Positive linear functional on an involutive Banach algebra

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Why is every positive linear functional on an involutive Banach algebra with a bounded approximate continuous?

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This is a result of Varopoulos. You'll find it in

R. S. Doran and V. A. Belfi, Characterizations of C${}^*$-algebras: The Gelfand–Naimark theorems. Monographs Textbooks Pure Appl. Math. 101.

on p. 69.