 Positive functional on a C-star algebra

A complex linear functional $\tau$ on a C-star algebra is positive when $\tau (a)\ge 0$ for every Positive element of a C-star algebra $a$. Such a linear functional is automatically bounded and satisfies the Cauchy–Schwarz inequality for positive C-star functionals:For a unital C-star algebra, positivity gives $\Vert \tau \Vert =\tau (1)$. Conversely, a bounded complex linear functional with $\Vert \tau \Vert =\tau (1)$ is positive. The unit assumption belongs to these formulas; it is not needed for the definition of positivity.

A concrete example on complex matrices is $\tau (a)=\mathrm{tr}(\rho a)$ with $\rho$ a positive semidefinite matrix. It is a state on a C-star algebra when $\mathrm{tr}\rho =1$. The positivity follows from $\mathrm{tr}(\rho b^{\ast }b)=\mathrm{tr}((b{\rho }^{1/2}{)}^{\ast }b{\rho }^{1/2})\ge 0$.