Counting Banach spaces C(K)

I will present some results concerning the following general problem:

How many isomorphism types of Banach spaces C(K) of real continuous functions on K do we have, for  K from a given class C of compacta.

The classical result of Bessaga and Pełczyński gives us a complete classification of C(K), for the class of countable compact spaces K; in particular, we have \omega_1 isomorphism types of such spaces C(K). On the other hand, Milutin's theorem says that, for the class of uncountable metrizable compact spaces K, we have only one isomorphism type of spaces C(K). 

I will discuss two well-known classes of compact spaces of weight \omega_1, for which the above problem is not decidable in ZFC.

The first of these classes is the class AU of compact spaces K generated by families of almost disjoint subsets of the set of natural numbers N, usually associated with the names of Mrówka, Isbell, Franklin, or Aleksandrov and Urysohn.  Assuming the continuum hypothesis, we have 2^c (c - continuum) isomorphism types of C(K), for K from AU. 

In turn, assuming Martin's axiom and negation of the continuum hypothesis, for all  K,L in AU  with weight w(K) = w(L) = \omega_1, the spaces C(K) and C(L) are isomorphic (joint results with R. Pol, F. Cabello S\'anchez, J. Castillo, G. Plebanek, A. Salguero-Alarc\'on).

The second class considered is the class SCL  of separable, compact linearly ordered spaces of weight \omega_1. Again, assuming the continuum hypothesis, we have 2^c (c - continuum) isomorphism types of C(K), for  K in SCL. On the other hand, assuming a certain axiom proposed by Baumgartner, we have only one isomorphism type of C(K), for K in SCL (joint results with Maciej Korpalski and Piotr Koszmider).

In general, we can prove (in ZFC) that we have 2^{\omega_1} isomorphism types of C(K), for  compact spaces K of weight \omega_1 (joint with Korpalski and Koszmider).