Tesis Doctoral
DOI
Documento
Autor
Nombre completo
Clayton Suguio Hida
Dirección Electrónica
Área de Conocimiento
Fecha de Defensa
Publicación
São Paulo, 2019
Director
Tribunal
Brech, Christina (Presidente)
Aurichi, Leandro Fiorini
Batista, Leandro Candido
Bianconi, Ricardo
Royer, Danilo
Título en inglés
Uncountable irredundant sets in nonseparable scattered C*-algebras
Palabras clave en inglés
Forcing
Irredundant sets
Scattered C*-algebras
Resumen en inglés
Given a C*-algebra $\A$, an irredundant set in $\A$ is a subset $\mathcal$ of $\A$ such that no $a\in \mathcal$ belongs to the C*-subalgebra generated by $\mathcal\setminus\{a\}$. Every separable C*-algebra has only countable irredundant sets and we ask if every nonseparable C*-algebra has an uncountable irredundant set. For commutative C*-algebras, if $K$ is the Kunen line then $C(K)$ is a consistent example of a nonseparable commutative C*-algebra without uncountable irredundant sets. On the other hand, a result due to S. Todorcevic establishes that it is consistent with ZFC that every nonseparable C*-algebra of the form $C(K)$, for a compact 0-dimensional space $K$, has an uncountable irredundant set. By the method of forcing, we construct a nonseparable and noncommutative scattered C*-algebra $\A$ without uncountable irredundant sets and with no nonseparable abelian subalgebras. On the other hand, we prove that it is consistent that every C*-subalgebra of $\B(\ell_2)$ of density continuum has an irredundant set of size continuum.
Título en inglés
Uncountable irredundant sets in nonseparable scattered C*-algebras
Palabras clave en inglés
Forcing
Irredundant sets
Scattered C*-algebras
Resumen en inglés
Given a C*-algebra $\A$, an irredundant set in $\A$ is a subset $\mathcal$ of $\A$ such that no $a\in \mathcal$ belongs to the C*-subalgebra generated by $\mathcal\setminus\{a\}$. Every separable C*-algebra has only countable irredundant sets and we ask if every nonseparable C*-algebra has an uncountable irredundant set. For commutative C*-algebras, if $K$ is the Kunen line then $C(K)$ is a consistent example of a nonseparable commutative C*-algebra without uncountable irredundant sets. On the other hand, a result due to S. Todorcevic establishes that it is consistent with ZFC that every nonseparable C*-algebra of the form $C(K)$, for a compact 0-dimensional space $K$, has an uncountable irredundant set. By the method of forcing, we construct a nonseparable and noncommutative scattered C*-algebra $\A$ without uncountable irredundant sets and with no nonseparable abelian subalgebras. On the other hand, we prove that it is consistent that every C*-subalgebra of $\B(\ell_2)$ of density continuum has an irredundant set of size continuum.

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Fecha de Publicación
2019-08-08