Von Neumann algebra

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.