Indag. Mathem., N.S., 7 (4), 489-502 December 16, 1996 Cones in Banach algebras by H. Raubenheimer and S. Rode Depurtment of’Mathematics, University @the Orangr Free State, P.O. Box 339, Bloernfimtein 9300. South Africcr Communicated by Prof. W.A.J. Luxemburg at the meeting of November 27, 1995 ABSTRACT We develop spectral theory for elements in an ordered Banach algebra I. INTRODUCTION It has been discovered around the turn of the century by 0. Perron and G. Frobenius that the spectral theory of positive matrices has certain special features. More generally, the study of the spectral theory of positive operators on ordered Banach spaces (or on Banach lattices), developed since, yields some interesting results. This theory is documented well in the monographs of H.H. Schaefer [32,29] and A.C. Zaanen [34]. We refer to this study, or in fact to the study of any generalisations of the above setup, as Perron-Frobenius the- ory. It has become an important field of modern operator th;ory having many practical and theoretical applications. We will show that Banach algebra techniques combined with ordered struc- tures yield new insights into the Perron-Frobenius theory. This has been ob- served in [17, 18, 19, 21, 81 and by others. Most of the well-known spectral the- oretical results in ordered structures have been proved in the operator algebra L(X) of bounded linear operators on an ordered Banach space X (cf. for ex- ample [ll, 22, 33]), or even on a Banach lattice X (cf. [29, 341). In this case, two cones are involved, namely a cone C of X and a cone K of C(X), where K is defined in terms of C as follows: K := {T E C(X) :TC c C}. In many cases the use of two cones tends to complicate matters. Apart from this problem. 489