Here is my precise question:

Let $A$ and $B$ be two $C^*$ algebras. Let $f: A \rightarrow B$ a homomorphism of $C^*$ algebras which is isometric (on its image). Is $f$ necessary a $*$-homomorphism ?

It sounds like a basic question, but I haven't found any counterexample nor any basic reference mentioning this kind of result, so I hope it is non trivial and suitable for MO.

Another equivalent question is the following:

Let $A$ be a $C^*$-algebra and $x$ an element of $A$ such that:

1) the spectrum of $x$ is included in $\mathbb{R}$

2) for any polynomial $P$ (with coefcients in $\mathbb{C}$) the norm of $P(x)$ is the supremum of $|P(t)|$ for $t \in \text{Spec}(x)$.

Is $x$ necessarily self-adjoint ?

Indeed if the answer to this second question is yes, then any isometric algebra homorphism send self adjoint element to self adjoint element hence is a $*$-homorphism, and conversely if the answer to the first question is yes then for such an element $x$ one can construct an isometric morphism from $\mathcal{C}(\text{Spec}(X))$ to $A$ which is hence a $*$-homomorphism and hence $x$ is self adjoint as the image of a self adjoint element.

Moreover, If I'm not mistaken the answer to these two questions is yes at least for finite dimensional algebras.

I don't really have a precise motivation for this question except curiosity, but it might be interesting to have such a "$*$-free" characterization of morphisms of $C^*$-algebras if one want to develop a satisfying analogue to $C^*$ algebras for other valued field than $\mathbb{R}$ and $\mathbb{C}$.

hermitianwhich makes no reference to an involution, just using the Banach algebra structure; and then if $A = H+iH$ where $H$ denotes the subset of "hermitian" elements, the V-P theorem says that $A$ is isometrically isomorphic to the underlying BA of a unital Cstar algebra, with the involution sending $h+ik$ to $h-ik$. $\endgroup$ – Yemon Choi Mar 16 '15 at 16:14