B. de Rijk The Order Bicommutant A study of analogues of the von Neumann Bicommutant Theorem, reflexivity results and Schur’s Lemma for operator algebras on Dedekind complete Riesz spaces

It this thesis we investigate whether an analogue of the von Neumann Bicommutant Theorem and related results are valid for Riesz spaces. Let H be a Hilbert space and D ⊂ Lb(H) a ∗-invariant subset. The bicommutant D ′ ′ equals P(D ′)′, where P(D ′) denotes the set of pro-jections in D ′. Since the sets D ′ ′ and P(D ′) ′ agree, there are multiple possibilities to define an analogue of bicommutant for Riesz spaces. Let E be a Dedekind complete Riesz space and A ⊂ Ln(E) a subset. Since the band generated by the projections in Ln(E) is given by Orth(E) and order projections in the commutant correspond bijectively to reducing bands, our approach is to define the bicommutant of A on E by U: = (A ′ ∩Orth(E))′. Our first result is that the bicommutant U equals {T ∈ Ln(E) : T is reduced by every A-reducing band}. Hence U is fully characterized by its reducing bands. This is the analogue of the fact that each von Neumann algebra in Lb(H) is reflexive. This result is based on the following two observations. Firstly, in Riesz spaces there is a one-to-one correspondence between bands and order projections, instead of a one-to-one correspondence between closed subspaces and projections. Secondly, every ∗-invariant subset of Lb(H) is reduced by each invariant sub