Positivity DOI 10.1007/s11117-016-0440-2 Positivity Spectral theory in ordered Banach algebras Sonja Mouton 1 · Heinrich Raubenheimer 2 Received: 1 October 2015 / Accepted: 9 August 2016 © Springer International Publishing 2016 Abstract We give a survey of the development of the spectral theory in ordered Banach algebras; from its roots in operator theory to the modern abstract context. Keywords Ordered Banach algebra · Spectral theory Mathematics Subject Classification 46H05 · 47A10 · 47B65 1 Introduction Let T be an n × n matrix with complex entries. One can view T as an operator of C n into C n in the normal way. It was discovered around the turn of the previous century that the spectrum of an n × n matrix with positive entries has certain special features. The first result was by Perron. Theorem 1.1 ([55]) Let T =[t ij ] be an n × n matrix with t ij > 0 for all i and j. Then: 1. T has strictly positive spectral radius r (T ). 2. r (T ) is a simple eigenvalue of T with strictly positive eigenvector. 3. T is primitive, i.e. r (T ) is the unique eigenvalue on the spectral circle |λ|= r (T ). B Sonja Mouton smo@sun.ac.za Heinrich Raubenheimer heinrichr@uj.ac.za 1 Department of Mathematical Sciences, Stellenbosch University, Private Bag X1, Matieland 7602, South Africa 2 Department of Mathematics, University of Johannesburg, P. O. Box 524, Auckland Park 2006, South Africa