In this paper, we extend the Kubo–Ando theory from operator means on C∗- algebras to a Hermitian Banach ∗-algebra A with a continues involution. For this end, 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.

Keywords