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Master's Dissertation
DOI
https://doi.org/10.11606/D.45.2014.tde-07072014-153504
Document
Author
Full name
Renata Akemi Maekawa
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2014
Supervisor
Committee
Monteiro, Martha Salerno (President)
Gonçalves, Daniel
Silva, Antonio Roberto da
Title in Portuguese
Uma descrição das aplicações de conexão em K-teoria de C*-álgebras usando cones
Keywords in Portuguese
aplicações de conexão
cone de uma aplicação
K-teoria de C*-álgebras
transformações naturais
Abstract in Portuguese
Dada uma aplicação f: B -> A entre duas C*-álgebras, o cone dessa aplicação, Cf, é o conjunto formado pelos pares (b,g) pertencentes à soma direta da C*-álgebra B com o cone CA tais que f(b) = g(0), sendo CA o cone de A. Neste trabalho estudamos o funtor determinado pela associação da sequência exata curta 0 -> SA -> Cf -> B -> 0 para cada *-homomorfismo f: B -> A, e demonstramos que esse funtor é exato. Caracterizamos as aplicações de conexão associadas à sequência exata 0 -> SA -> Cf -> B -> 0, mostrando que a aplicação do índice é dada por tAK1(f) e que a aplicação exponencial é dada por bAK0(f), sendo tA o isomorfismo entre K1(A) e K0(SA) e bA a aplicação de Bott. Por fim, usando que toda sequência exata curta de C*-álgebras pode ser vista na forma 0 -> Ker f -> B -> A -> 0, mostramos que as aplicações de conexão d1 e d0 associadas a cada sequência exata curta podem ser dadas por dn = Kn+1(j)-1 Kn+1(i) hn, em que j é a inclusão do núcleo de f em Cf, i é a inclusão da suspensão SA também em Cf, hn = bA e h1 = tA .
Title in English
A description of the connecting maps in K-theory for C*-algebras using cones
Keywords in English
connecting maps
K-theory for C*-algebras
mapping cones
natural transformations
Abstract in English
If f: B A is a map between the C*-algebras A and B, the mapping cone is the set of pairs (b,g) in the direct sum of B and CA such that f(b) = g(0), where CA is the cone of A. In this work, we study the functor determined by the assignment of the exact sequence 0 SA Cf B 0 to each *-homomorphism f: B -> A, and we show that this functor is exact. We characterize the connecting maps associated with the short exact sequence 0 SA Cf B 0 and we prove that its index map is tA K1(f) and that its exponential map is bA K0(f), where tA is the isomorphism between K1(A) and K0(SA), and bA is the Bott map. Finally, using that every short exact sequence of C*-algebras can be seen as 0 Ker f B (f ) A 0, we prove that the connecting maps, d1 and d0, associated with a short exact sequence are given by dn = Kn+1(j)-1 Kn+1(i) hn, where j is the inclusion of f's kernel in Cf, i is the inclusion of the suspension SA in Cf, hn = bA e h1 = tA .
 
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Publishing Date
2014-07-07
 
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