For any ∗-algebra A the
reducing ideal AR of A is the intersection of the kernels of all the ∗-representations
of A. Although the reducing ideal has been called the ∗-radical, and obviously
satisfies (A∕AR)R= {0}, it has not previously been shown to satisfy another
of the fundamental properties of an abstract radical except in the case of
hermitian Banach ∗-algebras where it equals the Jacobson radical. In this
paper we prove two extension theorems for ∗-representations. The more
important one states that any essential ∗-representation of a ∗-ideal of a
U∗-algebra (a fortiori, of a Banach ∗-algebra) has a unique extension to a
∗-representation of the whole algebra. These theorems show in particular that
(AR)R= AR if A is either a commutative ∗-algebra or a U∗-algebra. The
somewhat stronger statements which are actually proved, together with
previously known properties of the reducing ideal, show that the reducing ideal
defines a radical subcategory of each of the following three semi-abelian
categories:
(1) Commutative ∗-algebras and ∗homomorphisms.
(2) Banach ∗-algebras and continuous ∗-homomorphisms.
(3) Banach ∗-algebras and contractive ∗-homomorphisms.
