Compute the norm of the sum of two bounded linear functionals

Question

Let $\omega_1$ and $\omega_2$ be two states on a $C^*$-algebra $A$. Set $\tau=\omega_1+i\omega_2$.

Can we get the following conclusion: $\|\tau\|=\sqrt{2}$ if and only if $\omega_1=\omega_2$?

Answer 1

It seems to me that the conclusion is indeed correct.

"if" is trivial.

"only if": show that $\omega_1(x)=\omega_2(x)$ for every positive contraction of $A$ using the following inequality \begin{equation*} \sup_{x\in A_1^+}|\omega_1(x)^2-\omega_2(x)^2|^2+4 \leq \sup_{ x \in A_1}|\tau(x)|^4 = \|\tau\|^4 = 4, \end{equation*} where $A_1 = \{x\in A : \|x\|\leq 1\}$ is the unit ball in $A$, and $A_1^+\subseteq A_1$ is the set of positive contractions in $A$. It follows that $\omega_1(x)=\omega_2(x)$ for every positive element $x\in A$. Since every element of $A$ is a linear combination of four positive elements this ends the proof.