⧼exchistory⧽
3 exercise(s) shown, 0 hidden
SORT:
BBot
Apr 22'25
[math] \newcommand{\mathds}{\mathbb}[/math]

This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.

Look up and learn von Neumann's reduction theory, stating that given a von Neumann algebra [math]A\subset B(H)[/math], if we write its center as

[[math]] Z(A)=L^\infty(X) [[/math]]

then we have a decomposition as follows, with the fibers [math]A_x[/math] having trivial center,

[[math]] A=\int_XA_x\,dx [[/math]]

and then write down a brief account of what you learned.

BBot
Apr 22'25
[math] \newcommand{\mathds}{\mathbb}[/math]

This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.

Given two projections [math]P,Q\in B(H)[/math], with [math]H[/math] being infinite dimensional, find an elementary proof for the fact that we have, for any [math]x\in H[/math],

[[math]] (PQ)^nx\to (P\wedge Q)x [[/math]]

but the operators [math](PQ)^n[/math] do not necessarily converge in norm.

BBot
Apr 22'25
[math] \newcommand{\mathds}{\mathbb}[/math]

This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.

Given a commutative von Neumann algebra, written as

[[math]] A=L^\infty(X) [[/math]]

with [math]X[/math] being a measured space, write, by using the Gelfand theorem,

[[math]] A=C(\widehat{X}) [[/math]]

with [math]\widehat{X}[/math] being a compact space, and understand the correspondence [math]X\to\widehat{X}[/math].