A Banach space C(K) reading the dimension of K

Abstract

Assuming Jensen's diamond principle (⋄) we construct for every natural number n>0 a compact Hausdorff space K such that whenever the Banach spaces C(K) and C(L) are isomorphic for some compact Hausdorff L, then the covering dimension of L is equal to n. The constructed space K is separable and connected, and the Banach space C(K) has few operators i.e. every bounded linear operator T:C(K)→C(K) is of the form T(f)=fg+S(f), where g∈C(K) and S is weakly compact.