The operator algebra
Definition (operator algebra):
Let be a Banach space over the field or . Consider the set of bounded and linear functions from to itself. This
Topologies on a Banach space
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
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
Definition (von Neumann algebra):
A von Neumann algebra is a subalgebra which is closed under the weak operator topology.
Von Neumann bicommutant theorem