In Colmez's work on the p-adic local Langlands correspondence for ${\rm GL}_2(\mathbb{Q}_p)$, he works with ${\rm GL}_2(\mathbb{Q}_p)$-representations on $p$-adic Banach spaces which admit an invariant norm, so the reduction modulo $p$ makes sense. To each irreducible admissible representation of this kind (let's call these "unitary" Banach representations), he attaches a rank 2 $(\varphi, \Gamma)$-module, and hence a 2-dimensional p-adic representation of ${\rm Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$.

The unitary condition is quite strict -- it rules out all nontrivial finite-dimensional algebraic representations of ${\rm GL}_2$, for instance. Is there any natural way to extend the correspondence to *non-unitary* admissible Banach space representations of ${\rm GL}_2(\mathbb{Q}_p)$, and what sort of Galois-theoretic objects would these match up with?