HERMITIAN MORITA EQUIVALENCES BETWEEN MAXIMAL ORDERS IN CENTRAL SIMPLE ALGEBRAS

Let R be a Dedekind domain with quotient field K. That every maximal order in a finite dimensional central simple K-algebra A, (the algebra of nxn matrices over D), where D is separable over K, is Morita equivalent to every maximal order in D is a well known linear result. Hahn defined the notion of Hermitian Morita equivalence (HME) for algebras with anti-structure, generalizing previous work by Frohlich and McEvett. The question this paper investigates is the hermitian analogue of the above linear result. Specifically, when are maximal orders with anti-structure in A, HME to maximal orders with anti-structure in D in the sense of Hahn? Two sets of necessary and sufficient conditions are obtained with an application which provides the hermitian analogue under some conditions.