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.




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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 $c\_0$?**



Centralizers and containment of $c_0$
Cameron's answer indicates why $Z(X)$ rather than $X$ contains an isomorphic copy of $c\_0$. Using E. Behrends's function module representation theory, one actually obtains an isometric copy of $c\_0$ in $X$. This can be found explicitly in E. Behrends, $M$-Structure and the Banach-Stone Theorem; LNM 736 (1979), Prop. 4.22.


