It is well known that most topological spaces can be studied via their algebra of continuous real-valued (or complex-valued) functions. For instance, in the setting of compact Hausdorff spaces, there is a complete dictionary between topological properties of the space $X$ and corresponding algebraic properties of the algebra $C(X)=C(X,\mathbb{R})$, and two such spaces $X$ and $Y$ are homeomorphic if and only if their algebras $C(X)$ and $C(Y)$ are isomorphic. A similar theory is available in the locally compact Hausdorff case, by replacing $C(X)$ with the algebra of real-valued functions on $X$ which vanish at infinity. Furthermore, these algebras are precisely all the commutative C\*-algebras (at least if you take complex-valued functions). Thus if one wishes to study locally compact Hausdorff spaces, he might as well study commutative C\*-algebras.


Is there an analogous algebraic approach to the theory of "nice" metric spaces (e.g. compact, locally compact, connected, etc.)? More specifically, is there a natural function space one can attach to every nice metric space which essentially contains all data about the metric space, up to isometry?


I'll welcome any references related to this question.


**EDIT:** Many of the answers referred me to Nik Weaver's (and others') theory of "Lipschitz algebras". I took a look at his book and while very interesting (and well written), it doesn't seem to be quite what I am looking for. More specifically, it seems that his constructions of Lipschitz algebras only characterize the metric space up to bi-Lipschitz equivalence (which he calls a quasi-isometry), while I'm looking for natural function spaces that would characterize the space up to isometry. His results in section 1.8 of the book show that two (pointed) complete metric spaces have isomorphic Lipschitz function algebras if and only if they are bi-Lipschitz equivalent, rather than isometric. He does have a characterization up to isometry (using a different Lipschitz algebra) for the class of metric spaces of diameter less than 2 (section 1.7) , but his reduction of a general metric space (say complete) to a metric space of diameter $\le 2$ is of course far from preserving isometries (rather preserving Lipschitz mappings).


So it seems that I am looking for a somewhat more "rigid" construction. Any ideas about that?


Thanks.



Is there an algebraic approach to metric spaces?
A very complete reference is the book "Lipschitz Algebras" by Nik Weaver. In there you will find various types of spaces of Lipschitz functions that can be associated to a metric space, and several results of the kind you are asking about.


From the book's introduction:



> 
> Thus, there is a robust duality
>  between metric properties of $X$ and
>  algebraic properties of $Lip\_0(X)$.
>  The set of weak\* continuous
>  homomorphisms from $Lip\_0(X)$ into the
>  scalars can be isometrically
>  identified with the completion of $X$
>  (Theorem 4.3.2); Lipschitz maps from
>  $X$ into $Y$ correspond to weak\*
>  continuous homomorphisms from
>  $Lip\_0(Y)$ into $Lip\_0(X)$ (Corollary
>  4.2.9); closed subsets of $X$ correspond to weak\* closed ideals of
>  $Lip\_0(X)$ (Theorem 4.2.5); and 
>  nonexpansive images of $X$ correspond
>  to weak\* closed, self-adjoint
>  subalgebras of $Lip\_0(X)$ (Theorem
>  4.1.10).
> 
> 
> 


