On Sat, 22 Jan 2011 05:11:26 PM EST, Jeff Egger <jeffegger@yahoo.ca> responded to Andrej Bauer <andrej.bauer@andrej.com> as follows:
There is an analogous problem when trying to "extend" Gelfand duality to locally compact Hausdorff spaces and (not necessarily unital) C*-algebras: everything works fine on the object level, but there are many (not necessarily unital) *-homomorphisms C_0(1) --> C_0(2), but only one continuous map 2 --> 1. The standard solution, IIRC, is to restrict the class of *-homomorphisms to those which "preserve the approximate unit". ...
Another approach more closely resembles the "solution" that George Janelidze and I have pointed out for the Boolean problem. To sketch it, let me temporarily borrow the old Gelfand-Naimark terminology "normed ring" for commutative C*-algebras with unit, and use "normed rng" for their not-necessarily-unital counterparts. As in the Boolean setting, then, "normed rngs" is, to within equivalence, augmented "normed rings" (that is, the slice category "normed rings"|'C', where C is the "coefficient ring" -- probably the real or the complex field in most applications), whence as opposite to "normed rngs" one immediately deduces the category of pointed compact Hausdorff spaces (and *all* continuous base-point-preserving functions). And, as there also, while the complement of the base point in such a space may be locally compact, the passage to that complement is, again, far from functorial -- unless one is willing either to restrict one's attention, among maps of pointed compact spaces, to those that send *only* the base point to the base point, or to extend one's attention to certain only partially defined functions as maps of locally compact spaces. Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]