Order structure, multipliers, and Gelfand representation of vector-valued function algebras

Abstract

Abstract We continue the study begun by the third author of $C^*$-Segal algebra valued function algebras, with an emphasis on the order structure. Our main result is a characterization theorem for $C^*$-Segal algebra valued function algebras with an order unitization. As an intermediate step, we establish a function algebraic description of the multiplier module of arbitrary Segal algebra valued function algebras. We also consider Gelfand representation of these algebras in the commutative case