  In this paper, we investigate the relative operator entropies in the more
general settings of C*-algebras, real C*-algebras and JC-algebras. We show that
all the operator inequalities on relative operator entropies still hold in
these broader settings. In addition, we improve the lower and upper bounds of
the relative operator $(\alpha, \beta)$-entropy established by Nikoufar which
refined the bounds for the relative operator entropy obtained by Fujii and
Kamei.
