Re: Stone duality for generalized Boolean algebras

["Fred E.J. Linton" <fejlinton@usa.net>]

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



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