Existence of positive linear functionals such that $\varphi(a^*a)>0$

Question

Let $A$ be a $C^*$-algebra, $0\neq a\in A$. I want to show that there exists a positive linear functional $\varphi:A\to\mathbb{C}$ such that $\varphi(a^*a)>0$.

My definitions: positive means $\varphi(b^*b)\geq 0$ for all $b\in A$.

Answer 1

One can also do this without using a faithful representation (say, if you want to use the result to prove Gelfand-Naimark):

By Hahn-Banach, there exists a bounded linear functional on $A$ such that $\phi(a^\ast a)=\lVert \phi\rVert \lVert a\rVert^2$. We can assume $\lVert \phi\rVert=\lVert a\rVert=1$. Moreover, it is not hard to reduce the problem to the case of unital $C^*$-algebras, so I'll assume that $A$ has a unit.

Since $0\leq a^\ast a\leq 1$, we have $-1\leq 1-2a^\ast a\leq 1$, hence $\lVert 1-2a^\ast a\rVert\leq 1$. Let $\phi(1)=\alpha+i\beta$ with $\alpha,\beta\in \mathbb R$. On the one hand, $\lvert \phi(1)\rvert\leq 1$ implies $\alpha^2+\beta^2\leq 1$. On the other hand, $\lvert \phi(1)-2\phi(a^\ast a)\rvert\leq 1$ implies $(\alpha-2)^2+\beta^2\leq 1$. Then one can check that this is only satisfied for $\alpha=1$ and $\beta=0$, that is, $\phi(1)=1$.

To conclude, use that every unital functional of norm $1$ on a $C^\ast$-algebra is positive, hence a state.