Jump to content

Operator Algebrae/Printable version

From Wikibooks, open books for an open world


Operator Algebrae

The current, editable version of this book is available in Wikibooks, the open-content textbooks collection, at
https://en.wikibooks.org/wiki/Operator_Algebrae

Permission is granted to copy, distribute, and/or modify this document under the terms of the Creative Commons Attribution-ShareAlike 3.0 License.

Von Neumann algebrae

The operator algebra

[edit | edit source]

Definition (operator algebra):

Let be a Banach space over the field or . Consider the set of bounded and linear functions from to itself. This

Operator topologies

[edit | edit source]

Topologies on a Banach space

[edit | edit source]

Definition (weak topology):

Let be a Banach space, and let be its dual space. The weak topology on is defined to be the initial topology with respect to the maps , where ranges over .

Theorem (properties of the weak topology):

Topologies exclusively for operator spaces

[edit | edit source]

Proposition (bounded operators on a normed space form a Banach space under norm topology):

Let be a Banach space, and equip the space with

Definition (uniform topology):

Von Neumann algebrae, basic constructions

[edit | edit source]

Definition (von Neumann algebra):

A von Neumann algebra is a subalgebra which is closed under the weak operator topology.

Von Neumann bicommutant theorem

[edit | edit source]