Journal of Operator Theory

Volume 57, Issue 2, Spring 2007 pp. 243-250.

The geometric means in Banach $*$-algebras

Authors: Bao Qi Feng
Author institution: Department of Mathematical Sciences, Kent State University, Tuscarawas Campus, New Philadelphia, OH 44663, USA

Summary: The arithmetic-geometric-harmonic inequality has played a special role in elementary mathematics. During the past twenty five years (see [1], [2] and [8] etc.) a great many mathematicians have researched on various kinds of matrix versions of the arithmetic-geometric-harmonic inequality. It is interesting to see whether the arithmetic-geometric-harmonic inequality can be extended to the context of Banach $*$-algebras. In this article we will define the geometric means of positive elements in Banach $*$-algebras and prove that the arithmetic-geometric-harmonic inequality does hold in Banach $*$-algebras.

Keywords: Arithmetic mean, geometric mean, harmonic mean, Banach $*$-algebra

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