Hyperstonean cover and second dual extension
Let T be a completely regular space and \(C^*(T)\) denote the vector lattice of all bounded continuous real-valued functions on T. A vector lattice X is said to be an extension of \(C^*(T)\) in case there exists an injective vector lattice homomorphism from \(C^*(T)\) into X. Let \(C^*(T)'\) denote the vector lattice of all bounded Radon measures on T, and let \(C^*(T)''\) denote the vector lattice of all order-bounded functionals on \(C^*(T)'\). The author presents a purely topological construction of a completely regular space R that is the unique hyperstonian cover of T, starting from the Kelley ideals of T. Among his main results he shows that \(C^*(T)''\) and \(C^*(R)\) are isomorphic. He then derives various characterizations of \(C^*(T)''\) among all extensions of \(C^*(T)\).
scientific article; zbMATH DE number 4092315
scientific article