Operator Algebrae/Von Neumann algebrae

The operator algebraEdit

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 topologiesEdit

Topologies on a Banach spaceEdit

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 spacesEdit

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 constructionsEdit

Definition (von Neumann algebra):

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

Von Neumann bicommutant theoremEdit