Markov-Kakutani fixed-point theorem is usually stated as follows:
"Let $E$ be a locally convex topological vector space. Let $C$ be a compact convex subset of $E$. Let $S$ be a commuting family of self-mappings $T$ of $C$ which are continuous and affine, i.e. $T(tx +(1 – t)y) = tT(x) + (1 – t)T(y)$ for $t \in [0,1]$ and $x, y$ in $C$. Then the mappings have a common fixed point in $C$." (Wikipedia)
The proof that is given in wikipedia (following Reed-Simon) is based on the Hahn-Banach theorem which is where, I believe, the requirement of local convexity for $E$ is used. However, as far as I know there are other proofs of Markov-Kakutani theorem which avoid the Hahn-Banach theorem.
My question is whether the local convexity condition on $E$ is redundant in the statement of the theorem and it is included there only because of the preferred method of the proof.
Yes, the local convexity condition is precisely because of the nice parallel with the Hahn-Banach theorem (in fact, Kakutani himself has a paper showing the Markov-Kakutani theorem implies the Hahn-Banach theorem as well).
The Markov-Kakutani theorem holds in any general Hausdorff topological vector space (see, for example, Dunford & Schwartz "Linear Operators. Part 1" p. 456).