avatar