On $\alpha$-$z$-R\'{e}nyi divergence in the von Neumann algebra setting

TL;DR

This paper investigates the properties of the $oldsymbol{ ext{α-}z}$-Rényi divergence within the framework of von Neumann algebras, utilizing Haagerup non-commutative $L^p$-spaces, and establishes key properties for different parameter ranges.

Contribution

It extends the understanding of $oldsymbol{ ext{α-}z}$-Rényi divergence to the von Neumann algebra setting and proves its fundamental properties for $0<oldsymbol{ ext{α}}<1$ and some for $oldsymbol{ ext{α}}>1$.

Findings

Established almost all expected properties of the divergence for $0<α<1$.

Proved some properties of the divergence for $α>1$.

Provided an equality condition for generalized Hölder's inequality in Haagerup non-commutative $L^p$-spaces.

Abstract

We will investigate the $\alpha$-$z$-R\'{e}nyi divergence in the general von Neumann algebra setting based on Haagerup non-commutative $L^p$-spaces. In particular, we establish almost all its expected properties when $0 < \alpha < 1$ and some of them when $\alpha > 1$. In an appendix we also give an equality condition for generalized H\"{o}lder's inequality in Haagerup non-commutative $L^p$-spaces.