Speaker : Professor Cao, Peng Scattering Elements in the Banach Algebra 2012-09-19 (Wed) 10:30 - 12:00 Seminar Room 617, Institute of Mathematics (NTU Campus) 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 $A$.