A Remark On Proper Left H* — Algebras

Abstract

W. Ambrose gave the theory of proper H* -algebras and M. Smiley in (2) gave an example of a left H* -algebra which is not a two-sided H* -algebra. Then he modified some of the arguments of Ambrose which yield the structure of proper right H*-algebras. In fact he proved that a proper right H*-algebra is merely a proper H*-algebra in which the norm has been changed to a certain equivalent norm in each of the simple components. In this short paper, we define proper left H*-algebras and give two lemmas for these classes. Then we prove the main result that every proper left H*-algebra is a proper H*-algebra. Thus, in this paper, we prove that the following are equivalent: (i) Proper left H*-algebras. (ii) Proper right H*-algebras. (iii) Proper H*-algebras.