positive linear functionals are bounded in $C^*$-algebras

We say that a linear functional $f$ on a $C^*$-algebra $A$ is positive if $f(a^*a)\geq 0$ for all $a\in A$. Why must it be the case that every positive linear functional on a $C^*$-algebra is bounded?

Solution 1:

For self-adjoint elements $a$, we have the inequality $-\lVert a\rVert e\le a\le\lVert a\rVert e$, where $e$ is the identity. So if $f$ is a positive linear functional, $-\lVert a\rVert f(e)\le f(a)\le\lVert a\rVert f(e)$ follows; i.e., $\lvert f(a)\rvert\le\lVert a\rVert f(e)$. For non-selfadjoint $a$, write $a=b+ic$ with $b$ and $c$ selfadjoint and use the result just shown.

Solution 2:

The C*-algebra may not has identity. Your answer is in the following reference: C*algebras and operator theory by Gerard Murphy, page 88, thm 3.3.1