Some operator inequalities for Hermitian Banach *-algebras - Najafi, Hamed (Mathematica Scandinavica, 2020)
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.