Identities in algebras with involution

Abstract

One of the main features of the theory of polynomial identities is the existence (for anyn) of a division algebra of degreen, formed by adjoining quotients of central elements of the algebra of genericn×n matrices; this division algebra is extremely interesting and has been used by Amitsur (forn divisible by either 8 or the square of an odd prime) as an example of a non-crossed product central division algebra. The main object of this paper is to obtain, in a parallel method, division algebras with involution of the first kind, knowledge of which would answer some long-standing questions in the theory of division algebras with involution. One such question is, “Is every division algebra with involution of the first kind a tensor product of quaternion division algebras?” In the process, a theory of (polynomial) identities in algebras with involution is developed with emphasis on prime PI-algebras with involution.