Mackey topologies and mixed topologies in Riesz spaces (Q850598)

Let \(E\) be a vector space, \(\mathcal B\) a convex bornology on \(E\) and \(\tau\) a vector topology on \(E\) such that each \(B\in\mathcal B\) is \(\tau\)-bounded. The triple \((E,\mathcal B,\tau)\) is then called a mixed space and \(\tau \) a mixed topology. The finest locally convex topology \(\gamma_{\tau}(\mathcal B)\) coinciding with \(\tau\) on the sets from \(\mathcal B\) is an example of a mixed topology. In the paper under review, mixed topologies defined by the bornology of order bounded subsets of a Riesz space (vector lattice) are investigated. In particular, let \(E\) be a Riesz space such that the set \(E_n^\sim\) of all order continuous linear functionals on \(E\) separates the points of \(E\), and let \(\pi(E,E_n^\sim)\) be the topology of uniform convergence on the order weakly precompact subsets of \(E_n^\sim\). Then the topology \(\pi\) can be characterized as a mixed topology and as the so-called absolute Mackey topology. Results of this kind in the case of the spaces of bounded continuous functions are of a bit different nature, since the respective dual spaces may not separate the points of the given space. In the last section, results on weak compactness in symmetric operator spaces are used to show that Mackey topologies on non-commutative Banach function spaces can often be characterized as mixed topologies.