ABSTRACT.Using a very simple subharmonic argument we prove that a Banach Jordan algebra is Hermitian if and only if the sum of two positive elements is positive. We apply this result to give a characterization of Banach Jordan algebras with involution which are JB*-algebras for an equivalent norm.If B is a complex Banach algebra with involution, the Shirali-Ford theorem states that every selfadjoint element of B has real spectrum if and only if x*x has positive spectrum for every x in A. In particular this theorem implies that such an algebra has many Hilbert space representations.The corresponding problem in Banach Jordan algebras is also important as if a Hermitian algebra is symmetric it is possible to make use of the representation theory for JB-algebras given by Alfsen, Shultz and St0rmer in [1]. A solution to this problem was given by Behncke in [5] but the argument he used, while containing a nice idea, is rather long. The aim of this paper is to give a simple proof of Behncke's result using a subharmonic argument and to give an application to the characterization of JB*-algebras. Instead of working with real Banach Jordan algebras throughout this paper we shall let A denote a complex unital Banach Jordan algebra with involution and S shall denote the real linear subspace of selfadjoint elements of A. As for Banach algebras, if o G A one can define the spectrum and spectral radius of a; these will be denoted by Sp(a) and p(a), respectively. Some standard properties of Sp(o)
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.