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."