Banach spaces of weights on quasimanuals (Q1067162)

Section 1 is a brief introduction. Section 2 contains the basic definitions of quasimanuals, weights, and operational logics. The linear space \({\mathcal W}\) of all weights on a quasimanual \({\mathcal A}\) is introduced and given a norm. \({\mathcal W}\) with this norm is seen to be a Banach space. The subspace \({\mathcal V}\) of \({\mathcal W}\) generated by the positive cone of \({\mathcal W}\) is given the base norm and is also shown to be an Archimedian ordered Banach space with an additive norm. In Section 3 normal linear functionals on \({\mathcal V}^*\) are defined in analogy with normal linear functionals on \(W^*\) algebras. The space \({\mathcal V}\) is shown to be the set of normal functionals on \({\mathcal V}^*\) and we show \({\mathcal V}\) to be the unique partially ordered Banach space with a closed generating cone which is predual to \({\mathcal V}^*\). Next, weakly compact subsets of \({\mathcal W}\) are characterized in terms of eventwise convergence. This is the Hahn-Vitali-Saks theorem of classical measure theory in this noncommutative setting; several weak compactness results are drawn from this and compared with their classical counterparts. Section 4 introduces the ultraweak topology for \({\mathcal V}^*\) in analogy with the same for the trace class operators on Hilbert space. Here the condition for a compact base for the cone of \({\mathcal V}\) is examined and shown to be a poor and unnecessary hypothesis in many circumstances. Many connections with the existent literature are made and throughout the paper there are many examples and open questions.