Convex inequalities in Hilbert $C^*$-modules

TL;DR

This paper extends the H"older-McCarty inequalities, originally in Hilbert spaces, to the setting of Hilbert $C^*$-modules, broadening their applicability and exploring their properties in this more general context.

Contribution

The paper introduces a convex inequality in Hilbert $C^*$-modules and investigates the H"older-McCarty inequalities within this framework, expanding existing mathematical theory.

Findings

Extended convex inequality to Hilbert $C^*$-modules

Analyzed properties of H"older-McCarty inequalities in this setting

Provided foundational results for future research in operator algebras

Abstract

The H$\ddot{{\rm o}}$lder-McCarty inequalities are originally derived in the Hilbert space case and have been generalized via a convex inequality. The main purpose of this paper is to extend this convex inequality to the Hilbert $C^*$-module case, and meanwhile to make some investigations on the H$\ddot{{\rm o}}$lder-McCarty inequalities in the Hilbert $C^*$-module case.