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

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

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

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.

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Publishing Date

2019-08-08

CeTI-SC/STI

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