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 A with a continuous involution. For this purpose, we show that if a and b are self-adjoint elements in A with spectra in an interval J such that a≤b, then f(a)≤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.