Quasi-Jordan Banach Algebras

Abstract

We initiate a study of quasi-Jordan normed algebras. It is demonstrated that any quasi-Jordan Banach algebra with a norm $1$ unit can be given an equivalent norm making the algebra isometrically isomorphic to a closed right ideal of a unital split quasi-Jordan Banach algebra; the set of invertible elements may not be open; the spectrum of any element is nonempty, but it may be neither bounded nor closed and hence not compact. Some characterizations of the unbounded spectrum of an element in a split quasi-Jordan Banach algebra with certain examples are given in the end.