C*-like modules and matrix $p$-operator norms
Alessandra Calin, Ian Cartwright, Luke Coffman, Alonso Delfín, Charles Girard, Jack Goldrick, Anoushka Nerella, Wilson Wu
TL;DR
The paper advances the theory of $L^p$-operator algebras by formulating $L^p$-modules with an $A$-valued pairing and defining a C*-like property via norm recovery from this pairing. It proves that row-column modules over block-diagonal subalgebras of $M_d^p(\mathbb{C})$ (and for $A=M_d^p(\mathbb{C})$) are C*-like, and provides block-structure lemmas that reduce general norm questions to sub-block computations. It also identifies clear counterexamples showing that C*-likeness can fail when the block-diagonal structure is lost or when approximate units are absent, including non-unital and certain unital subalgebras; these results illuminate the sensitivity of Hölder-duality-type norm recovery to the underlying algebra. Collectively, this work extends Hilbert C*-module techniques to the $L^p$-setting, clarifies when duality-like identifications persist, and highlights computational and structural challenges in $p$-operator norms for matrix algebras.
Abstract
We present a generalization of Hölder duality to algebra-valued pairings via $L^p$-modules. Hölder duality states that if $p \in (1, \infty)$ and $p^{\prime}$ are conjugate exponents, then the dual space of $L^p(μ)$ is isometrically isomorphic to $L^{p^{\prime}}(μ)$. In this work we study certain pairs $(\mathsf{Y},\mathsf{X})$, as generalizations of the pair $(L^{p^{\prime}}(μ), L^p(μ))$, 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.