Timeline for Is there an operator algebraic reformulation of the invariant subspace problem?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Oct 30, 2013 at 10:01 | vote | accept | Sebastien Palcoux | ||
| Aug 4, 2013 at 14:07 | comment | added | Sebastien Palcoux | See this post : Is there a proof that the $C^{∗}$-algebras don't see the invariant subspace problem? | |
| Aug 2, 2013 at 22:03 | comment | added | Sebastien Palcoux | @YemonChoi, you wrote: "if $T$ is an ISP counter-example, then so is $S^{−1}TS$ with $S$ bijective, however $C^{*}(T)$ and $C^{*}(S^{−1}TS)$ can be highly different". I would be very interested by an explicit proof of "the $C^{*}$-algebras don't see the ISP" (as for the $W^{*}$). Do you have one ? If so, please write it as an answer. | |
| Aug 2, 2013 at 21:44 | comment | added | Yemon Choi | I think your last comment is more likely to be the correct reading. All these vague and hopeful questions about the STAR-algebras generated by T seem far less convincing than approaches via non-self-adjoint op alg theory, for the reasons hinted at in my previous comment | |
| Aug 2, 2013 at 21:41 | comment | added | Sebastien Palcoux | Now, perhaps, the $C^{*}$-algebras do not see the ISP, and so it would be more relevant to investigate $C(T)$ or $W(T)$ which is known to be reductive : it's an important property, because RAD implies ISP (see Mike answer below). | |
| Aug 2, 2013 at 21:35 | comment | added | Sebastien Palcoux | @YemonChoi : intuitively, the question is to know if the $C^{*}$-algebras see the ISP. If $T$ is an ISP counter-example, then $T$ is irreducible, noncompact-commuting, nonnormal and with spectrum strictly continuous. We could deduce some properties of $C^{*}(T)$. Unfortunately, this list here is not sufficient, because there are weight shifts checking this list and the ISP. It's the purpose of my post about banded operators, if they all check the ISP, we could then add non-banded to the list, and there is still not a candidate checking this completed list. | |
| Aug 2, 2013 at 20:56 | comment | added | Yemon Choi | It seems that you are looking for properties of C^*(T) which are, in particular, invariant under replacement of T by any S^{-1}TS. Perhaps you could obtain a more focused question by adding some specific properties you think might be relevant? (E.g. commutativity is not such a property, and I suspect nuclearity is not.) | |
| Jul 31, 2013 at 11:26 | comment | added | Sebastien Palcoux | If $T \in B(H)$ is an ISP counter-example then $W^{*}(T) = B(H)$, but what's the list of all the properties of $C^{*}(T)$ ? Conversely, if $C^{*}(T)$ verifies this list, is $T$ an ISP counter-example ? (idem with $W(T) = \overline{\langle T \rangle }^{wot} $). | |
| Jul 29, 2013 at 19:12 | history | edited | Sebastien Palcoux |
I replace the tag "banach-space" by "c-star-algebras".
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| Jul 29, 2013 at 13:55 | answer | added | Mike Jury | timeline score: 4 | |
| Jul 29, 2013 at 9:25 | history | asked | Sebastien Palcoux | CC BY-SA 3.0 |