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.