The Jacobson property in rings and Banach algebras

Abstract

In a Banach algebra A it is well known that the usual spectrum has the following property: σ(ab)\\{0}=σ(ba)\\{0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sigma (ab) \\setminus \\{0\\} = \\sigma (ba) \\setminus \\{0\\} \\end{aligned}$$\\end{document}for elements a, b in A. In this note we are interested in subsets of A that have the Jacobson Property, i.e. X subset A such that for a, b in A: 1-ab∈X⇒1-ba∈X.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} 1 - ab \\in X \\implies 1 - ba \\in X. \\end{aligned}$$\\end{document}We are interested in sets with this property in the more general setting of a ring. We also look at the consequences of ideals having this property. We show that there are rings for which the Jacobson radical has this property.