Some operator inequalities for Hermitian Banach $*$-algebras

In this paper, we extend the Kubo-Ando theory from operator means on C$^{*}$-algebras to a Hermitian Banach $*$-algebra $\mathcal {A}$ with a continuous involution. For this purpose, we show that if $a$ and $b$ are self-adjoint elements in $\mathcal {A}$ with spectra in an interval $J$ such that $a \leq b$, then $f(a) \leq f(b)$ for every operator monotone function $f$ on $J$, where $f(a)$ and $f(b)$ are defined by the Riesz-Dunford integral. Moreover, we show that some convexity properties of the usual operator convex functions are preserved in the setting of Hermitian Banach $*$-algebras. In particular, Jensen's operator inequality is presented in these cases.