Uncountable irredundant sets in nonseparable scattered C*-algebras

• Ano de defesa: 2019
• Autor(a) principal: Clayton Suguio Hida
• Orientador(a): Christina Brech
• Banca de defesa: Leandro Fiorini Aurichi, Leandro Candido Batista, Ricardo Bianconi, Danilo Royer
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Universidade de São Paulo
• Programa de Pós-Graduação: Matemática
• País: BR
• Link de acesso: https://doi.org/10.11606/T.45.2019.tde-05082019-165942

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.