# Operator Algebrae/Von Neumann algebrae

## The operator algebra

Definition (operator algebra):

Let ${\displaystyle X}$  be a Banach space over the field ${\displaystyle \mathbb {K} =\mathbb {R} }$  or ${\displaystyle \mathbb {C} }$ . Consider the set ${\displaystyle B(X)}$  of bounded and linear functions from ${\displaystyle X}$  to itself. This

## Operator topologies

### Topologies on a Banach space

Definition (weak topology):

Let ${\displaystyle X}$  be a Banach space, and let ${\displaystyle X^{*}}$  be its dual space. The weak topology on ${\displaystyle X}$  is defined to be the initial topology with respect to the maps ${\displaystyle x\mapsto x^{*}(x)}$ , where ${\displaystyle x^{*}}$  ranges over ${\displaystyle X^{*}}$ .

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 ${\displaystyle X}$  be a Banach space, and equip the space ${\displaystyle B(X)}$  with

Definition (uniform topology):

## Von Neumann algebrae, basic constructions

Definition (von Neumann algebra):

A von Neumann algebra is a subalgebra ${\displaystyle A\leq B(H)}$  which is closed under the weak operator topology.