Redirected from Talk:Locally compact
Sure, if you think the other concept is more useful, let's change it. AxelBoldt, Sunday, June 9, 2002
I moved some of the examples over to compact space.
Is it actually true that all compact spaces are locally compact? Just checking :-)
Tychonoff | nonregular | |
---|---|---|
Compact | Examples! | Examples! |
LC, not C | Examples! | Examples! |
WLC, not LC | nonexistent | Examples! | Not WLC | Examples! | too generic to bother mentioning |
I moved the following example:
I believe the left order topology on R is locally compact. If x∈R, then ]-∞, x+1/n] for n∈N is a local neighborhood base for x consisting only of compact sets.
Heh, so much for tomorrow. I'll be gone until Monday evening — then I'll work this mess out. In the meantime, visitors to the main article should be sufficiently warned. — Toby Bartels, Friday, June 28, 2002
Maybe we can avoid confusing the beginner by first explaining the standard locally compact notion for Hausdorff spaces (which is what most people need and where the definitions all agree), with all the common examples and properties, and then have a separate section where the various definitions for non-Hausdorff spaces are compared.
The reason I edited the examples was the same: I find it more important that the reader first gets a good grasp of the concept by studying simple motivating examples, rather than find themselves in an overwhelming glut of tiny distinctions that are really only important for the specialists working in the field. AxelBoldt, Friday, June 28, 2002
OK, I've written a new version now with these principles in mind. (Note that I also added the Hausdorffness condition to Wikipedia's definition of topological vector space; the simplicity that this brought to the discussion of TVSs in this article is characteristic of the reasons for that requirement in the definition.) Have at it. — Toby Bartels, Wednesday, July 3, 2002
Some thoughts inspired by Axel's recent changes: I've always felt that the complex numbers were a red herring in functional analysis — when you get down to it, you can do most of this stuff over any commutative real C*algebra, so ¿what's so special about C? — which explains my focus on real-valued functions as the simplest example. But I agree that the complex-valued case is so common that it merits a mention here — especially as regards the Gelfand-Naimark theorem, the real version of which exists but is not quite as I had stated it. I do object to using notation like "C0" for the complex case as a default, but I agree that that's pretty common too. — Toby Bartels, Saturday, July 6, 2002
Is the classification of the commutative C* algebras really called "Gelfand-Naimark Theorem"? I seem to recall that name for the result that any C* algebra is a *-subalgebra of the algebra of linear continuous operators on some Hilbert space. AxelBoldt, Saturday, July 6, 2002
What you are citing is called "Gelfand Naimark Segal". Specifically, the Gelfand Naimark Segal construction takes a C* algebra and an element of the algebra and constructs from that a Hilbert space and a representation of the C* algebra on the Hilbert space. Your statement, the Gelfand Naimark Segal theorem, follows because the element of the algebra can be chosen so that the representation is faithful. A quick web search for "gelfand" AND "naimark" AND "theorem" brings up many references to my theorem but no references to your theorem that don't also have the name "Segal" attached. There do seem to be generalisations of the GN theorem to noncommutative C* algebras, representing the C* algebra as an algebra of functions defined on a noncommutative space (see the introduction to http://nyjm.albany.edu:8000/PacJ/1998/184-1-5 for a survey), but this is again different from the GNS theorem — although I wouldn't be surprised if the GNS theorem couldn't be made a corollary of some of these generalisations, nor would I be surprised if the generalisations made use of the GNS construction. Still, the basic idea seems clear: a GN theorem represents the C* algebra as bounded functions on a topological space (or generalisation thereof), while a GNS theorem represents the C* algebra as bounded operators on a Hilbert space. — Toby Bartels, Saturday, July 6, 2002
The first paragraph has a link to "local base," which describes them as a collection of open sets. Doesn't this contradict the parenthetical sentence which asserts that the neigborhoods need not be open? Bill Kielhorn, Mar 31, 2003.
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