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

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 $L^p(\hat{I}¼)$ is isometrically isomorphic to $L^{p^{\prime }}(\hat{I}¼)$. In this work we study certain pairs $(\mathsf{Y},\mathsf{X})$, as generalizations of the pair $(L^{p^{\prime }}(\hat{I}¼),L^p(\hat{I}¼))$, 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.

AMSLaTeX; 19 pages. V2: Final version accepted in the Annals of Functional Analysis