An involution of an algebra over
a field of characteristic different from two is called scalar if the sum of each element
and its involute is a scalar (multiple of the unit). Certain algebras having scalar
involutions have played an important role in the construction of metaplectic
representations and the applications of that theory to problems in number theory and
automorphic forms. They also arise in an analytic context related to homomorphic
discrete series and in questions about invariants of classical groups. This paper deals
with determining the structure of the most general algebras having scalar
involutions.
