We discuss three problems of Koszmider on the structure of the spaces of continuous functions on the Stone compact $K_{ \mathcal A}$ generated by an almost disjoint family $\mathcal A$ of infinite subsets of $\omega$ -- we present a solution to two problems and develop a previous results of Marciszewski and Pol answering the third one. We will show, in particular, that assuming Martin's axiom the space $C(K_{ \mathcal A})$ is uniquely determined up to isomorphism by the cardinality of $\mathcal A$ whenever $|{ \mathcal A}|<{ \mathfrak c}$, while there are $2^{ \mathfrak c}$ nonisomorphic spaces $C(K_{ \mathcal A})$ with $|{ \mathcal A}|= { \mathfrak c}$. We also investigate Koszmider's problems in the context of the class of separable Rosenthal compacta and indicate the meaning of our results in the language of twisted sums of $c_0$ and some $C(K)$ spaces.
Categories: math.FA