C*-like modules and matrix $p$-operator norms

We present a generalization of H\"older duality to algebra-valued pairings via $L^p$-modules. H\"older duality states that if $p \in (1, \infty)$ and $p^{\prime}$ are conjugate exponents, then the dual space of $L^p(\mu)$ is isometrically isomorphic to $L^{p^{\prime}}(\mu)$. In this work we study certain pairs $(\mathsf{Y},\mathsf{X})$, as generalizations of the pair $(L^{p^{\prime}}(\mu), L^p(\mu))$, that have an $L^p$-operator algebra valued pairing $\mathsf{Y} \times \mathsf{X} \to A$. When the $A$-valued version of H\"older duality still holds, we say that $(\mathsf{Y},\mathsf{X})$ is C*-like. We show that finite and countable direct sums of the C*-like module $(A,A)$ are still C*-like when $A$ is any block diagonal subalgebra of $d \times d$ matrices. We provide counterexamples when $A \subset M_d^p(\mathbb{C})$ is not block diagonal.