Every positive functional achieves its norm on the identity - converse?

Question

Let $X$ be a unital $C^*$ algebra. I can show that if $f$ is a positive linear functional, i.e. $f(xx^*)\geq0$, then $f$ is continuous and achieves its norm at the identity. Is the converse also true - is a continuous linear functional that achieves its norm at the identity positive?

Answer 1

See Lemma 3.2 on p130 in Takesaki: https://books.google.at/books?id=dTnq4hjjtgMC&lpg=PA139&vq=faithful&hl=de&pg=PA130#v=onepage&q&f=false