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

We present a generalization of Hölder duality to algebra-valued pairings via $L^p$-modules. Hölder duality states that if $p\in (1,\mathrm{\infty })$ and $p^{\mathrm{\prime }}$ are conjugate exponents, then the dual space of $L^p(\mu )$ is isometrically isomorphic to $L^{p^{\mathrm{\prime }}}(\mu )$. In this work we study certain pairs $(\mathsf{Y},\mathsf{X})$, as generalizations of the pair $(L^{p^{\mathrm{\prime }}}(\mu ),L^p(\mu ))$, that have an $L^p$-operator algebra valued pairing $\mathsf{Y}\times \mathsf{X}\to A$. When the $A$-valued version of Hölder 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.