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.