From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6490 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: Stone duality for generalized Boolean algebras Date: Sun, 23 Jan 2011 10:38:17 -0500 Message-ID: Reply-To: "Fred E.J. Linton" 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 1295877371 3410 80.91.229.12 (24 Jan 2011 13:56:11 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 24 Jan 2011 13:56:11 +0000 (UTC) Cc: Jeff Egger To: Original-X-From: majordomo@mlist.mta.ca Mon Jan 24 14:56:06 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 1PhMtp-0004jp-Gk for gsmc-categories@m.gmane.org; Mon, 24 Jan 2011 14:56:05 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:33241) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PhMtP-0003e9-Mj; Mon, 24 Jan 2011 09:55:39 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PhMsf-0000zH-8i for categories-list@mlist.mta.ca; Mon, 24 Jan 2011 09:54:54 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6490 Archived-At: On Sat, 22 Jan 2011 05:11:26 PM EST, Jeff Egger resp= onded to Andrej Bauer as follows: > There is an analogous problem when trying to "extend" Gelfand duality t= o > locally compact Hausdorff spaces and (not necessarily unital) C*-algeb= ras: > 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.=A0 The standard solution, IIRC, is to restrict = the > class of *-homomorphisms to those which "preserve the approximate unit".=A0... 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 rin= g" = 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 space= s. Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]