A characterisation of $C^*$-algebras through positivity of functionals

Abstract

‎We show that a unital involutive Banach algebra‎, ‎with identity of‎ ‎norm one and continuous involution‎, ‎is a $C^*$-algebra‎, ‎with the‎ ‎given involution and norm‎, ‎if every continuous linear functional‎ ‎attaining its norm at the identity is positive‎.