$L^p$-modules, $L^p$-correspondences, and their $L^p$-operator algebras

Abstract: In this paper, we introduce an $L^p$-operator algebraic analogue of Hilbert C*-modules. We initiate the theory of concrete $L^p$-modules, their morphisms, and basic constructions such as countable direct sums and tensor products. We then define $L^p$-correspondences together with their Fock representations and the $L^p$-operator algebras generated by these. We present evidence that well-known $L^p$-operator algebras can be constructed from $L^p$-correspondences via covariant Fock representations. In particular, for $p \in (1,\infty)$ and $q$ its H\"older conjugate, we show that the $L^p$-module $(\ell_d^p, \ell_d^q)$ gives rise to an $L^p$-correspondence over $\Bbb{C}$ whose $L^p$-operator algebra is isometrically isomorphic to $\mathcal{O}_d^p$, the $L^p$-analogue of the Cuntz algebra $\mathcal{O}_d$ introduced by N.C. Phillips in 2012. As a second example, we fix a nondegenerate $L^p$-operator algebra $A$ with a bicontractive approximate identity together with an isometric automorphism $\varphi_A \in \mathrm{Aut}(A)$. We then show that there is a contractive map from the crossed product $F^p(\Bbb{Z}, A, \varphi_A)$ to the $L^p$-operator algebra generated by the covariant Fock representation of the $L^p$-correspondence $(A, A, \varphi_A)$.