This question relates to one on topology and C^\*-algebras that was asked two days ago, namely at the link: [C\*-algebras with no nontrivial endomorphisms](https://mathoverflow.net/questions/109772/c-algebras-with-no-nontrivial-endomorphisms) [link text](https://mathoverflow.net/questions/109772/c-algebras-with-no-nontrivial-endomorphisms)


Let D be the closed unit disk in the plane. Let C(D) be the unital ring of continuous complex-valued functions on D. Then, C(D) is naturally a Banach algebra with pointwise addition and multiplication as the ring operations. Furthermore, the "star-operation" on elements of C(D) can be defined by: $g\*(x)$ to be the complex conjugate (pointwise) of g(x), any x in D, and for any function g in C(D).


The question in reference 1 above was related to injective star-endomorphisms of some $C\*$ -algebras. Without saying so, I think the famous commutative Gelfand-Naimark theorem and the Gelfand representation figured "in the background", because of the interplay of commutative $C\*$-algebras and topology on compact spaces ...
If $\alpha$ is an injective star-morphism of C(D) to itself, is it possible for $\alpha$(C(D)) to be a proper (i.e. a `strict') star-sub-algebra of the $C\*$-algebra C(D) ?


If so, I cannot find such a special \*-morphism $\alpha$, hence my question.



injective *morphisms of C(D), the C^* algebra of continuous functions on the closed unit disk D
Assume $D$ lies in $\mathbb{R}^2$ and define $f:D\rightarrow D$ by $f(x):=2x$ if $\|x\|\leq 1/2$ and $f(x):=\frac{x}{\|x\|}$ if $\|x\|\geq 1/2$. Then, $f$ is onto and continuous, but it is not injective. It is easy to see that $f^\ast :C(D)\rightarrow C(D)$ is an injective $^\ast$-homomorphism, but it is not onto.


