  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.
