This is a follow up question to [this one](https://mathoverflow.net/questions/120421/).


If $X$ is a metric space, denote by $C\_u(X)$ the $C^\ast$-algebra of all bounded, uniformly continuous functions on $X$ (with the sup-norm).



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> Do we have $C\_u(X\_1 \times X\_2) = C\_u(X\_1) \hat{\otimes} C\_u(X\_2)$?
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> 


(Maybe it is more natural to use uniform spaces instead of metric spaces, but I think that the answer does not depend on this.)



Tensor product of C*-algebras of bounded, uniformly continuous functions on metric spaces
The spectrum of the $C^\ast$-algebra of bounded, uniformly continuous functions on a uniform space is the Samuel compactification. So your query can be restated in the form:
is the Samuel compactification of a product naturally identifiable with the product of the Samuel compactifications of the individual spaces. This is almost certainly wrong. The corresponding result for the Stone-Cech compactification of completely regular spaces is about as wrong as it could be. Presumably, you can give an explicit counterexample in the uniform case by using the fine uniformity. This is yet another example of the fact that if you want to extend duality (Riesz representation theory, Gelfand duality) from the compact case to the non-compact one (for completely regular spaces, uniform spaces, etc.) then you are well advised to leave categories of Banach spaces or algebras and use more general ones (mixed topologies, Saks spaces and algebras). A fairly systematic account can be found in the monograph "Saks Spaces and Applications to Functional Analysis".


