Let $A$ be a unital $C^\*$-algebra. Suppose $\rho\_1$ and $\rho\_2$ are two states on $A$. If $\rho\_1=\rho\_2$, we have $\|\rho\_1+i\rho\_2\|=\sqrt{2}$.


If we have $\|\rho\_1+i\rho\_2\|=\sqrt{2}$, can we conclude that $\rho\_1=\rho\_2$?



Calculation of the norm of linear combinitation of two states on a $C^*$-algebra
The answer is YES.
Consider any $a\in A$ with $0\le a\le 1$ and
put $\alpha:=(\rho\_1+i\rho\_2)(a)$ and $\beta:=(\rho\_1+i\rho\_2)(1-a)=(1+i)-\alpha$.
Then $x:=(\bar{\alpha}/|\alpha|)a+(\bar{\beta}/|\beta|)(1-a)$ has norm at most $1$ and so
$$|1+i|\geq|(\rho\_1+i\rho\_2)(x)|=|\alpha|+|(1+i)-\alpha|.$$
Hence $\alpha$ must be on the line connecting $0$ and $1+i$, that is to say,
$\rho\_1(a)=\rho\_2(a)$.


