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

Assuming Jensen’s diamond principle ($\mathrm{♢}$) 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 $T:C(K)\to C(K)$ is of the form $T(f)=fg+S(f)$, where $g\in C(K)$ and $S$ is weakly compact.