Timeline for Is the Fell-Doran problem trivial in a topological setting?
Current License: CC BY-SA 2.5
11 events
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| Nov 18, 2009 at 19:44 | comment | added | Yemon Choi | @Bruce: Just to offer a drive-by nitpick - I think it's slightly misleading to think of preservation of algebraic structure per se as the issue. The point is that in a C*-algebra the link between the *-structure and the norm is atypically tight and powerful; ultimately this is what gets you Kaplansky density | |
| Nov 17, 2009 at 12:57 | comment | added | Andrew Stacey | One of the pluses of MO is that you don't have to ask precise questions because it's people answering rather than computers. But for the record (since these questions are left public) you ought to edit your question so that the accepted answer is clearly the answer to the question asked! (Ideally in such a way that the original question is clear as well). However, life's short and all these comments should make it clear anyway! Your penance is to write a nice nlab page about it all. | |
| Nov 17, 2009 at 10:49 | comment | added | Bruce Bartlett | By the way, I admit my mistake, indeed I thought the thing that was missing was that you needed the algebra $A$ to be topological in some way and the rep to be continuous, but it seems it's more important to try to preerve algebraic structure like a $*$-structure. | |
| Nov 17, 2009 at 10:42 | comment | added | Bruce Bartlett | Hi Andrew, ok you got me on that one. Also I don't know much about the difference "locally convex topological vector spaces" vs "locally convex spaces"; I got this from a seminar yesterday and I think he meant the former, as you point out. I don't know much hard-core functional analysis as you know, it just seemed like a natural looking question, "When is the image of a representation dense?", that I was surprised the answer wasn't known apparantly even for Hilbert spaces! This breaks with the "karma" of math, so I wanted to know where the sneaky disclaimer is,though I didn't ask it that way | |
| Nov 17, 2009 at 10:27 | comment | added | Andrew Stacey | This is going to sound a bit whingey, but that's not what you asked, Bruce! You specifically asked about topology on the algebra. Also, you asked about a locally convex (topological vector?) space and this is for a Hilbert space where everything is almost always simpler than for other spaces. | |
| Nov 17, 2009 at 10:16 | comment | added | Bruce Bartlett | Do I understand this correctly? Mathhew is saying above that in the case of a $*$-structure preserving representation $T$ of a $*$-algebra on a seperable Hilbert space $H$, the Fell-Doran problem indeed trivializes in the sense that it follows from the "Kaplansky Density Theorem" and the "von Neumann bicommutant theorem" that the image of $T$ is then operator dense inside $L(H)$. That was spirit of my quest: I wanted to know to what extent the Fell-Doran is a "pathological" problem (having to do with the fact that one doesn't require the rep to preserve relevant structure) or a "natural" one. | |
| Nov 17, 2009 at 9:58 | vote | accept | Bruce Bartlett | ||
| Nov 16, 2009 at 20:01 | comment | added | Andrew Stacey | Bleugh. Now you say it, I remember getting all confused over the vast array of names for all of these different topologies! I have no problem with the actual topologies, just their names. Okay, now this sounds a lot like the approximation question which is known to be false. | |
| Nov 16, 2009 at 19:42 | comment | added | Matthew Daws | Yes! It's just the topology given by the seminorms $T\mapsto \|T(x)\|$ as $x$ varies over $H$. | |
| Nov 16, 2009 at 19:39 | comment | added | Andrew Stacey | Then I've misunderstood what "strong operator topology" means. I interpreted it as norm convergence, now you're saying that it's pointwise convergence (but pointwise with respect to the norm convergence on the space), is that correct? | |
| Nov 16, 2009 at 19:36 | history | answered | Matthew Daws | CC BY-SA 2.5 |