Thesis (M. Phil.)--University of Hong Kong, 2002. / Includes bibliographical references (leaves 51-52).

published_or_final_version / abstract / toc / Mathematics / Master / Master of Philosophy

Thesis (testi di perfezionamento)--Scuola normale superiore, Pisa, 1969. / At head of title: Scuola normale superiore, Pisa. Classe de scienze. Includes bibliographical references (p. 151-153).

The purpose of the paper is to prove that the Lp spaces, p ≥ 1, of equivalence classes of functions are Banach spaces.

No description available.

by Lo Chui-sim. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 91-93). / Abstract --- p.i / Introcution --- p.1 / Chapter 1 --- Preliminaries --- p.3 / Chapter 1.1 --- Preliminaries on von-Neumann algebra --- p.3 / Chapter 1.2 --- Modular theory --- p.6 / Chapter 2 --- Abstract Lp Spaces --- p.10 / Chapter 2.1 --- "Preliminaries on dual action, dual weights and extended positive part" --- p.10 / Chapter 2.2 --- Abstract LP spaces associated with von-Neumann algebras --- p.20 / Chapter 2.3 --- "LP(M) is a Banach space for p E [1, ∞ ]" --- p.25 / Chapter 2.4 --- Independence of the choice of ψ --- p.32 / Chapter 3 --- Spatial Lp Spaces --- p.34 / Chapter 3.1 --- Definition and elementary properties of spatial derivative --- p.35 / Chapter 3.2 --- Modular properties of spatial derivatives --- p.47 / Chapter 3.3 --- Spatial Lp spaces --- p.51 / Chapter 4 --- LP Spaces constructed by using complex interpolation method --- p.60 / Chapter 4.1 --- The complex interpolation space --- p.60 / Chapter 4.2 --- LP space with respect to a faithful normal state --- p.71 / Chapter 4.3 --- LP spaces with respect to a normal faithful semifinite weight . . --- p.78 / Chapter 4.4 --- Equivalence to spatial LP spaces --- p.87 / Bibliography --- p.91

Thesis (Ph. D.)--Oregon State University, 1994. / Typescript (photocopy). Includes bibliographical references (leaves 41-45). Also available on the World Wide Web.

Nous étudions des propriétés de rigidité et des propriétés de non-rigidité forte d'actions de groupes sur des espaces Lp non-commutatifs. Récemment, des variantes de la propriété (T) de Kazhdan et de la propriété de point fixe (FH) ont été introduites, appelées respectivement propriété (TB) et propriété (FB), et énoncées en termes de représentations orthogonales sur un espace de Banach B. Nous nous intéressons au cas où B est un espace Lp non-commutatif Lp(M), associé à une algèbre de von Neumann M. Dans un premier temps, nous montrons qu'un groupe possédant la propriété (T) possède la propriété (TLp(M)) pour toute algèbre de von Neumann M. On en déduit que les groupes de rang supérieur ont la propriété (FLp(M)). Nous montrons que pour certaines algèbres, comme par exemple M=B(H), les propriétés (T) et (TLp(M) sont équivalentes. A l'opposé, nous caractérisons les groupes possédant la propriété (Tlp), et montrons que cette classe de groupes est strictement plus grande que celle avec la propriété (T). Dans un second temps, nous introduisons des variantes de la propriété (H) de Haagerup, les propriétés (HLp(M)) et l' a-FLp(M)-menabilité, définies en termes d'actions sur l'espace Lp(M). Nous décrivons les liens entre la propriété (H) et sa variante (HLp(M)) suivant l'algèbre M considérée. Nous montrons que les groupes possédant (H) sont a-FLp(M)-menables pour certaines algèbres M, comme par exemple le facteur II infini hyperfini. / We studied rigidity properties and strong non-rigidity properties for group actions on non-commutative Lp spaces. Recently, variants of Kazhdan's property (T) and fixed-point property (FH) were introduced, respectively called property (TB) and property (FB), and described in terms of orthogonal representations on a Banach space B. We are interested in the case where B is a non-commutative Lp space Lp(M), associated to a von Neumann algebra M. In a first part, we show that if a group has property (T), then it has property (TLp(M)) for any von Neumann algebra M. We deduce that higher rank groups have property (FLp(M)). We show that for some algebras, such as M=B(H), properties (T) and (TLp(M)) are equivalent. By contrast, we characterize groups with property (Tlp), and show that this class of groups is larger than the one with property (T). In a second part, we introduce variants of the Haagerup property (H), namely properties (HLp(M)) and a-FLp(M)-menability, defined in terms of actions on the space Lp(M). We describe relationships between property (H) and its variant (HLp(M)) for different algebras M. We show that groups with property (H) are a-FLp(M)-menable for some algebras M, such as the hyperfinite II infinite factor.

In this thesis we examine different types of convergence for sequences of functions and how these are related. The functions considered are real valued Lebesgue measurablefunctions defined on a subset of R. We also devote a chapter to explore when continuity of a sequence of functions is preserved under pointwise convergence, and see that this happens precisely when the convergence is quasi uniform. / I denna uppsats utforskar vi olika typer av konvergens för funktionsföljder för att se hur de är besläktade. Funktionerna i fråga är reellvärda Lebesguemätbara funktioner definierade på delmängder av R. Vi ägnar också ett kapitel åt att undersöka när kontinuitet hos en följd av funktioner bevaras under punktvis konvergens och ser att detta händer precis då konvergensen är kvasilikformig.

Dados um espaço de medida (X;A;m) e números reais p,q>1 com 1/p+1/q=1, o Teorema de Representação de Riesz afirma que Lq(X;A;m) é o dual topológico de Lp(X;A;m) e que Loo(X;A; m) é o dual topológico de L1(X;A;m) se o espaço (X;A;m) for sigma-finito. Observamos que a sigma-finitude de (X;A;m) é condição suficiente mas não necessária para que Loo(X;A;m) seja o dual de L1(X;A;m). Os contra-exemplos tipicamente apresentados para essa última identificação são \"triviais\", no sentido de que desaparecem se \"consertarmos\" a medida , transformando-a numa medida perfeita. Neste trabalho apresentamos condições sufcientes mais fracas que sigma-finitude a fim de que Loo(X;A;m) e o dual de L1(X;A;m) possam ser isometricamente identificados. Além disso, introduzimos um invariante cardinal para espaços de medida que chamaremos a dimensão do espaço e mostramos que se o espaço (X;A;m) for de medida perfeita e tiver dimensão menor ou igual à cardinalidade do continuum então uma condição necessária e suficiente para Loo(X;A;m) seja o dual de L1(X;A;m) é que X admita uma decomposição. / Given a measure space (X;A;m) and real numbers p,q>1 with 1/p+1/q=1, the Riesz Representation Theorem states that Lq(X;A;m) is the topological dual space of Lp(X;A;m) and that Loo(X;A; m) is the topological dual space of L1(X;A;m) if (X;A; m) is sigma-finite. We observe that the sigma-finiteness of (X;A;m) is a suficient but not necessary condition for Loo(X;A;m) to be the dual of L1(X;A;m). The counter-examples that are typically presented for Loo(X;A;m) = L1(X;A;m)* are \"trivial\", in the sense that they vanish if we fix the measure , making it into a perfect measure. In this work we present suficient conditions weaker than sigma-finiteness in order that Loo(X;A; m) and L1(X;A;m)* can be isometrically identified. Moreover, we introduce a cardinal invariant for measure spaces which we call the dimension of the space and we show that if the space (X;A;m) has perfect measure and dimension less than or equal to the cardinal of the continuum then a necessary and suficient condition for Loo(X;A;m) = L1(X;A;m)* is that X admits a decomposition.