Commutatively ordered Banach algebras

Abstract

Spectral theory in ordered Banach algebras (OBAs) has been investigated and several authors have made contributions. However, the results are not applicable to non-commutative C^*-algebras, since a non-commutative C^*-algebra is not an OBA. In this paper we introduce a more general structure, called a commutatively ordered Banach algebra (COBA), which includes the class of OBAs. Every C^* - algebra is a COBA. We will give the basic properties of COBAs and show how known results in OBAs can be generalized to the COBA setting. We will then discuss two spectral problems regarding COBA elements. The results obtained, of course, hold true in an OBA as well. These results extend the theory of COBAs and OBAs.