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The construction of a non-separable reflexive Banach space on which every operator is the sum of a scalar multiple of the identity operator and an operator of separable range is presented. Using a result of Rao, a sufficient condition is given for Banach spaces with smooth norms to be decomposable. It is shown that operators on Banach spaces of co-dimension one in their biduals are the sum of a scalar multiple of the identity operator and a weakly compact operator. The Banach spaces of bounded operators L(1¹, 1^p) (1<p<ꝏ) and L(1^p, 1^r), 1 < p ≤ r ≤ p¹ < ꝏ, where 1/p + 1/p¹ = 1, are shown to be primary. The spaces of bounded diagonal operators and compact diagonal operators on a seminormalized Schauder basis β, the multiplier algebras L<sup>d</sub>(X, β) and K_d(X, β), are introduced and studied. New examples of these multiplier algebras are presented and a theorem of Sersouri is extended. A necessary and sufficient condition is given for c_o to embed in K_d(X, β). A sufficient condition is given on a semi-normalized Schauder basis β of a reflexive hereditarily indecomposable Banach space Y to ensure that K_d(Y, β) has the RNP. It is shown that the algebra L_d(X, β) is semisimple and that on the algebra K_d(X, β) derivations are automatically continuous. By representing diagonal operators as stochastic processes a general method of constructing multiplier algebras is given. A non trivial multiplier invariance for the normalized Haar basis of L¹[0,1] is proved.

Types were introduced by Krivine and Maurey, in a refinement of a result by Aldous showing that infinite dimensional subspaces of L_r 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 L_ϕ(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 L_p(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(Ω_p). 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(L_2k). 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.

Thesis (Ph. D.)--Lehigh University, 1997. / Includes vita. Includes bibliographical references (leaf 46).

Thesis (Ph. D.)--University of Missouri-Columbia, 1997. / Typescript. Vita. Includes bibliographical references (leaves 38-40). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1997. / Typescript. Vita. Includes bibliographical references (leaves 38-40). Also available on the Internet.

Thesis (Ph. D.)--University of Florida, 1980. / Description based on print version record. Typescript. Vita. Includes bibliographical references (leaves 61-62).