On the $l_p$-equivalence of ultrafilters

We show for $n,m \geq 1$ and $\{u_1, \ldots, u_n, v_1, \ldots, v_m\} \subseteq \omega^*$ that $C_p(\bigoplus_{i=1}^n \omega_{u_i})$ and $C_p(\bigoplus_{i=1}^m \omega_{v_i})$ are linearly homeomorphic if and only if $n = m$ and there is a permutation $\pi: \{1,\ldots,n\} \rightarrow \{1,\ldots,n\}$ such that for every $i \leq n$, $\omega_{u_i}$ and $\omega_{v_{\pi(i)}}$ are homeomorphic. This generalizes a result by Gul'ko. We will also show that for $n,m \geq 1$, $\{u_1, \ldots, u_n\} \subseteq \omega^*$ and countable spaces $Y_1, \ldots, Y_n$ with only one non-isolated point, if $C_p(\bigoplus_{i=1}^n \omega_{u_i})$ and $C_p(\bigoplus_{i=1}^m Y_i)$ are linearly homeomorphic, then $m \leq n$. Moreover, $m =n$ if and only if each $Y_i$ is homeomorphic to $\omega_{v_i}$ for some $v_i \in \omega^*$.