# Operator Algebrae/Printable version

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# 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.