This is Proposition 13.3 on Page 78 in An Introduction to Operator Algebras by Kehe Zhu. The statement is as follows:
Proposition 13.3. If $\varphi$ is a Hermitian linear functional on a $C^*$-algebra $A$, then $\Vert \varphi \Vert = \sup\{\varphi(x):x = x^*, \Vert x\Vert \leq 1\}.$
where a linear functional $\varphi: A \rightarrow \mathbb{C}$ on a $C^*$-algebra $A$ is Hermitian if $\varphi(x^*) = \overline{\varphi(x)}$.
I wonder whether this can be generalized to the case that the "target" of the linear map $\varphi$ is a general $C^*$-algebra, say, $B$? I give the definition below:
- A linear map $\varphi: A \rightarrow B$ between two $C^*$-algebras $A$ and $B$ is said to be Hermitian if $\varphi(x^*) = \varphi(x)^*$.
and I guess Proposition 13.3 can be generalized as the following statement:
If $\varphi: A \rightarrow B$ is a Hermitian linear map between two $C^*$-algebras $A$ and $B$, then $\Vert \varphi \Vert = \sup\{\Vert\varphi(x)\Vert:x = x^*, \Vert x\Vert \leq 1\}$.
It is evident that in the above statement, the right-hand side is less than or equal to the left-hand side, because by definition $\Vert\varphi\Vert = \sup\{\Vert\varphi(x)\Vert: \Vert x\Vert \leq 1\} $ (without the extra limitation $x = x^*$).
However, I have no idea how to obtain the inequality in the reverse direction, because in Zhu's book he involved the properties of complex numbers such as rotations, etc.
Could anyone present a proof of this statement or a counterexample to disprove this statement (although I may feel a bit disappointed)? Any help is appreciated.