Fractional powers on noncommutative $L_p$ for $p<1$

TL;DR

This paper establishes that fractional power functions exhibit a specific Hölder continuity property on self-adjoint elements within noncommutative Lp-spaces, extending previous inequalities to a broader noncommutative setting.

Contribution

It proves the $ heta$-Hölder continuity of the homogeneous functional calculus for fractional powers on noncommutative Lp-spaces, generalizing classical inequalities.

Findings

Proves $ heta$-Hölder continuity of fractional power functions on noncommutative Lp-spaces.

Extends classical inequalities of Birman, Koplienko, and Solomjak to noncommutative setting.

Applicable for all $0<p extless= ext{infinity}$ with values in $L_{p/ heta}$.

Abstract

We prove that the homogeneous functional calculus associated to $x\mapsto |x|^\theta$ or $x\mapsto {\rm sgn}\, (x) |x|^{\theta}$ for $0<\theta<1$ is $\theta$-H\"older on selfadjoint elements of noncommutative $L_p$-spaces for $0<p\leq\infty$ with values in $L_{p/\theta}$. This extends an inequality of Birman, Koplienko and Solomjak also obtained by Ando.