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

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