Algebraic Absolutely Invertible Elements in Archimedean Riesz Algebras

Abstract

Let A be an Archimedean Riesz algebra with a positive unit element e. An element f ∈ A is said to be algebraic if P (f) = 0 for some non-zero polynomial P with real coefficients. Moreover, f is called an e-step function in A if there exist pairwise disjoint components p1, . . . , pn of e and real numbers α1, . . . , αn such that

e = p1 + · · · + pn and f = α1p1 + · · · + αnpn.

First, we shall prove that if A is an f-algebra, then f is algebraic if and only if f is an e-step function. This leads to the main result of this paper, which asserts that if f is an absolutely invertible element (i.e., |f| is invertible and its inverse |f|−1 is positive) in an arbitrary Archimedean Riesz algebra with positive identity, then f is algebraic if and only if f has an e-step function power in A. As a consequence, we obtain all previous results by Boulabiar, Buskes, and Sirotkin who investigated the special case of disjointness preserving operators on Archimedean Riesz spaces.