Talk:C*-algebra

From Wikipedia, the free encyclopedia

WikiProject Mathematics
Mathematics Portal
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: B Class High Priority  Field: Analysis
WikiProject Physics This article is within the scope of WikiProject Physics, which collaborates on articles related to physics.
B This article has been rated as B-Class on the assessment scale.
??? This article has not yet received an importance rating within physics.

Help with this template

Comments:

What is the point of the spaces introduced by User:Deflog, some of which change the paragraph structure?CSTAR 19:36, 14 May 2004 (UTC)

This article seems to suggest ||x*||=||x|| for B* algebras. Phys 23:42, 3 Aug 2004 (UTC)

Yes it does. Indeed, this is the commonly accepted definition. Cf Dixmier 1.2.1 CSTAR 23:56, 3 Aug 2004 (UTC)

Contents

[edit] C*-algebra

It really sucks that we have to write C-star-algebra!CSTAR 03:01, 8 Sep 2004 (UTC)

You don't like writing C* algebra or C*-algebra, then? :) Lupin 03:35, 8 Sep 2004 (UTC)
In the title?CSTAR
We could move the page... Lupin 04:00, 8 Sep 2004 (UTC)


Is that possible? I though asterisks were not permissiblle characters in titles.CSTAR 04:08, 8 Sep 2004 (UTC)
Well I guess they are possible. Maybe we should move others: In particular, Spectrum of a C-star-algebra to Spectrum of a C*-algebra. CSTAR 04:18, 8 Sep 2004 (UTC)

[edit] NPOV

Ya know, saying that a book is "exciting and insightful" is not a statement that belongs in an encyclopedia without some careful couching ("commonly held to be..." etc). So I think that the opinionated comments about the references do violate NPOVness. Lupin 03:32, 8 Sep 2004 (UTC)

Granted, possibly the phrasing is in bad taste. But to call this an opininiated comment? The reference we are talking about is a major work by a Fields medalist. This book is indeed widely regarded as an exciting and insightful book. Various instances of courses at major univetrsities (UC Berkeley UCLA) have been based on this book; this has has opened new areas of research. Moreovrr to use NPOV (which can be a highly subjective criterion) as a bludgeon to correct some other problem, e.g. a problem of style or taste is really a bad idea.
The whole point of a collabortaive effort, if you feel more careful "couching" is required, is to by all means put that couching in. But don't simply wipe some piece of text out entirely if some justification is possible, and I believe I have given an argument as to why the book is insightful and exciting.CSTAR 03:48, 8 Sep 2004 (UTC)
Well that comment (and the others) may not necessarily be "wrong", but it certainly does come across as opinionated to me. Unfortunately I don't myself feel qualified to speak with any authority about the perceived merits of the references. Perhaps you could expand on your comments? They still sound inappropriate in an encyclopedia to me. Apologies if my "wiping" ruffled feathers. Lupin 03:59, 8 Sep 2004 (UTC)

[edit] Moving to C*-algebra

This is currently a redirect page. Please voice objections soon. VERY soon. CSTAR 04:25, 8 Sep 2004 (UTC)

[edit] type I C*-algebras

I have changed the statement 'A is type I iff every factor representation generates a type I factor' to 'A is type I iff every representation generates a type I von Neumann algebra'. If I'm not mistaken, the latter is equivalent with the former (or is a separability condition necessary?)? I get my definition (indirectly) from Arveson's book. So I'll add the first statement as a relaxation of the type I condition. I think this approach is nicer, since then the definition is easier and more general to state.

They are clearly equivalent for separable A. According to Sakai's book, Theorem 4.6.4 it's true in general.--CSTAR 04:52, 16 September 2006 (UTC)

[edit] Uniqueness

What's the problem with claiming "A is a C*-algebra in a unique way"? A C*-algebra (involutive algebra with a complete C*-norm) has a unique complete C*-norm. Am I missing something?--CSTAR 18:10, 18 September 2007 (UTC)

good to see you still around, CSTAR :-). C* norms are unique in general, for every C*-algebra. that sentence about quotient algebras just seemed kinda funny: "The algebraic quotient of a C*-algebra by a closed proper two-sided ideal is a C*-algebra in a unique way". it might suggest that, somehow in some weird way, one should expect more than one C*-structure on the quotient algebra but this turns out not to be the case.
it's more appropriate to point out here instead that the natural norm turns out to be a C*-norm, hence the only one, IMHO. Mct mht 23:13, 18 September 2007 (UTC)