### On strictly semi-simple Banach algebras

Pacific Journal of Mathematics, Vol. 9, No. 2 (1959), 551–554.

Introduction

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.