Trabalhamos com funções definidas em Rn que tomam valores numa C*-álgebra A. Consideramos o conjunto SA (Rn) das funções de Schwartz, (de decrescimento rápido), com norma dada por ||f||2 = ||?f(x)*f(x)dx||½. Denotamos por CB?(R2n,A) o conjunto das funções C? com todas as suas derivadas limitadas. Provamos que os operadores pseudo-diferenciais com símbolo em CB?(R2n,A) são contínuos em SA(Rn) com a norma || ? ||2, fazendo uma generalização de [10]. Rieffel prova em [1] que CB?(Rn,A) age em SA(Rn) por meio de um produto deformado, induzido por uma matriz anti-simétrica, J, como segue: LFg(x)=F×Jg(x) = ?e2?iuvF(x+Ju)g(x+v)dudv, (integral oscilatória). Dizemos que um operador S é Heisenberg-suave se as aplicações z |-> T-zSTz e ? |-> M-?SM?, z,? E Rn, são C? onde Tzg(x)=g(x-z) e M?g(x)=ei?xg(x). No final do capítulo 4 de [1], Rieffel propõe uma conjectura: que todos os operadores \"adjuntáveis\" em SA(Rn), Heisenberg-suaves, que comutam com a representação regular à direita de CB?(Rn,A), RGf = f×JG, são os operadores do tipo LF. Provamos este resultado para o caso A=|C, ver [14], usando a caracterização de Cordes (ver [17]) dos operadores Heisenberg-suaves em L2(Rn) como sendo os operadores pseudo-diferenciais com símbolo em CB?(R2n). Também é provado neste trabalho que, se vale uma generalização natural da caracterização de Cordes, a conjectura de Rieffel é verdadeira. / We work with functions defined on Rn with values in a C*-algebra A. We consider the set SA(Rn) of Schwartz functions (rapidly decreasing), with norm given by ||f||2 = ||?f(x)*f(x)dx||½ . We denote CB?(R2n,A) the set of functions which are C? and have all their derivatives bounded. We prove that pseudo-differential operators with symbol in CB?(R2n,A) are continuous on SA(Rn) with the norm || · ||2, thus generalizing the result in [10]. Rieffel proves in [1] that CB?(Rn,A) acts on SA(Rn) through a deformed product induced by an anti-symmetric matrix, J, as follows: LFg(x)=F×Jg(x) = ?e2?iuvF(x+Ju)g(x+v)dudv (an oscillatory integral). We say that an operator S is Heisenberg-smooth if the maps z |-> T-zSTz and ? |-> M-?SM?, z,? E Rn are C?; where Tzg(x)=g(x-z) and where M?g(x)=ei?xg(x). At the end of chapter 4 of [1], Rieffel proposes a conjecture: that all ”adjointable” operators in SA(Rn) that are Heisenberg-smooth and that commute with the right-regular representation of CB?(Rn,A), RGf = f×JG, are operators of type LF . We proved this result for the case A = |C in [14], using Cordes\' characterization of Heisenberg-smooth operators on L2(Rn) as being the pseudo-differential operators with symbol in CB?(R2n). It is also proved in this thesis that, if a natural generalization of Cordes\' characterization is valid, then the Rieffel conjecture is true.

This dissertation explores aspects of the representation theory for tensor algebras, which are non-selfadjoint operator algebras Muhly and Solel introduced in 1998, by developing a cohomology theory for completely bounded Hilbert modules. Similar theories have been developed for Banach modules by Johnson in 1970, for operator modules by Paulsen in 1997, and for Hilbert modules over the disc algebra by Carlson and Clark in 1995. The framework presented here was motivated by a desire to further understand the completely bounded representation theory for tensor algebras on Hilbert spaces. The focal point of this thesis is the first Ext group, Ext1, which is defined as equivalence classes of short exact sequences of completely bounded Hilbert modules. Alternate descriptions of this group are presented. For general operator algebras, Ext1 can be realized as the collection completely bounded derivations equivalent up to an inner derivation. When the operator algebra is a tensor algebra, Ext1 can be described as a quotient space of intertwining operators, a description analogous to a result of Ferguson in 1996 in the case of the classical disc algebra. A theorem of Sz.-Nagy and Foias from 1967, concerning contractions in triangular form, is applied to analyze derivations that are off-diagonal corners of completely contractive representations. It is proved that, in some cases, this analysis determines when all derivations must be inner or suggests ways to construct non-inner derivations. In the third chapter, a characterization is given of completely bounded representations of a tensor algebra in terms of similarities of contractive intertwiners. Also proven is that for a Csup*;-correspondence X over a Csup*;-algebra A and the Toeplitz algebra T(X), Mn(T(X))= T(Mn(X)). The analogous statement for tensor algebras is deduced as a corollary. In the final chapter, a brief survey of non-abelian category theory is provided. Extensions of completely bounded Hilbert modules over operator algebras are defined. Theorems asserting the projectivity of isometric modules and injectivity of coisometric modules by Carlson, Clark, Foias, and Williams in 1995 are generalized to the noncommutative setting of tensor algebras using commutant lifting. A result of Popesecu in 1996 for noncommutative disc algebras is also covered in the general framework of this thesis.

This thesis discusses the topic of Boolean algebras. In order to build intuitive understanding of the topic, research began with the investigation of Boolean algebras in the area of Abstract Algebra. The content of this initial research used a particular notation. The ideas of partially ordered sets, lattices, least upper bounds, and greatest lower bounds were used to define the structure of a Boolean algebra. From this fundamental understanding, we were able to study atoms, Boolean algebra isomorphisms, and Stone’s Representation Theorem for finite Boolean algebras. We also verified and proved many properties involving Boolean algebras and related structures. We then expanded our study to more thoroughly developed theory. This comprehensive theory was more abstract and required the use of a different, more universal, notation. We continued examining least upper and greatest lower bounds but extended our knowledge to subalgebras and families of subsets. The notions of cardinality, cellularity, and pairwise disjoint families were investigated, defined, and then used to understand the Erdös-Tarski Theorem. Lastly, this study concluded with the investigation of denseness and incomparability as well as normal forms and the completion of Boolean algebras.

No description available.

Bibliography: p. 155-161.

Bibliography: p. 199-224. / xviii, 224 p. : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / This thesis reviews models of the simple cell receptive field profile (RFP) and its variation over the simple cell population, and uses artifical neural networks to investigate the multi-dimensional signal processing role of the RFP in the formation of a cortical representation of the visual image. / Thesis (Ph.D.)--University of Adelaide, Dept. of Electrical and Electronic Engineering, 1994?

Bibliography: p. 155-161. / x, 161 p. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 2001

Thesis (Ph.D.)--McMaster University, 2005. / Supervisor: Manfred Kolster. Includes bibliographical references (leaves 93-96).

Representations of SO(4,2;R) are constructed using 4 x 4 and 2 x 2 matrices with elements in H' ��� C . Using 2 x 2 matrix representations of C and H', the 4 x 4 representation is interpreted in terms of 16 x 16 real matrices. Finally, the known isomorphism between the conformal group and SO(4,2;R) is written explicitly in terms of the 4 x 4 representation. The 4 x 4 construction should generalize to matrices with elements in K' ��� K for K any normed division algebra over the reals and K' any split algebra over the reals. / Graduation date: 2013

by Charles Rockland. / "June 1985." / Bibliography: p. 111-113. / Army Research Office Grant (DAAG29-84-K-0005)