We study the geometric classi&#64257;cation of Banach spaces via Lipschitz, uniformly continuous, and coarse mappings. We prove that a Banach space which is uniformly homeomorphic to a linear quotient of lp is itself a linear quotient of lp when p<2. We show that a Banach space which is Lipschitz universal for all separable metric spaces cannot be asymptotically uniformly convex. Next we consider coarse embedding maps as de&#64257;ned by Gromov, and show that lp cannot coarsely embed into a Hilbert space when p> 2. We then build upon the method of this proof to show that a quasi-Banach space coarsely embeds into a Hilbert space if and only if it is isomorphic to a subspace of L0(??) for some probability space (&#937;,B,??).

Thesis (Ph. D.)--Lehigh University, 2000. / Includes vita. Includes bibliographical references (leaves 49-50).

Thèse de doctorat : Mathématiques : Bordeaux 1 : 2009. Thèse de doctorat : Mathématiques : Univerzita Karlova (Prague) : 2009. / Titre provenant de l'écran-titre.

Title from PDF of title page (University of Missouri--Columbia, viewed on Feb. 20, 2010). The entire thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file; a non-technical public abstract appears in the public.pdf file. Dissertation advisor: Professor Nigel J. Kalton. Vita. Includes bibliographical references.

The purpose of this paper is two fold. Firstly, to introduce and study the Silov Boundary of a Banach algebra of continuous complex valued functions defined on a compact Hausdorff space X. Secondly, to apply the definition of Silov Boundary to more general families of functions, namely, to linear spaces and semi-groups of functions. It will be seen that in these latter cases, existence of the Silov Boundary depends on certain restricting assumptions about the linear space or semi-group. [...]

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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.