Topological radicals. VI: Scattered elements in Banach Jordan and associative algebras (Q2833675)

The scattered radical of a Banach algebra \(A\) was introduced in [\textit{Yu. V. Turovskii} and \textit{V. S. Shulman}, Funct. Anal. Appl. 35, No. 4, 312--314 (2001; Zbl 1045.46033); translation from Funkts. Anal. Prilozh. 35, No. 4, 88--91 (2001)]. An ideal of \(A\) is said to be scattered if all of its elements have countable spectrum, and the scattered radical, \(R_s(A)\), is the largest scattered ideal of \(A\). One of the main results of the paper under review is that the scattered radical exists for each Banach Jordan algebra. Further, the scattered radical is closed and the map \(A\mapsto R_s(A)\) is a topological radical on the class of all Banach Jordan algebras. The authors also discuss the so-called sub-Banach Jordan algebras, which are a variation of the Banach Jordan algebras.NEWLINENEWLINE The previous parts of this series are [\textit{V. S. Shulman} and \textit{Y. V. Turovskii}, Banach Cent. Publ. 67, 293--333 (2005; Zbl 1090.46038)], [\textit{V. S. Shulman} and \textit{Yu. V. Turovskii}, in: Elementary operators and their applications. 3rd international workshop, Queen's University Belfast, Belfast, Northern Ireland, April, 14--17, 2009. Basel: Birkhäuser. 45--114 (2011; Zbl 1253.46053)], [\textit{V. S. Shulman} and \textit{Yu. V. Turovskii}, Funct. Anal. Appl. 46, No. 4, 287--304 (2012; Zbl 1319.46038); translation from Funkts. Anal. Prilozh. 46, No. 4, 61--82 (2012)], [\textit{E. Kissin} et al., Integral Equations Oper. Theory 74, No. 1, 51--121 (2012; Zbl 1270.46043)] and [\textit{V. S. Shulman} and \textit{Y. V. Turovskii}, in: Algebraic methods in functional analysis. The Victor Shulman anniversary volume. Basel: Birkhäuser. 171--280 (2014; Zbl 1321.46050)].