---
tag: F12V24NF4R
kind: definition
name: Von Neumann algebra
msc: [46L10]
defines: PSFWQVZ0FS
uses: [HM2HHE3J51, BMDG4JQPEH, 6HSJBJ8TTQ, 4XM1VN1C1N]
sources:
  - url: https://en.wikipedia.org/wiki/Von_Neumann_algebra
    locator: Definitions
---

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $B(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$, with the [[adjoint|6HSJBJ8TTQ]] $T \mapsto T^*$.
A *von Neumann algebra* on $H$ is a subset $M \subseteq B(H)$ such that:

1. $M$ is a subalgebra of $B(H)$ that contains the identity operator $I$;
2. $T^* \in M$ for every $T \in M$;
3. $M$ is closed in the [[weak operator topology|4XM1VN1C1N]].

## Remarks

Example: let $(X, \mu)$ be a $\sigma$-finite measure space and $H = L^2(X, \mu)$.
For $f \in L^\infty(X, \mu)$ let $M_f$ be multiplication by $f$.
The set $\set{M_f : f \in L^\infty}$ contains $I = M_1$, is closed under products and under $M_f^* = M_{\bar f}$, and one checks that it equals its own commutant, so it is weakly closed.
Other examples are $B(H)$ itself and the matrix algebras $M_n(\C)$ acting on $\C^n$.

Non-example: on $H = L^2[0,1]$, the multiplication operators $M_f$ with $f$ continuous satisfy clauses 1 and 2 and form a C*-algebra.
But clause 3 fails.
The indicator function of a subinterval is a pointwise limit of uniformly bounded continuous functions.
By dominated convergence, its multiplication operator is a weak limit of elements of the algebra, but it is not continuous.
The weak closure of this algebra is the whole multiplication algebra of $L^\infty[0,1]$.

By the von Neumann bicommutant theorem, clause 3 can be replaced by closedness in the strong operator topology, or by the condition $M = M''$, where $M'$ is the commutant.
Every von Neumann algebra is a C*-algebra.
By Sakai's theorem, an abstract C*-algebra is isomorphic to a von Neumann algebra exactly when it is a dual Banach space.
Such algebras are called W*-algebras.