This thesis consists of studies in the separate fields of operator algebras and non-associative algebras. Two natural operator algebra structures, A ⊗_max B and A ⊗_min B, exist on the tensor product of two given unital operator algebras A and B. Because of the different properties enjoyed by the two tensor products in connection to dilation theory, it is of interest to know when they coincide (completely isometrically). Motivated by earlier work due to Paulsen and Power, we provide conditions relating an operator algebra B and another family {C_i}_i of operator algebras under which, for any operator algebra A, the equality A ⊗_max B = A ⊗_min B either implies, or is implied by, the equalities A ⊗_max C_i = A ⊗_min C_i for every i. These results can be applied to the setting of a discrete group G pre-ordered by a subsemigroup G⁺, where B ⊆ C*_r(G) is the subalgebra of the reduced group C*-algebra of G generated by G⁺, and C_i = A(Q_i) are digraph algebras defined by considering certain pre-ordered subsets Q_i of G. The 16-dimensional algebra A₄ of real sedenions is obtained by applying the Cayley-Dickson doubling process to the real division algebra of octonions. The classification of subalgebras of A₄ up to conjugacy (i.e. by the action of the automorphism group of A₄) was completed in a previous investigation, except for the collection of those subalgebras which are isomorphic to the quaternions. We present a classification of quaternion subalgebras up to conjugacy.

This thesis consists of studies in the separate fields of operator algebras and non-associative algebras. Two natural operator algebra structures, A ⊗_max B and A ⊗_min B, exist on the tensor product of two given unital operator algebras A and B. Because of the different properties enjoyed by the two tensor products in connection to dilation theory, it is of interest to know when they coincide (completely isometrically). Motivated by earlier work due to Paulsen and Power, we provide conditions relating an operator algebra B and another family {C_i}_i of operator algebras under which, for any operator algebra A, the equality A ⊗_max B = A ⊗_min B either implies, or is implied by, the equalities A ⊗_max C_i = A ⊗_min C_i for every i. These results can be applied to the setting of a discrete group G pre-ordered by a subsemigroup G⁺, where B ⊆ C*_r(G) is the subalgebra of the reduced group C*-algebra of G generated by G⁺, and C_i = A(Q_i) are digraph algebras defined by considering certain pre-ordered subsets Q_i of G. The 16-dimensional algebra A₄ of real sedenions is obtained by applying the Cayley-Dickson doubling process to the real division algebra of octonions. The classification of subalgebras of A₄ up to conjugacy (i.e. by the action of the automorphism group of A₄) was completed in a previous investigation, except for the collection of those subalgebras which are isomorphic to the quaternions. We present a classification of quaternion subalgebras up to conjugacy.

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-02-17Bitstream added on 2014-06-13T18:06:51Z : No. of bitstreams: 1 roque_mrd_me_sjrp.pdf: 11300361 bytes, checksum: 634655b9889665fb4488c7076d5db292 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Nesta dissertação, estuda-se estruturas matemáticas relacionadas à álgebra dos sedenions de Cayley-Dickson. O conceito de funções sedeniônicas do tipo f(z) = zn, z 2 S e n 2 N, é desenvolvido a partir da distância jf(y)¡f(x)j, com o objetivo de obter-se uma generalização. A este tipo de mapeamentos trata-se por funções quaseconformes, ou seja, mapeamentos que não preservam a magnitude dos ângulos. Em particular, através de métodos de resolução, apresenta-se e discute-se polinômios de 2n graus com coeficientes sedeniônicos com o intuito de enfatizar o valor da k-dilatação causada quando trabalha-se com o número sedeniônico em coordenadas esféricas. Por fim, ilustra-se geometricamente os cortes produzidos em hiperesferas B(x; r) quando submetidas às transformações do tipo z2 e z3. / In this work, we propose to study the mathematical construction related with algebra of Cayley-Dickson sedenions. We will present the concept of sedenions functions of f(z) = zn type, z 2 S and n 2 N, developing jf(y) ¡ f(x)j distance, with the objective of creating a generalization. This type of mappings is known as quasiconformal functions, that is, mapping that don't preserve the magnitude of angles. Specially, by means of resolution methods, we will discuss polynomials of 2n degrees with sedenions coefficients focused on highlighting the value of the k-dilation caused when we work with the sedenion number in spherical coordinates. Finally, it is illustrated geometrically the cuts produced in hiperspheres B(x; r) when submitted to the transformations of the type z2 and z3.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-29T16:36:38Z No. of bitstreams: 1 arquivototal.pdf: 3935354 bytes, checksum: f5f2372b545760005e2b3da81e164bed (MD5) / Approved for entry into archive by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2017-08-30T11:44:57Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 3935354 bytes, checksum: f5f2372b545760005e2b3da81e164bed (MD5) / Made available in DSpace on 2017-08-30T11:44:57Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 3935354 bytes, checksum: f5f2372b545760005e2b3da81e164bed (MD5) Previous issue date: 2015-08-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this paper it will be studied the imaginary numbers and how their evolution over time occurred. Such evolution has occurred at a slow pace until it reached at what is known today as the imaginary number i. However, the creation of the complex was not the end of the study of imaginary numbers. These studies have introduced even more comprehensive concepts creating sets as quaternions, extension of four dimensions of the complex. It will be concluded, with the extensions of eight and sixteen dimensions of the complex numbers, known as octonions and sedenions, respectively. Additionally, it will be submitted some applications of these extensions, also known as hypercomplex numbers. / Neste trabalho serão estudados os números imaginários e como se deu a sua evolução ao longo do tempo. Evolução esta que ocorreu de forma bem lenta, até se chegar no que é conhecido hoje como o número imaginário i. Entretanto, a criação dos complexos não foi o ponto nal do estudo dos números imaginários. Estudos seguintes introduziram conceitos ainda mais abrangentes criando conjuntos como os quatérnios, extensão de quatro dimensões dos complexos. Finaliza-se o trabalho, com as extensões de oito e dezesseis dimensões dos complexos, conhecidas como octônios e sedênios, respectivamente. Além de ser apresentado algumas aplicações dessas extensões, também conhecidas como números hipercomplexos.

Orientador: Manoel Ferreira Borges Neto / Banca: Masoyoshi Tsuchida / Banca: José Arnaldo Frutuoso Roveda / Resumo: Nesta dissertação, estuda-se estruturas matemáticas relacionadas à álgebra dos sedenions de Cayley-Dickson. O conceito de funções sedeniônicas do tipo f(z) = zn, z 2 S e n 2 N, é desenvolvido a partir da distância jf(y)¡f(x)j, com o objetivo de obter-se uma generalização. A este tipo de mapeamentos trata-se por funções quaseconformes, ou seja, mapeamentos que não preservam a magnitude dos ângulos. Em particular, através de métodos de resolução, apresenta-se e discute-se polinômios de 2n graus com coeficientes sedeniônicos com o intuito de enfatizar o valor da k-dilatação causada quando trabalha-se com o número sedeniônico em coordenadas esféricas. Por fim, ilustra-se geometricamente os cortes produzidos em hiperesferas B(x; r) quando submetidas às transformações do tipo z2 e z3. / Abstract: In this work, we propose to study the mathematical construction related with algebra of Cayley-Dickson sedenions. We will present the concept of sedenions functions of f(z) = zn type, z 2 S and n 2 N, developing jf(y) ¡ f(x)j distance, with the objective of creating a generalization. This type of mappings is known as quasiconformal functions, that is, mapping that don't preserve the magnitude of angles. Specially, by means of resolution methods, we will discuss polynomials of 2n degrees with sedenions coefficients focused on highlighting the value of the k-dilation caused when we work with the sedenion number in spherical coordinates. Finally, it is illustrated geometrically the cuts produced in hiperspheres B(x; r) when submitted to the transformations of the type z2 and z3. / Mestre