Algebraic Absolutely Invertible Elements in Archimedean Riesz Algebras

Let A be an Archimedean Riesz algebra with a positive unit element e. An element f is an element of 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 p(1),..., p(n) of e and real numbers alpha(1),..., alpha(n) such that e = p(1) + ...+ p(n) and f = alpha(1)p(1) + ... + alpha(n)p(n). 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 vertical bar f vertical bar(-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.