let C(T) denote the set of all continuous functions on the unit circle T. We know this to be a c*-algebra. Let m be the normalised arc length measure on T.

Show that the functional \varphi:C(T)\rightarrow \mathbb{C}, f\mapsto \int fdm
is positive, ie \varphi(C(T)^+)\subseteq \mathbb{C}^+ where the "+" denoted the positive elements in the set under question.