Operator Algebrae/Von Neumann algebrae

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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]