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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?

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    $\begingroup$ Continuous functions on $[0,1]$ act on the space of measures on $[0,1]$. The latter is not the space of sections of some bundle of Banach spaces. $\endgroup$
    – André Henriques
    Aug 26, 2022 at 22:00
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    $\begingroup$ What is the quantifier on $p$? $\endgroup$
    – Mikael de la Salle
    Aug 27, 2022 at 10:08

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