The noncommutative $\ell_1-\ell_2$ inequality for Hilbert C*-modules and the exact constant

TL;DR

This paper extends the classical $ ext{L}_1$-$ ext{L}_2$ inequality to Hilbert C*-modules, providing a method to compute exact constants and a formula for the continuous case, with applications to Kasparov's integral.

Contribution

It introduces a way to compute the exact constant in the noncommutative $ ext{L}_1$-$ ext{L}_2$ inequality for Hilbert C*-modules, generalizing previous results.

Findings

Derived an explicit formula for the exact constant in the inequality.

Provided a method to compute a positive element $c_x$ for certain tuples.

Applied the results to Kasparov's integral and continuous inequalities.

Abstract

Let $\mathcal{A}$ be a unital C*-algebra. Then the theory of Hilbert C*-modules tells that \begin{align*} \sum_{i=1}^{n}(a_ia_i^*)^\frac{1}{2}\leq \sqrt{n} \left(\sum_{i=1}^{n}a_ia_i^*\right)^\frac{1}{2}, \quad \forall n \in \mathbb{N}, \forall a_1, \dots, a_n \in \mathcal{A}. \end{align*} By modifications of arguments of Botelho-Andrade, Casazza, Cheng, and Tran given in 2019, for certain tuple $x=(a_1, \dots, a_n) \in \mathcal{A}^n$, we give a method to compute a positive element $c_x$ in the C*-algebra $\mathcal{A}$ such that the equality \begin{align*} \sum_{i=1}^{n}(a_ia_i^*)^\frac{1}{2}=c_x \sqrt{n} \left(\sum_{i=1}^{n}a_ia_i^*\right)^\frac{1}{2}. \end{align*} holds. We give an application for the integral of G. G. Kasparov. We also derive the formula for the exact constant for the continuous $\ell_1-\ell_2$ inequality.