prove a linear functional on a unital $C^*$-algebra is positive

Question

Suppose $A$ is a unital $C^*$-algebra and $p$ is a projection in $A$. Suppose there is a state $\tau$ on $A$ such that $\tau(pa)=\tau(ap)$ for all $a\in A$. Can we conclude that $\tau(xp)\geq 0$ for all $x\in A^{+}$, where $A^{+}$ is the set of all positive elements in $A$.

Answer 1

It's true even if "projection" is replaced with "positive". That is, if $b\geq0$ and $\tau(ab)=\tau(ba)$ for all $a$, then you first go to $\tau(ab^n)=\tau(b^na)$ for all $n$, then $\tau(f(b)a)=\tau(af(b))$ for all polynomials $f$, and taking limits for all continuous functions $f$. Then, if $a\geq0$, $$ \tau(ab)=\tau(b^{1/2}ab^{1/2})\geq0, $$ since $b^{1/2}ab^{1/2}\geq0$.

Answer 2

Yes. Note that $pxp$ is positive whenever $x$ is, and we have $$\tau(xp)=\tau(xp^2)=\tau(pxp)\geq0.$$ The first equality follows from $p$ being a projection, the second from the given property of $\tau$, and the inequality from the fact that $\tau$ is a state.