tag:blogger.com,1999:blog-6912603287930240451.post1551386532557331727..comments2023-08-16T08:02:29.630+00:00Comments on Noncommutative Geometry: Be wise, quantize!Unknownnoreply@blogger.comBlogger12125tag:blogger.com,1999:blog-6912603287930240451.post-50765930952242760212007-03-07T03:52:00.000+00:002007-03-07T03:52:00.000+00:00This is in response to Urs's last question. Yes, i...This is in response to Urs's last question. Yes, it makes sense, but one has to be a bit careful about the class of states on L(H) (=B(H), in your notation). For any positive trace class operator p with Tr (p)=1, <BR/>\phi (a)=Tr (ap) is a state one L(H). It is pure iff p is a rank one projection which in this case then is identified with a unit vector v. It is however not true that all statesAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-5258148471973540122007-03-06T22:05:00.000+00:002007-03-06T22:05:00.000+00:00Several years ago I had a go at formulating the GN...Several years ago I had a go at formulating the GNS embedding that arises by taking the direct product of <B>all GNS reps</B> over <B>all states</B> as a left adjoint, but couldn't get the details to work. (I think this is what Urs' original post was aiming at.)<BR/><BR/>One problem, if I recall correctly, is that the uniqueness of factorization needed in the definition of left adjoint doesn't Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-76102134843068265322007-03-05T17:53:00.000+00:002007-03-05T17:53:00.000+00:00Hi,okay, good. Thanks for you patience with me!Hop...Hi,<BR/><BR/>okay, good. Thanks for you patience with me!<BR/><BR/>Hopefully without straining that patience too much once again, let me try to suggest a slight modification of the category of quadruples which you mentioned, a modification that comes closer to the original motivation I had:<BR/><BR/>You suggest a category whose objects are quadruples consisting of<BR/><BR/> H -- a Hilbert space<Unknownhttps://www.blogger.com/profile/01006460812296046328noreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-52300211450506944872007-03-05T03:20:00.000+00:002007-03-05T03:20:00.000+00:00That is good; at least now we have a common starti...That is good; at least now we have a common starting point. I don't think an interesting functor from Hilb back to the category of pairs really exits (but see below). In fact Hilb is not the target category for the GNS (unless you are willing to forget a lot by composing it with a forgetful functor to Hilb). I can now elaborate on my first response to your question. Given a pair (A, \phi),Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-50327937514123590832007-03-04T13:05:00.000+00:002007-03-04T13:05:00.000+00:00You write:"So, the initial data for the GNS is rea...You write:<BR/><BR/>"So, the initial data for the GNS is really a pair (A, \phi) of a C*-algebra and a state on it and the role of \phi is by no means auxiliary."<BR/><BR/>Ah, thanks. I did not appreciate the role played by the states, properly.<BR/><BR/>So, let me see if I can come up with a well-formed version of my question then.<BR/><BR/>I presume we have a category of pairs (A,\phi), whose Unknownhttps://www.blogger.com/profile/01006460812296046328noreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-32464672917201568652007-03-03T17:50:00.000+00:002007-03-03T17:50:00.000+00:00hi again,I think before discussing the functorial...hi again,<BR/>I think before discussing the functoriality property of the GNS construction it is important to realize that the Hilbert space that one defines through GNS, in general, depends on the state. For different states \phi on the same C*-algebras you can get Hilbert spaces of different dimensions. Example: start with a commutative algebra A =C(X), or A =C[0, 1] for that matter. Then aAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-22253325851206385472007-03-02T21:24:00.000+00:002007-03-02T21:24:00.000+00:00Hi again,it was me who asked the above question (I...Hi again,<BR/><BR/>it was me who asked the above question (I did not mean to remain anonymous) -- many thanks for the reply!<BR/><BR/>There is a choice involved in the GNS construction, that of a state. This alone need not mean that we don't get an adjoint functor. It is quite common for adjoint functors to require us making lots of choices.<BR/><BR/>What happens if I choose a different state andAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-14275648874051956502007-03-02T21:09:00.000+00:002007-03-02T21:09:00.000+00:00oops! in the above comment I meant to say `irreduc...oops! in the above comment I meant to say `irreducible' instead of `faitthful' (thanks to Arup!)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-46283788804232365702007-03-02T20:28:00.000+00:002007-03-02T20:28:00.000+00:00Dear Anonymous,Thanks for your question. In fact ...Dear Anonymous,<BR/>Thanks for your question. In fact there is no such functor from C^*-algebras, per se, to Hilbert spaces. The GNS construction starts with a C^*-algebra and a STATE on it and produces a Hilbert space as well as a C^*-morphism from the given C^*-algebra into the algebra of bounded operators on that Hilbert space. This representation may very well fail to be faithful, but it isAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-5165778819280433932007-03-02T19:46:00.000+00:002007-03-02T19:46:00.000+00:00Since in the above entry it says"I would like to i...Since in the above entry it says<BR/><BR/>"I would like to invite all those who are interested to contribute to the following issues or to a related topic of their choice. [...] 7. The role of operator algebras"<BR/><BR/>I might maybe dare to audaciously go ahead and post a question belonging to that topic:<BR/><BR/>on Hilb -- the category of Hilbert spaces and isomorphism between them -- we haveAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-90434322769346520792007-02-25T17:53:00.000+00:002007-02-25T17:53:00.000+00:00Hi,I am glad to see a blog like this here come int...Hi,<BR/><BR/>I am glad to see a blog like this here come into existence. Will be very interested in what is going on here.<BR/><BR/>I am from over at the <A HREF="http://golem.ph.utexas.edu/category/" REL="nofollow">n-category Cafe</A>, where we like to think about what one might call "<A HREF="http://golem.ph.utexas.edu/category/2007/02/qft_of_charged_nparticle_dynam.html" REL="nofollow">Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-66404736390084206572007-02-15T21:08:00.000+00:002007-02-15T21:08:00.000+00:00This topic of "quantization and NCG" is very relev...This topic of "quantization and NCG" is very relevant. The word `quantum', from the beginning, is not so much related to `non-commutativity' but rather to `integrality'. In the word quantum' there is really this discovery by Planck, of the formula<BR/>for blackbody radiation, from which he understood that energy had<BR/>to be quantized in quanta of $\hbar \nu$. There is a confusion, created byAChttps://www.blogger.com/profile/02309452726336306635noreply@blogger.com