Reflexive Banach algebras

Abstract

1. This paper contains a generalization in the commutative case of the structure theorem of W. Ambrose for H*-algebras. An immediate result of Ambrose's axioms is that the orthogonal complement of an ideal is an ideal of the same nature. This fact suggests thinking in terms of a Banach algebra whose conjugate space is a Banach algebra such that the annihilator of an ideal in one algebra is an ideal in the other algebra. This is the essential idea in the definition of a commutative, semi-simple, reflexive Banach algebra given in ?2. This section shows that such an algebra is essentially an 1,-space in which a product has been defined by means of co-ordinate multiplication. ?3 contains a few remarks and examples concerning the author's conjecture that a similar structure theorem must exist in the noncommutative case. The term annihilator will be constantly needed in both its ring and vector space sense. The phrase ring annihilator will be used to distinguish the use of the term in the ring sense from its use in the vector space sense. B' will be used to denote the conjugate space of the given algebra B. Greek letters e, 8, and p will be used to denote complex numbers. Elements in B will be denoted by letters x, y, and z, while elements in B' will be denoted by f, g, and h. The value of the linear functionalf at the point x will be denoted by (x, f). Projections in B will be denoted by means of the letter E, while those in B' will be denoted by means of E'. The following theorems from the general theory will be used.