### On radicals and continuity of homomorphisms into Banach algebras

Pacific Journal of Mathematics, Vol. 9, No. 3 (1959), 755–758

Introduction

All Banach algebras considered are over the real
field and all homomorphisms considered are algebraic (real-linear). An
algebra is called semi-simple, strongly semi-simple, or strictly semi-simple, if its Jacobson radical, Segal radical, or strict radical, respectively, is the zero ideal; that is, if its regular maximal right
ideals, its regular maximal two-sided ideals, or those of its two-sided
ideals which are regular maximal right ideals, intersect in the zero ideal.
Rickart proved that a semi-simple commutative Banach
algebra has the property that every homomorphism of a Banach algebra
into it is continuous. Call an algebra with this property an absolute
algebra. Yood proved that every homomorphism of a
Banach algebra onto a dense subset of a strongly semi-simple Banach
algebra is continuous. Thus a strongly semi-simple Banach algebra is
"almost" absolute. The question arose: Is a (noncommutative) semi-simple or strongly semi-simple Banach algebra necessarily absolute?
A negative answer is furnished in the present note. It is shown that
in order for a Banach algebra to be absolute it is sufficient that it be
strictly semi-simple and necessary that it have zero as its only nilpotent
element. The latter condition is shown to be sufficient for some special
Banach algebras to be absolute.