Re: Stone duality for generalized Boolean algebras

[Jeff Egger <jeffegger@yahoo.ca>]

Hi Andrej,

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".  [As it turns out: every (not necessarily unital) C*-algebra has an "approximate unit" (even a canonical one); and, for a *-homomorphism between unital  C*-algebras, preserving the approximate unit is equivalent to preserving the unit.]  

In any event, I have found it (paradoxically) illuminating to think of: locally compact Hausdorff spaces and proper maps as a subcategory, via the one-point compactification functor (here denoted ( )+1), of  compact Hausdorff spaces; and, (not necessarily unital) C*-algebras as a subcategory, via the free functor (also denoted ( )+1), of unital C*-algebras.  Since C(X+1)=C_0(X)+1 holds at the level of objects (where = means isomorphic), it remains to reverse-engineer the correct classes of arrows  in order to piggyback the desired statement off the usual duality theorem.  Of course, it's also possible to consider the b.o./f.f. factorisations of the two ( )+1 functors: that results in some class of partial maps on the topological side, as you suggest.

I expect that something similar happens in the case of Stone duality and GBAs.  Hope this helps!

Cheers,
Jeff.

--- On Fri, 1/21/11, Andrej Bauer <andrej.bauer@andrej.com>  wrote:

> From: Andrej Bauer <andrej.bauer@andrej.com>
> Subject: categories: Stone duality for generalized Boolean algebras
> To: "categories list" <categories@mta.ca>
> Received: Friday, January 21, 2011, 8:19 AM
> The well known Stone duality says
> that there is an equivalence between
> Boolean algebras (BA) and the opposite of Stone spaces and
> continuous
> maps. Here a Stone space is a Hausdorff zero-dimensional
> compact
> space. Furthermore, Boolean algebras correspond to Boolean
> rings with
> unit.
> 
> How exactly does this extend to generalized Boolean
> algebras? A
> generalized Boolean algebra (GBA) is an algebra with 0,
> binary meet,
> binary join, and relative complement in which meets
> distribute over
> joins. Equivalently it is a Boolean ring (possibly without
> a unit). I
> have seen it stated that the dual to these are (the
> opposite of)
> locally compact zero-dimensional Hausdorff spaces and
> proper maps,
> e.g., it is stated in Benjamin Steinberg: "A groupoid
> approach to
> discrete inverse semigroup algebras", Advances in
> Mathematics 223
> (2010) 689-727.
> 
> Another source to look at is Givant & Halmos
> "Introduction to Boolean
> algebras", but there  this material is covered in exercises
> and the
> duality is stated separately for objects and for morphisms,
> and I
> can't find an exercise that treats the morphisms, so I
> wouldn't count
> that as a reliable  reference. Stone's original work does
> not seem to
> speak about morphisms very clearly (to me).
> 
> Unless I am missing something very obvious, it cannot be
> the case that
> GBA's correspond to locally compact 0-dimensional Hausdorff
> spaces and
> proper maps, for the following  reason.
> 
> The space which corresponds to the GBA 2 = {0,1} is the
> singleton. The
> space which corresponds to the four-element GBA 2 x  2 is
> the two-point
> discrete space 2. There are _four_ GBA homomorphisms from 2
> to 2 x 2
> (because a GBA homomorphism preserves 0 but it need not
> preserve 1),
> but there is only one continuous map from 2  to 1. Or to put
> it another
> way, there are _four_ ring homomorphisms from Z_2 to Z_2 x
> Z_2
> (because they need not preserve 1), but there is only one
> continuous
> map from 2 to 1. So, either the spectrum of a GBA is not
> what I think
> it is (namely maximal ideals), or we should be taking a
> more liberal
> notion of maps on the topological side. For example, there
> are _four_
> partial maps from 2 to 1.
> 
> If someone knows of a reliable reference, one would be
> much
> appreciated. I won't object to a direct proof of duality
> either.
> 
> With kind regards,
> 
> Andrej
> 
> 
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