Doctoral Thesis
DOI
Document
Author
Full name
Clayton Suguio Hida
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2019
Supervisor
Committee
Brech, Christina (President)
Aurichi, Leandro Fiorini
Batista, Leandro Candido
Bianconi, Ricardo
Royer, Danilo
Title in English
Uncountable irredundant sets in nonseparable scattered C*-algebras
Keywords in English
Forcing
Irredundant sets
Scattered C*-algebras
Abstract in English
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.
Title in English
Uncountable irredundant sets in nonseparable scattered C*-algebras
Keywords in English
Forcing
Irredundant sets
Scattered C*-algebras
Abstract in English
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.