It is well known that, if $\pi:A\to \mathbb B(\mathcal H)$ is a $^*$-representation of a type I $C^*$-algebra, then $\pi$ is unitarily equivalent to a direct integral of irreducible representations.
My problem is the following, is every Banach space representation $\pi:\mathcal C_0(X)\to \mathbb B(\mathcal E)$ equivalent to a direct integral of irreducible representations? (ie. equivalent to a representation on the $L^p$ space of cross-sections of some Banach bundle $\{\mathcal E_x\}_{x\in X}$)
If this is not known, is there any progress towards a solution?