Convex inequalities in Hilbert $C^*$-modules

TL;DR

This work extends the Hölder–McCarty inequalities from Hilbert spaces to Hilbert $C^*$-modules by incorporating the Mond–Pečarić convex framework. It proves a key convex-inequality lemma in the module setting and generalizes it using states $\rho$ on $\mathfrak{A}$ to obtain $f(\rho(\langle t x,x\rangle))\leq \rho(\langle f(t)x,x\rangle)$. It then establishes Hilbert $C^*$-module Hölder–McCarty inequalities and analyzes when these inequalities hold for commutative versus noncommutative $C^*$-algebras, showing positive results for commutative bases and counterexamples in $\mathbb{B}(H)$ and $\mathbb{K}(H)$. These results clarify the scope of convex-inequality methods in operator module settings and highlight fundamental differences between commutative and noncommutative bases.

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.

Theorems & Definitions (18)

• Lemma 1.1
• Lemma 1.2
• Theorem 3.1
• proof
• Corollary 3.2
• proof
• Corollary 4.1
• proof
• Theorem 4.2
• proof