Quotients of indecomposable Banach spaces of continuous functions

Rogério Augusto dos Santos Fajardo¹

¹ Instituto de Matemática e Estatística Universidade de São Paulo Rua do Matão, 1010 São Paulo, SP, Brazil

Studia Mathematica, Tome 212 (2012) no. 3, pp. 259-283

Résumé

Assuming $\diamondsuit$, we construct a connected compact topological space $K$ such that for every closed $L\subset K$ the Banach space $C(L)$ has few operators, in the sense that every operator on $C(L)$ is multiplication by a continuous function plus a weakly compact operator. In particular, $C(K)$ is indecomposable and has continuum many non-isomorphic indecomposable quotients, and $K$ does not contain a homeomorphic copy of $\beta \mathbb N$.Moreover, assuming CH we construct a connected compact $K$ where $C(K)$ has few operators and $K$ contains a homeomorphic copy of $\beta \mathbb N$.

Détail

DOI : 10.4064/sm212-3-4

Keywords: assuming diamondsuit construct connected compact topological space every closed subset banach space has few operators sense every operator multiplication continuous function plus weakly compact operator particular indecomposable has continuum many non isomorphic indecomposable quotients does contain homeomorphic copy beta mathbb moreover assuming construct connected compact where has few operators contains homeomorphic copy beta mathbb