Are self-adjoints elements norming?

Question

Let $\mathsf A$ be a C*-algebra and let $\mathsf{A}^*$ be its dual space. Is it true that for $f\in \mathsf A^*$ we have

$$\|f\|=\sup\{|f(x)|\colon\; \|x\|=1\mbox{ and }x^* = x\}?$$

Answer 1

No. Let $A:=M_2(\Bbb C)$, this splits as $\mathcal S\oplus \mathcal A$ where $\mathcal S=\{U \,\mid\, U^*=U\}$ and $\mathcal A=\{V\,\mid\, V^*=-V\}$. Then consider the projection $\pi_{\mathcal A}:A\to A$ to $\mathcal A$ along $\mathcal S$ composed by e.g. $\langle -,u \rangle$ for an $u\in \mathcal S$ with $\|u\|=1$. For this the right hand side is $0$, but its norm is $1$.