Isomorphisms of H*-algebras

Abstract

H*-algebras were introduced and studied by Ambrose [1] in the associative case, and the theory has been extended to such particular classes of non-associative algebras as Lie [18, 19], Jordan[20, 21, 7], alternative [11] and non-commutative Jordan [6] algebras. In all these cases the core of the matter is showing that every H*-algebra (in the given class) with zero annihilator is the closure of the orthogonal sum of its minimal closed ideals (each of which is a topologically simple H*-algebra), and then listing all the topologically simple H*-algebras in the class. In fact every nonassociative H*-algebra with zero annihilator is the closure of the orthogonal sum of its minimal closed ideals [6, theorem 2·7], so the problem of the classification of topologically simple non-associative H*-algebras becomes interesting. In relation with this problem the question arises whether, once an algebra A has been structured as a topologically simple H*-algebra, every H*-algebra structure on A is (up to a positive multiple of the inner product) totally isomorphic to the given one (see [3] and [11, section 4]). As a consequence of the results in this paper we give a general affirmative answer to this question.