Let $a,b$ be elements of a unital C*-algebra $A$ with $0\leq a,b\leq 1$ (e.g., $a,b$ are projections). Is it the case there is a state $\tau$ on $A$ such that $|\tau(ab)|=\|ab\|$?
If $ab$ is normal (e.g., $a$ and $b$ commute), then this is a standard fact. Also, if $\|a\|=\|b\|=1$ and $\tau$ is a state satisfying $\tau(a)=\tau(b)=1$, then $\tau$ will work, but I don't see how to come up with such a $\tau$ if $a$ and $b$ don't commute.
Edit: I've reasked this question on MO: https://mathoverflow.net/questions/203411/existence-of-state-on-a-c-algebra-satisfying-tauab-ab