Trivolutions on Banach algebras

Let A be a Banach algebra. The notion of trivolution introduced and studied in [?]. In this talk we characterize a general study of trivolutions on Banach algebras. Also we study positive linear functionals on a trivolutive Banach algebra and we investigate relationship between the positive linear functionals on A and A♯. We show that, unlike C* -algebras, Gelfand-Naimark-Segal theorem does not hold even the range of trivolution is a C* -algebra.