Crossed product functors associated to $\ell^p$-pseudofunctions

We show that the $\ell^p$-pseudofunctions, which were recently shown to lead to exotic completions of group $C^*$-algebras by Wiersma and the second named author, can be used to construct well-behaved crossed product functors in the sense of Buss, Echterhoff and Willett. The construction proceeds via introducing certain Banach algebras, related to operators acting on Hilbert valued $\ell^p$-spaces, which a priori depend on the choice of a Hilbert space representation of the underlying C*-algebra. We prove that, in fact, the resulting algebras are isomorphic (with the isomorphism constant depending only on $p$), and hence their C*-envelopes are isometrically isomorphic. This, in particular, means that the construction genuinely generalises the one studied earlier in the group case. The tools we develop allow us to show that for certain non-amenable actions, the resulting crossed product completions must indeed be exotic.