von Neumann Double Commutant Theorem
0.1 Introduction
Definition
1
Blah
commutant The commutant of a set \(S\) of operators is the set of \(T\in B(\mathcal{H})\) such that \(\forall s\in S,\; Ts=sT\).
Definition
2
weak_op_closure For \(A\subseteq B(\mathcal{H})\), the weak-operator closure \(\overline{A}^{\mathrm{WOT}}\) is the closure of \(A\) in the weak operator topology on \(B(\mathcal{H})\).
Definition
3
self_adj_star_subalg A \(*\)-subalgebra \(A\subseteq B(\mathcal{H})\) is self-adjoint if \(T\in A \Rightarrow T^*\in A\).
Theorem
4
von Neumann 1929
double_commutant def:commutant def:weak_op_closure def:self_adj_star_subalg The weak operator closure of a self-adjoint \(*\)-subalgebra of \(B(\mathcal{H})\) equals its double commutant.