Topological radicals, VI. Scattered elements in Banach Jordan and associative algebras

A Jordan or associative algebra is called scattered if it consists of elements with countable spectrum (so called scattered elements). It is proved that for sub-Banach, Jordan or associative, algebras there exists the largest scattered ideal and it is closed. Accordingly, this determines the scattered topological radical. The characterization of the scattered radical is given, and the perturbation class of scattered elements is considered.