C∗-Segal algebras with order unit

We introduce the notion of a (noncommutative) C∗-Segal algebra as a Banach algebra (A,‖⋅‖A) which is a dense ideal in a C∗-algebra (C,‖⋅‖C), where ‖⋅‖A is strictly stronger than ‖⋅‖C on A. Several basic properties are investigated and, with the aid of the theory of multiplier modules, the structure of C∗-Segal algebras with order unit is determined.