Positivity 1: 305–317, 1997. 305 c 1997 Kluwer Academic Publishers. Printed in the Netherlands. More Spectral Theory in Ordered Banach Algebras S. MOUTON (N ´ EE RODE) 1 and H. RAUBENHEIMER 2 1 Department of Mathematics, University of Stellenbosch, Stellenbosch 7600, South Africa 2 Department of Mathematics, University of the Free State, Bloemfontein 9300, South Africa (Received: 13 December 1996; Accepted: 20 August 1997) Abstract. We continue our development of spectral theory for positive elements in an ordered Banach algebra. In particular we provide a suitable version of the Krein-Rutman theorem, obtain some results concerning the peripheral spectrum of a positive element and provide a characterisation of positive quasi inessential elements, in the context of an ordered Banach algebra. Mathematics Subject Classification (1991): 46H05, 46H10, 47B60 Keywords: ordered Banach algebra, positive element 1. Preliminaries In this paper we continue our development of spectral theory for positive elements in an ordered Banach algebra, starting with a discussion of the Krein-Rutman the- orem. This theorem provides some answers to the question of when the spectral radius of a positive element will be an eigenvalue of that element, with a positive eigenvector. The original version of this theorem (theorem 3.1) was in terms of operators. We shall present different versions of the Krein-Rutman theorem, con- cluding with a suitable version of this theorem in the context of an ordered Banach algebra (theorem 3.7). We shall also consider the peripheral spectrum of a posi- tive element in an ordered Banach algebra. Some interesting results are obtained, including a useful domination property (theorem 4.3). While being interesting in itself, the peripheral spectrum can also be used in the study of positive quasi inessential elements (the Banach algebra analogue of quasi compact operators on a Banach space) . Amongst other things a characterisation of positive quasi inessen- tial elements in an ordered Banach algebra is obtained (theorem 5.2), and it is shown that under certain circumstances the property of a positive element being quasi inessential is inherited by positive elements dominated by (corollary 5.4). Before we continue, some preliminary notation and definitions are provided. Throughout (or ) will denote a complex Banach algebra with unit 1. If and are Banach algebras, then a linear operator : is called a homomorphism if ( ) and 1 1. The spectrum of an element in will be denoted by and the spectral radius of in by . Whenever there is no ambiguity, we shall drop the in and . We denote the radical of by Rad and the set of all complex valued functions which are analytic in a