Relative operator entropies and Tsallis relative operator entropies in JB-algebras

TL;DR

This paper extends the study of relative operator entropies and Tsallis relative operator entropies to JB-algebras, establishing their properties, inequalities, and bounds with novel techniques suited for this algebraic setting.

Contribution

It introduces the concepts of relative operator entropies in JB-algebras and extends key inequalities, improving bounds previously known in Hilbert space operators.

Findings

Established basic properties of entropies in JB-algebras

Extended operator inequalities to JB-algebras

Improved bounds of relative operator $(eta, eta)$-entropy

Abstract

We initiate the study of relative operator entropies and Tsallis relative operator entropies in the setting of JB-algebras. We establish their basic properties and extend the operator inequalities on relative operator entropies and Tsallis relative operator entropies to this setting. In addition, we improve the lower and upper bounds of the relative operator $(\alpha, \beta)$-entropy in the setting of JB-algebras that were established in Hilbert space operators setting by Nikoufar [18, 20]. Though we employ the same notation as in the classical setting of Hilbert space operators, the inequalities in the setting of JB-algebras have different connotations and their proofs requires techniques in JB-algebras.