Some problems in Banach space theory
Types were introduced by Krivine and Maurey, in a refinement of a result by Aldous showing that infinite dimensional subspaces of Lr contain Ωp for some 1≤pꝏ) . A synthesis of these ideas was provided by Garling whose representation of types as random measures was the motivation for much of this work. This thesis aims to investigate the structure of the representation, and to provide concrete representations for differing Banach spaces. Chapter one contains the necessary preliminaries for the later chapters, and finishes by introducing the representation due to Garling of types on Lϕ(X) as random measures on τ(X) The second chapter consists of two parts. In the first part we examine the structure of the map between types on Lp(X) and random measures on τ(X) . We show that convolution is preserved by the mapping, and give an explicit representation of the space of types on L1(Ωp). The second part is concerned with representations of τ(X) . We give conditions for the decomposition of τ(X) into X*S(X) , and derive representations for the space of types on L1(L2k). The third chapter studies differentiability of types. We extend differentiability from X to τ(X) , and develop ideas that will be used in the study of uniqueness. In chapter four we consider questions concerning the uniqueness of measures and random measures on X and τ(X) . We construct spaces where the representation of types as random measures is not in uniquely determined. We prove that if a certain uniqueness property for measures on X fails then Ωn1 embeds in X.