Some operator inequalities for Hermitian Banach $*$-algebras

  • Hamed Najafi

Abstract

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.

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Published
2020-03-29
How to Cite
Najafi, H. (2020). Some operator inequalities for Hermitian Banach $*$-algebras. MATHEMATICA SCANDINAVICA, 126(1), 82-98. https://doi.org/10.7146/math.scand.a-115624
Section
Articles