Agnes Scott College

Edith H. Luchins

On strictly semi-simple Banach algebras
Pacific Journal of Mathematics, Vol. 9, No. 2 (1959), 551–554.


Define the strict radical of an algebra to be the intersection of just those of its two-sided ideals which are regular maximal right ideals. Call the algebra strictly semi-simple (sss) if its strict radical is the zero ideal. This note proves that the strict radical of a real Banach algebra B contains the set of topologically nilpotent elements of B. Also, it gives a condition which is both necessary and sufficient for B to be sss.