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Category measures, the dual of the Dedekind completion of C(K) spaces and hyper-Stonean spaces

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Professor Jan Harm van der Walt from the University of Pretoria has just delivered his lecture at the RGOSA seminar series, titled “Category measures, the dual of {C(K)^\delta} and hyper-Stonean spaces”.

The abstract of the talk is here:

For a compact Hausdorff space {K}, we give descriptions of the dual of {C(K)^\delta}, the Dedekind completion of the Banach lattice {C(K)} of continuous, real-valued functions on {K}. We characterize those functionals which are {\sigma}-order continuous and order continuous, respectively, in terms of Oxtoby’s category measures. As applications, we give a purely topological characterization of hyper-Stonean spaces, and characterize those spaces {K} for which {C(K)} admits a strictly positive order continuous functional.

Please find the seminar video attached, and the presentation slides below.