Abstract:
 It is well known that in a Banach algebra, if the set of quasinilpotent
elements is a linear subspace, or a semigroup, then this set equals the
Jacobson Radicals. In this talk, we will consisder the similar case for the
elements with at most countable spectrum, which are called scattered
elements. Firstly, we will give the definition of scattered radical, and
show that the scattered radical has many properties as the Jacobson Radical.
Then we will give some equilavent conditions:
(i) $S(A)+S(A)\subset S(A);$
(ii) $S(A)S(A)\subset S(A);$
(iii) $[S(A),A]\subset R_{sc}(A).$
where, $S(A)$ means the set of scattered elements in the Banach algebras $A$
, and $R_{sc}(A)$ means the scattered radical of $A$.
