Quotients of indecomposable Banach spaces of continuous functions
Volume 212 / 2012
Studia Mathematica 212 (2012), 259-283
MSC: Primary 46E15; Secondary 46B26, 03E65, 03E35, 54D05.
DOI: 10.4064/sm212-3-4
Abstract
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 $\beta \mathbb N$.
