Integral representation. Choquet theory for linear operators on function spaces (Q6094106)

The present book is a very deep and profound contribution to the Choquet representation theory. The origin of this theory can be found in the paper [C. R. Acad. Sci., Paris 243, 699--702 (1956; Zbl 0071.10702)], where \textit{G.~Choquet} established an integral representation theorem for positive linear functionals on the space of continuous affine functions on a compact convex metrizable set. It is shown there that any such functional can be represented by a regular measure that is supported by the set of extreme points. This theorem was later generalized to nonmetrizable compact spaces by \textit{E.~Bishop} and \textit{K.~de Leeuw} [Ann. Inst. Fourier 9, 305--331 (1959; Zbl 0096.08103)]. This classical approach to integral representation theory is well described in the excellent books by \textit{E.~M. Alfsen} [Compact convex sets and boundary integrals. Berlin: Springer-Verlag (1971; Zbl 0209.42601)], \textit{R.~R. Phelps} [Lectures on Choquet's theorem. Berlin: Springer (2001; Zbl 0997.46005)] and in the survey paper by \textit{V.~P. Fonf} et al. [in: Handbook of the geometry of Banach spaces. Volume~1. Amsterdam: Elsevier. 599--670 (2001; Zbl 1086.46004)]. The classical setting deals with a subspace \(H\) of the space \(C(K,\mathbb{R})\) of all continuous real functions on a compact Hausdorff space \(K\) such that \(H\) separates the points of \(K\) and contains the constant functions. A natural attempt is to generalize this framework to a complex or even a vector-valued function space \(H\), even for a general subspace \(H\). This was achieved in a series of papers due to O.~Hustad, R.~Fuhr and R.~R. Phelps, P.~Saab and M.~Talagrand. The approach to the integral representation in the present book leans on ideas by the author in [J. Lond. Math. Soc., II.~Ser. 34, 81--96 (1986; Zbl 0609.46004)] and their generalization due to \textit{C.~J.~K. Batty} [Proc. Lond. Math. Soc. (3) 60, No.~3, 530--548 (1990; Zbl 0672.46016)]. The author improves the Choquet theory in such a way that it encompasses locally compact domains, spaces of vector-valued continuous functions and the representation of linear operators. In order to achieve this goal, a suitable integration theory for vector-valued functions with respect to operator-valued measures is introduced. The first part of the text is thus dedicated to the development of a suitable integration theory. The main result of the first part then establishes an integral representation for continuous linear operators on spaces of vector-valued functions. The second part of the book then uses this theory in order to generalize the Choquet theory to the framework mentioned above. Here is a rough description of the contents (partly borrowing formulations from the book) which is meant as an invitation to look more closely at the profound material offered by the book. The first chapter compiles general prerequisities and frequently used terminology and notation. Spaces of vector-valued functions and the concept of function space neighborhoods are introduced as well as particular topologies on suitable spaces of continuous vector-valued functions on locally compact spaces. Several aspects of continuous linear operators on locally convex topological vector spaces, different operator space topologies and notions of compactness in such spaces are reviewed. As a preparation for the development of operator-valued measures locally convex cones are introduced. The second chapter investigates an integral representation of linear operators from a space of vector-valued continuous functions into a locally convex Hausdorff space. Thus, the chapter starts with operator-valued measures and measurability of vector-valued functions. These concepts are then used for the definition of integrals of vector- and set-valued functions. Next general convergence theorems are presented. This part culminates in a general theorem on integral representation for linear operators on function spaces (this theorem recovers e.g. the Bartle-Dunford-Schwartz theorem). The third chapter extends the classical Choquet theory to spaces of linear operators on function spaces utilizing the results of the previous chapters. This extension is far from being trivial, as the text deals with locally compact domains, vector-valued functions and linear operators (instead of functionals). The technique presented uses the notion of a \(C(X)\)-convex set in a suitable tensor product. A Choquet order on the space of continuous linear operators from the space of vector-valued continuous functions into locally convex space is established which then provides the sought minimal representing elements. Analogues of envelopes and the Choquet boundary are introduced and a localization of the support of the representation measures of minimal operators is presented. The chapter concludes with special cases and examples of the general theory. The presented theory is a very interesting development of Choquet theory in a highly nontrivial framework. Thus this book is recommended to everyone interested in Choquet representation theory.