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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?

greg
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    Lots of information also on this related thread. Also, note that two compact Hausdorff spaces $X,Y$ are homeomorphic iff $C(X)$ and $C(Y)$ are -isomorphic. This is why one usually sees the field of $C^$-algebras as the study of "noncommutative topological spaces". With the particular subset of von Neumann algebras seen as "noncommutative measure spaces". – Julien Jul 16 '13 at 14:08

2 Answers2

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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 $h'(f):=f \circ h$.

Edoardo Lanari
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  • What is this suitable category? – Martin Jul 20 '13 at 03:41
  • At the beginning I misread the question and I thought that the domain category was that of banach spaces and bounded linear mappings, so that such an induced map would have been exactly that of the contr.hom-functor.. – Edoardo Lanari Jul 20 '13 at 07:52
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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.

Hui Yu
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    The key-words are "Yoneda embedding" – Edoardo Lanari Jul 16 '13 at 10:39
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    @Lano I got the feeling that the PO is more interested in analysis so I mentioned Gelfand-Naimark, which has a part of the flavour of Yoneda. – Hui Yu Jul 16 '13 at 10:41
  • I am interested in GN, in particular model theoretic applications. Here's a follow up question: if $h$ is instead a quotient map, does it induce a quotient of Banach spaces? – greg Jul 16 '13 at 11:12
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    For simplicity, let's take $X$ and $Y$ to be compact Hausdorff. If $\pi : X \to Y$ is a surjection (e.g., if $Y$ is a sufficiently nice quotient of $X$, and $\pi$ is the canonical surjection), then the induced map $C(\pi) : C(Y) \to C(X)$ is actually an injection, so that in the Gelfand--Naimark picture, (nice) quotient spaces correspond to (unital $C^\ast$-)subalgebras. – Branimir Ćaćić Jul 16 '13 at 13:20