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. / 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.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Este trabalho à baseado no artigo "Uniquely Covered Groups" de M. A. Brodie, que investiga grupos finitos que possuem uma Ãnica cobertura irredundante por subgrupos prÃprios. O resultado principal obtido por M. A. Brodie assegura que um grupo finito e nÃo nilpotente G à unicamente coberto se, e somente se, G/Z(G) à um grupo nÃo abeliano de ordem pq, onde p e q sÃo primos distantes e {x,Z(G) à cÃclico para todo x â G. Nosso propÃsito à apresentar a demonstraÃÃo e uma aplicaÃÃo deste teorema. / This work is based on the article "Uniquely Covered Groups" due to M. A. Brodie, which investigates finite groups that have a single irredundante coveraging by subgroups. The main result obtained by M. A. Brodie asserts that a non-nilpotent finite group G is uniquely covered if and only if, G/Z(G) is a non-Abelian group of order pq, where p and q are distinct primes and {x,Z(G) is cyclic for every x â G. Our purpose is to present the proof and application of this theorem.