Do maps between topological spaces somehow induce maps between Banach spaces?

Question

If $X,Y$ are topological spaces and $h:X\rightarrow Y$ is a continuous map, is there some sort of induced map \begin{align*} h':C_b(X)\rightarrow C_b(Y) \end{align*} (or in the other direction) where $C_b(X)$ is the Banach space of bounded, continuous, real-valued functions on $X$?

If $h:X\rightarrow Y$ is a quotient map, is there an induced (quotient) map between the associated Banach spaces?

Answer 1

There is such an induced map. However it is in the other direction.

If $h:X\to Y$ is a continuous map between topological spaces, then for each $f\in C(Y)$, we have $f\circ h\in C(X)$ since compositions of continuos maps are themselves continuous. This induces a map \begin{equation} f\in C(Y)\xrightarrow{h'}f\circ h\in C(X). \end{equation}

This is the starting point of so much wonderful mathematics, such as differential topology, differential geometry, spectral theory, and non-commutative geometry. Sorry to get emotional here, but I still remember the excitement and joy when I first read about Gelfand-Naimark theorem in my junior year. Really, they just studied the induced map of this induced map.

According to my shallow knowledge, this shows a link between the geometry of a set and the geometry of the function space over the set. When we know much about the set, we can use this link to study the functions. When we know more function theory, then this sheds light on properties of the underlying set.

Answer 2

What about reversing the arrows? You get the canonical "contravariant hom-functor" (since $C_b(X) =Hom(X,\mathbb{R})$ in a suitable category) so that $h:X \to Y$ induces $h':C_b(Y) \to C_b(X)$ with $h'(f):=f \circ h$.