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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 $\mathbb{C}$.

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    $\begingroup$ If your $C^{\ast}$-algebras are unital and you assume that your morphism preserves the unit then the answer is yes, because unital contractions between $C^{\ast}$-algebras are positive and positive maps are self-adjoint (i.e. preserve $*$). Probably you may use unitisations to conclude it in the general case but I am not sure how to do that. $\endgroup$
    – Mateusz Wasilewski
    Commented Mar 3, 2015 at 17:39
  • $\begingroup$ Good point, That was indeed trivial. thank you. $\endgroup$
    – Simon Henry
    Commented Mar 3, 2015 at 17:48
  • $\begingroup$ Regarding your final comment: unfortunately I don't think this will help you get a star-free characterization, because the star (and the Cstar identity) is implicitly needed in Mateusz's argument. I am fairly sure the result is false for other classes of unital Banach star-algebras $\endgroup$
    – Yemon Choi
    Commented Mar 16, 2015 at 11:07
  • $\begingroup$ @Yemon Choi : My point was that this gives a star-free characterization of morphisms of C* algebras between C* algebras. It shows that the question of whether there exists a star-free characterization of C*-algebras among Banach algebras makes sense, because the category of C*-algebra is a full subcategory of the category of Banach algebra and homomorphisms of norm smaller than one, but it does not answer it. $\endgroup$
    – Simon Henry
    Commented Mar 16, 2015 at 15:12
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    $\begingroup$ @SimonHenry OK, I see now; thanks for clarifying. After I wrote my comment I recalled that there may be something along the lines you want: have you come across the Vidav-Palmer theorem? In a unital Banach algebra one can define a notion of hermitian which 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
    Commented Mar 16, 2015 at 16:14

1 Answer 1

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This fact is proved in greater generality in propositions A.5.8 and A.5.9 in Operator algebras and their modules. An operator space approach. D. P. Blecher, C. Le Merdy

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