Let $X$ and $Y$ be $2$ topological spaces and let $C_b(X)$ and $C_b(Y)$ denote the set of all continuous and bounded functions on X and Y, respectively, to the space of complex numbers. It is a well-known fact that $C_b(X)$ and $C_y(Y)$ are $C^*$ algebras (under the usual operations: pointwise addition, pointwise multiplication, pointwise scalar multiplication, sup norm, and the star operation being conjugation)
The following theorem is rather easy to show:
Theorem: If there exists a continuous and surjective map $F:Y \to X$, then there exists an injective $*$-homomorphim $H:C_b(X) \to C_b(Y)$.
One can just define $H(f)=f \circ F$ and show that this $H$ has the desired properties.
I am interested in the converse, namely:
Question: If there exists an injective $*$-homomorphim $H:C_b(X) \to C_b(Y),$ does there exist a continuous and surjective map $F:Y \to X?$ How should one define $F?$
Recall that a $*$-homomorphism is an algebra homomorphism that also preserves the norm and the star operation.
Any help would be greatly appreciated.
PS. If you are not familiar with $C^*$ algebras, then disregard that part of the question and regard this problem as a question about algebras.
No, not without additional hypotheses on the spaces $X$ and $Y$. For instance, let $Y$ be a single point and let $X$ be an indiscrete space (or any other space with no non-constant complex-valued functions) with more than one point. Then $C_b(X)\cong C_b(Y)\cong\mathbb{C}$, but there does not exist any surjection $Y\to X$.
It is true if you assume $X$ and $Y$ are compact Hausdorff. In that case, by Gelfand duality, $X$ is canonically homeomorphic to the space of $*$-homomorphisms $C_b(X)\to\mathbb{C}$ (with the topology of pointwise convergence), by identifying $x\in X$ with the $*$-homomorphism given by evaluation at $x$, and similarly for $Y$. Any $*$-homomorphism $H:C_b(X)\to C_b(Y)$ induces a continuous map $F:Y\to X$ by composition: take a point of $Y$, consider it as a $*$-homomorphism to $\mathbb{C}$, compose it with $H$ to get a $*$-homomorphism $C_b(X)\to\mathbb{C}$, and then identify this $*$-homomorphism with an element of $X$. Note that conversely, $H$ can similarly be recovered from $F$ by composition: $H$ is the map $C_b(X)\to C_b(Y)$ which takes a function on $X$ and composes it with $F$ to get a function on $Y$. (All of this is part of Gelfand duality, which says the functor $C_b$ gives an equivalence of categories between the category of compact Hausdorff spaces and the opposite category of commutative unital C*-algebras.)
In particular, if $H$ is injective, this means that a continuous function on $X$ is uniquely determined by its composition with $F$. This implies that $F$ is surjective (otherwise by Urysohn's lemma there would exist a nonzero continuous function on $X$ which vanishes on the proper closed subset $F(Y)$).
If $X$ is an arbitrary space, then $C_b(X)$ is naturally isomorphic to $C_b(\beta X)$, where $\beta X$ is the Stone-Cech compactification of $X$. So if $X$ and $Y$ are arbitrary topological spaces, all we can get from an injective $*$-homomorphism $C_b(X)\to C_b(Y)$ is a continuous surjection $\beta Y\to \beta X$.