tag:blogger.com,1999:blog-6912603287930240451.post2715013977315975296..comments2024-03-29T00:47:51.081+00:00Comments on Noncommutative Geometry: On Gelfand-Naimark TheoremsUnknownnoreply@blogger.comBlogger13125tag:blogger.com,1999:blog-6912603287930240451.post-91355939777678857152009-10-06T03:14:53.759+00:002009-10-06T03:14:53.759+00:00I just learned that Israel Gelfand is no longer wi...I just learned that <a href="http://en.wikipedia.org/wiki/Israel_Gelfand" rel="nofollow" rel="nofollow">Israel Gelfand</a> is <a href="http://aclinks.wordpress.com/2009/10/06/israel-gelfand-1913-2009-r-i-p/" rel="nofollow" rel="nofollow">no longer with us</a>. R.I.P.Successful Researcherhttp://aclinks.wordpress.comnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-64664978842750020992008-06-08T19:08:00.000+00:002008-06-08T19:08:00.000+00:00Dear Pedro, Thanks a lot for your comments and ref...Dear Pedro, Thanks a lot for your comments and references. It looks indeed things get fairly complicated as soon as one passes the `commutative line'. This is, of course, a general principle.Masoud Khalkhalihttps://www.blogger.com/profile/03769072750559219167noreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-75643606723582626742008-06-06T06:50:00.000+00:002008-06-06T06:50:00.000+00:00Thanks a lot, Pedro, this seems very interesting. ...Thanks a lot, Pedro, this seems very interesting. I've ordered Stone's book, and I'll try to understand all this better and read the references you give.Fabien Besnardhttps://www.blogger.com/profile/00975996576182366143noreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-62885227016391635802008-06-05T22:57:00.000+00:002008-06-05T22:57:00.000+00:00Dear Masoud and Fabien,There is a lattice theoreti...Dear Masoud and Fabien,<BR/><BR/>There is a lattice theoretic (I should say locale theoretic) version of CGNT based on the idea that any Hausdorff space can be completely recovered up to homeomorphism from its locale of open sets (a locale is a complete lattice in which binary infs distribute over arbitrary sups); in particular, the points correspond, of course, to the maximal open sets. This Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-30796488623960524802008-05-14T03:16:00.000+00:002008-05-14T03:16:00.000+00:00Dear Tom,Thanks for your comments and for pointing...Dear Tom,<BR/>Thanks for your comments and for pointing out the book by Johnstone.<BR/>By the way, the NC Gelfand-Naimark theorem is not a duality result. Not clear to me at least how it could be. So if one just say `G-N duality' would be clear and points to the right theorem I think. May be you have something else in mind... My comment about categories really just meant as a truism and Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-17972220192479921822008-05-12T16:42:00.000+00:002008-05-12T16:42:00.000+00:00The connections between the Gelfand-Naimark theore...The connections between the Gelfand-Naimark theorems, Stone's representation theorem, and many other kinds of duality are thoroughly explored in Peter Johnstone's excellent book "Stone Spaces". <BR/><BR/>Masoud, a question. If one wants to use the word "duality" rather than "theorem", how should one refer to the duality between compact Hausdorff spaces and commutative C*-algebras? "CommutativeAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-50078941826513272142008-04-29T19:55:00.000+00:002008-04-29T19:55:00.000+00:00Dear Fabien,As you know, one definition of a Boole...Dear Fabien,<BR/>As you know, one definition of a Boolean algebra (or Boolean ring ), the one that actually enters the Stone theorem I mentioned, is a ring in which every element is an idempotent. This of course automatically implies that the ring is commutative and allows no noncommutative generalization. The equivalent lattice theoretic version, however, lends itself to a ``noncommutative" Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-84269761874972680992008-04-27T10:41:00.000+00:002008-04-27T10:41:00.000+00:00Dear Masoud,yes, it is certainly in the same vein....Dear Masoud,<BR/>yes, it is certainly in the same vein. I don't know if one can recover Stone's representation theorem with the CGNT, but let's imagine we can do it. Then there should be a commutative C*-algebra canonically associated to a boolean algebra. The boolean algebra would then be recovered as the lattice of projections of the C*-algebra, I guess. We can even think that this boolean Fabien Besnardhttps://www.blogger.com/profile/00975996576182366143noreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-72983311020814948182008-04-25T17:52:00.000+00:002008-04-25T17:52:00.000+00:00Dear Fabien,Thanks for your comment and for pointi...Dear Fabien,<BR/>Thanks for your comment and for pointing to your paper in the ArXive. There is an old result of Stone which says that the category of sets is anti-equivalent to a certain sub category of the category of Boolean algebras. I wonder if this result can be derived from the CGNT? It is certainly in the same vein....Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-78196912980234582892008-04-23T07:42:00.000+00:002008-04-23T07:42:00.000+00:00Thank you for this clarification Masoud. I personn...Thank you for this clarification Masoud. I personnaly use the name "Gelfand-Naimark theorem" for commutative GN theorem, and "Gelfand-Naimark-Segal" for noncommutative one. I think this is a rather common convention. Incidentally, it is quite easy to generalize the (commutative) GN theorem to the category of compact ordered spaces, and I've just put a paper on the archives about this !Fabien Besnardhttps://www.blogger.com/profile/00975996576182366143noreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-67462714681621122162008-04-16T07:29:00.000+00:002008-04-16T07:29:00.000+00:00The theorem is a very nice one, of clear historic ...The theorem is a very nice one, of clear historic significance, and much worthy of discussion, but why is it at all important "how it should be called"?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-22500891646940050382008-04-13T13:42:00.000+00:002008-04-13T13:42:00.000+00:00Dear Jamie,Thanks for your comment. One thing that...Dear Jamie,<BR/>Thanks for your comment. One thing that I did not mention, and I should have, is the spectral radius formula (Gelfand-Beurling) which is valid in any Banach algebra. Coupled with the C* identity it shows immediately that the norm of a C* algebra can be characterized algebraically as || x|| = square root of the spectral radius of x*x, and in particular is unique. The result onAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-7361580599672179822008-04-12T11:18:00.000+00:002008-04-12T11:18:00.000+00:00Dear Masoud,Thanks for that interesting post. I th...Dear Masoud,<BR/><BR/>Thanks for that interesting post. I think calling these two theorems the commutative and noncommutative Gelfand-Naimark theorem. Maybe somebody should try and get this change made on Wikipedia.<BR/><BR/>The result about involutive algebra maps being continuous is amazing, I hadn't come across that before! It's surprising just how powerful these involtions are... I wonder if Jamie Vicaryhttps://www.blogger.com/profile/10598376252445460712noreply@blogger.com