From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6487 Path: news.gmane.org!not-for-mail From: Jeff Egger Newsgroups: gmane.science.mathematics.categories Subject: Re: Stone duality for generalized Boolean algebras Date: Sat, 22 Jan 2011 10:47:49 -0800 (PST) Message-ID: Reply-To: Jeff Egger NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1295734035 15750 80.91.229.12 (22 Jan 2011 22:07:15 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 22 Jan 2011 22:07:15 +0000 (UTC) To: categories list , Andrej Bauer Original-X-From: majordomo@mlist.mta.ca Sat Jan 22 23:07:09 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Pglbw-0004qf-Rt for gsmc-categories@m.gmane.org; Sat, 22 Jan 2011 23:07:09 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:59679) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Pglbq-0007hs-UH; Sat, 22 Jan 2011 18:07:02 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PglbY-0003Vc-0U for categories-list@mlist.mta.ca; Sat, 22 Jan 2011 18:06:46 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6487 Archived-At: Hi Andrej,=0A=0AThere is an analogous problem when trying to "extend" Gelfa= nd 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 c= ontinuous map 2 --> 1.=A0 The standard solution, IIRC, is to restrict the c= lass of *-homomorphisms to those which "preserve the approximate unit".=A0 = [As it turns out: every (not necessarily unital) C*-algebra has an "approxi= mate unit" (even a canonical one); and, for a *-homomorphism between unital= C*-algebras, preserving the approximate unit is equivalent to preserving t= he unit.]=A0 =0A=0AIn any event, I have found it (paradoxically) illuminati= ng to think of: locally compact Hausdorff spaces and proper maps as a subca= tegory, via the one-point compactification functor (here denoted ( )+1), of= compact Hausdorff spaces; and, (not necessarily unital) C*-algebras as a s= ubcategory, via the free functor (also denoted ( )+1), of unital C*-algebra= s.=A0 Since C(X+1)=3DC_0(X)+1 holds at the level of objects (where =3D mean= s isomorphic), it remains to reverse-engineer the correct classes of arrows= in order to piggyback the desired statement off the usual duality theorem.= =A0 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 th= e topological side, as you suggest.=0A=0AI expect that something similar ha= ppens in the case of Stone duality and GBAs.=A0 Hope this helps!=0A=0ACheer= s,=0AJeff.=0A=0A--- On Fri, 1/21/11, Andrej Bauer = wrote:=0A=0A> From: Andrej Bauer =0A> Subject: ca= tegories: Stone duality for generalized Boolean algebras=0A> To: "categorie= s list" =0A> Received: Friday, January 21, 2011, 8:19 AM= =0A> The well known Stone duality says=0A> that there is an equivalence bet= ween=0A> Boolean algebras (BA) and the opposite of Stone spaces and=0A> con= tinuous=0A> maps. Here a Stone space is a Hausdorff zero-dimensional=0A> co= mpact=0A> space. Furthermore, Boolean algebras correspond to Boolean=0A> ri= ngs with=0A> unit.=0A> =0A> How exactly does this extend to generalized Boo= lean=0A> algebras? A=0A> generalized Boolean algebra (GBA) is an algebra wi= th 0,=0A> binary meet,=0A> binary join, and relative complement in which me= ets=0A> distribute over=0A> joins. Equivalently it is a Boolean ring (possi= bly without=0A> a unit). I=0A> have seen it stated that the dual to these a= re (the=0A> opposite of)=0A> locally compact zero-dimensional Hausdorff spa= ces and=0A> proper maps,=0A> e.g., it is stated in Benjamin Steinberg: "A g= roupoid=0A> approach to=0A> discrete inverse semigroup algebras", Advances = in=0A> Mathematics 223=0A> (2010) 689-727.=0A> =0A> Another source to look = at is Givant & Halmos=0A> "Introduction to Boolean=0A> algebras", but there= this material is covered in exercises=0A> and the=0A> duality is stated se= parately for objects and for morphisms,=0A> and I=0A> can't find an exercis= e that treats the morphisms, so I=0A> wouldn't count=0A> that as a reliable= reference. Stone's original work does=0A> not seem to=0A> speak about morp= hisms very clearly (to me).=0A> =0A> Unless I am missing something very obv= ious, it cannot be=0A> the case that=0A> GBA's correspond to locally compac= t 0-dimensional Hausdorff=0A> spaces and=0A> proper maps, for the following= reason.=0A> =0A> The space which corresponds to the GBA 2 =3D {0,1} is the= =0A> singleton. The=0A> space which corresponds to the four-element GBA 2 x= 2 is=0A> the two-point=0A> discrete space 2. There are _four_ GBA homomorp= hisms from 2=0A> to 2 x 2=0A> (because a GBA homomorphism preserves 0 but i= t need not=0A> preserve 1),=0A> but there is only one continuous map from 2= to 1. Or to put=0A> it another=0A> way, there are _four_ ring homomorphism= s from Z_2 to Z_2 x=0A> Z_2=0A> (because they need not preserve 1), but the= re is only one=0A> continuous=0A> map from 2 to 1. So, either the spectrum = of a GBA is not=0A> what I think=0A> it is (namely maximal ideals), or we s= hould be taking a=0A> more liberal=0A> notion of maps on the topological si= de. For example, there=0A> are _four_=0A> partial maps from 2 to 1.=0A> =0A= > If someone knows of a reliable reference, one would be=0A> much=0A> appre= ciated. I won't object to a direct proof of duality=0A> either.=0A> =0A> Wi= th kind regards,=0A> =0A> Andrej=0A> =0A> =0A> [For admin and other informa= tion see: http://www.mta.ca/~cat-dist/ ]=0A> =0A=0A=0A [For admin and other information see: http://www.mta.ca/~cat-dist/ ]