Vol. 43, No. 1, 1972

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The reducing ideal is a radical

Theodore Windle Palmer

Vol. 43 (1972), No. 1, 207–219

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 (AAR)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.

Mathematical Subject Classification 2000
Primary: 46K05
Received: 30 March 1971
Published: 1 October 1972
Theodore Windle Palmer