S. Mouton

More Spectral Theory in Ordered Banach Algebras

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.

Ruston elements and fredholm theory relative to arbitrary homomorphisms

We extend some of the Fredholm theory by, among other things, developing the theory of Ruston and almost Ruston elements and spectra relative to an arbitrary homomorphism. In addition, we provide a number of applications and generalise certain well-known results.

A condition for spectral continuity of positive elements

Let a be an element of a Banach algebra A. We introduce a compact subset T(a) of the complex plane, show that the function which maps a onto T(a) is upper semicontinuous and use this fact to provide a condition on a which ensures that if (an) is a sequence of positive elements converging to a, then the sequence of the spectral radii of the terms an converges to the spectral radius of a in the case that A is partially ordered by a closed and normal algebra cone and a is a positive element.

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.