From: toby@ugcs.caltech.edu (Toby Bartels) Subject: Re: path integral - sum over all paths? Date: 5 Jun 2000 19:16:51 GMT Newsgroups: sci.physics.research Summary: [missing] John Baez wrote: >Toby Bartels wrote: >>I have seen books that call it "Gelfand-Naimark-Segal"; >>I even have one that goes so far as to abbreviate it "GNS". >I think you may be mixing up the Gelfand-Naimark theorem and the >Gelfand-Naimark-Segal construction. Yep, you're right. >>If A is all of C(X), then isn't Y the Stone Cech compactification of X? >Yeah, but don't people only talk about the Stone-Cech compactification >of a locally compact space? The original definition was definitely limited in some way. But I *know* I've seen people extend it as follows: The Stone Cech compactification functor from the category of topological spaces to the category of compact topological spaces is the left adjoint of the inclusion functor. I first saw this in Categories for the Working Mathematician. -- Toby toby@ugcs.caltech.edu ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: path integral - sum over all paths? Date: Tue, 6 Jun 2000 23:03:15 GMT Newsgroups: sci.physics.research In article <8hc0t9$kgs@gap.cco.caltech.edu>, Toby Bartels wrote: >John Baez wrote: >>Suppose X is a topological space and A is some algebra of >>bounded continuous complex-valued functions on X that's closed >>under pointwise complex conjugation. Then there's an obvious >>way to complete A and obtain a commutative C*-algebra. By the >>Gelfand-Naimark theorem this C*-algebra is isomorphic to C(Y), >>the algebra of all continuous complex-valued functions on Y, for >>some compact Hausdorff space Y which is unique up to canonical >>isomorphism. There's a continuous map i: X -> Y since any point of X >>determines a maximal ideal in C(Y). So in some very rough sense, >>Y is a "compactification" (and "Hausdorffification") of X. >If A is all of C(X), then isn't Y the Stone Cech compactification of X? Some extra comments here... First of all, you really meant to ask: "If A is all of the *bounded* continuous functions on X, then isn't Y the Stone Cech compactification of X?" - unless you are secretly using C(X) to stand for bounded continuous functions on X. We need boundedness for A to be a C*-algebra. And then the answer is: "Morally speaking yes, but most people use the term `compactification' only if the original topology on X matches the induced topology it gets by thinking of it as a subspace of Y using the map i: X -> Y. And this will be true only if some conditions hold. I think the usual condition people consider is that X be locally compact, but your favorite functional analysis textbook (by Conway) seems to favor the condition that X be completely regular, which seems to be necessary and sufficient."