I have this question also in MSE (see: https://math.stackexchange.com/questions/666053/centralizers-and-containment-of-c-0), but I have not got an answer there. So I thought I try my luck here.

Let $X$ be a Banach space over $\mathbb{R}$ or $\mathbb{C}$.

By a multiplier on $X$ we mean a bounded linear operator $T$ on $X$ such that every extreme point of $B_{X^*}$ becomes an eigenvector for $T^*.$ Given a multiplier $T$ on $X$, and an extreme point $p$ of $B_{X^*}$, there exists a unique number $a_T(p)$ satisfying $p\circ T^* = a_T(p)p$.

The centralizer $Z(X)$ of $X$ is the set of those multipliers $T$ on $X$ such that there exists a multiplier $S$ on $X$ satisfying $a_S(p)=\overline{a_T(p)}$ for every extreme point $p$ of $B_{X^*}$. (For more one can find in Harmand, Werner, Werner, "M-ideals in Banach spaces and Banach algebras.")

I wonder if the answer to the following question is known:

If $Z(X)$ is infinite-dimensional does then $X$ contain an isomporphic copy of $c_0$?