Timeline for The weak-star closure of closed left ideals corresponding to pure states
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Feb 14, 2016 at 5:56 | comment | added | ABB | Everting that I know about: We have always $1-$supp$\phi$=supp$N_{\tilde{\phi}}$ where supp$N_{\tilde{\phi}}$ is the supremum of all positive elements in the unit ball of $N_{\tilde{\phi}}$. Since $\phi$ is a pure state then its support is a minimal projection, say $e$ and so $N_{\tilde{\phi}}=A^{**}(1-e)$ is a maximal $w^*$-closed left ideal in $A^{**}$. It seems that $N_{\phi}=A^{**}(1-e)\cap A$ (which is a maximal closed left ideal in $A$). So the question is : If $e$ is a minimal projection in $A^{**}$ then $$A^{**}(1-e)=\overline{A^{**}(1-e)\cap A}^{w^*}$$ | |
| Feb 13, 2016 at 12:38 | comment | added | Simon Henry | I hav'nt been able to find a satisfying answer, but I have the impression that the situation is very different depending on if in the GNS representation associated to $\phi$ there is or not operator in the image of $A$ (so roughly on whether if algebra is type I/post-liminal or not). My guess would be that it is true for type I/post-liminal algebras and not necessarily for more general algebra... Have you tried to consider non type I examples ? | |
| S Feb 13, 2016 at 11:33 | history | suggested | Chris Ramsey | CC BY-SA 3.0 |
Added tags and cleaned up presentation.
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| Feb 13, 2016 at 11:19 | review | Suggested edits | |||
| S Feb 13, 2016 at 11:33 | |||||
| Feb 13, 2016 at 9:34 | comment | added | Simon Henry | In commutative algebra it also holds for finite linear combination of characters I think. In $K(H)$ if you take some injective trace class operator it produces a state $\phi$ such that $N_{\overline{\phi}} = 0$ and hence statisfies your property without being pure or finite combination of pure states. | |
| Feb 13, 2016 at 5:53 | history | asked | ABB | CC BY-SA 3.0 |