# Operator Algebrae/Von Neumann algebrae

## The operator algebra

Definition (operator algebra):

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

## Operator topologies

### Topologies on a Banach space

Definition (weak topology):

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

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 $X$  be a Banach space, and equip the space $B(X)$  with

Definition (uniform topology):

## Von Neumann algebrae, basic constructions

Definition (von Neumann algebra):

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