 Positive functional on a C-star algebra

A bounded linear functional $\tau$ on a C-star algebra is positive when $\tau (x)\ge 0$ for every positive element of a C-star algebra. Positivity impliesand $\Vert \tau \Vert =\tau (1)$; conversely, $\Vert \tau \Vert =\tau (1)$ implies positivity.