---
tag: PSFWQVZ0FS
kind: concept
name: Von Neumann algebra
msc: [46L10]
sources:
  - url: https://en.wikipedia.org/wiki/Von_Neumann_algebra
same_as:
  wikidata: Q1970172
  wikipedia: Von Neumann algebra
---

## Summary

A von Neumann algebra is an algebra of bounded operators on a Hilbert space that contains the identity and is closed under taking adjoints.
It must also be closed in the [[weak operator topology|4XM1VN1C1N]].
It is a special kind of C*-algebra, one that is rich in projections and admits a measurable functional calculus.
Commutative von Neumann algebras are the algebras $L^\infty$ of measure spaces, so the theory is often called noncommutative measure theory.
John von Neumann introduced them and studied them with Francis Murray.
They are basic objects in operator algebras, quantum physics and representation theory.

## Explanations

### L1 human

Take some operators on a Hilbert space and keep adding sums, products, adjoints and limits.
If the limits are taken in the norm, one gets a C*-algebra.
If they are taken in the much weaker sense of matrix coefficients, one gets a von Neumann algebra.
The weaker limits add many projections, the operator analogues of indicator functions of sets.
For example, the continuous functions on $[0,1]$, acting by multiplication on $L^2[0,1]$, do not form a von Neumann algebra.
Adding the weak limits gives all bounded measurable functions, which do.
The matrix algebras $M_n(\C)$ are the simplest examples.