Continuous deformations of Fredholm operators in B(H)
Let X be a compact Hausdorff topological space. The K-group of X, denoted by K(X), is the Grothendieck group associated to the commutative monoid of isomorphism classes of complex vector bundles over X, equipped with the Whitney sum. Let H be an infinite dimensional Hilbert space and F(H) be the set...
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| Tipo de recurso: | tesis de maestría |
| Estado: | Versión publicada |
| Fecha de publicación: | 2021 |
| País: | Brasil |
| Institución: | Universidade de São Paulo (USP) |
| Repositorio: | Biblioteca Digital de Teses e Dissertações da USP |
| Idioma: | inglés |
| OAI Identifier: | oai:teses.usp.br:tde-09082021-231927 |
| Acceso en línea: | https://www.teses.usp.br/teses/disponiveis/45/45131/tde-09082021-231927/ |
| Access Level: | acceso abierto |
| Palabra clave: | Fredholm index Fredholm operators Index theory Índice de Fredholm K-teoria K-teoria de espaços compactos K-theory K-theory of compact spaces Operadores de Fredholm Teoria do índice |
| Sumario: | Let X be a compact Hausdorff topological space. The K-group of X, denoted by K(X), is the Grothendieck group associated to the commutative monoid of isomorphism classes of complex vector bundles over X, equipped with the Whitney sum. Let H be an infinite dimensional Hilbert space and F(H) be the set of Fredholm operators on H. The Atiyah-Jänich Theorem states that the families-index is a natural isomorphism between the monoid of homotopy classes of functions from X into F(H) and the group K(X). In case X is a singleton, the families-index is the classic Fredholm index, and the Atiyah-Jänich Theorem states that the arcwise connected components of F(H) are characterized by the Fredholm index. In this work, we give a detailed exposition of the Atiyah-Jänich Theorem, studying the necessary elements to understand the construction of the K-group of a compact Hausdorff topological space, the definition of the families-index and giving a proof that such an index is the mentioned isomorphism. |
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