Timeline for Self-adjoint bounded operator, resolution of the identity, def. of the diagonal
Current License: CC BY-SA 3.0
3 events
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| Aug 2, 2011 at 2:12 | comment | added | Dima Shlyakhtenko | Perhaps you are right, in any case it was not very clear what the original question wants in terms of convergence. The original question stated that working out what $Diag(B)$ means for $B$ a spectral projection is enough, which makes me think that the original question required some form of convergence that would make $Diag$ a normal map. | |
| Aug 1, 2011 at 11:37 | comment | added | Mikael de la Salle | Dima: what makes you think that, if it exists, the limit will be normal? Are you assuming that the limit is in the norm topology? Because in the point-weak* topology, the net $Diag_\theta$ is relatively compact (because the $Diag_\theta$'s are unital completely positive), and hence has (at least) one cluster point. What is clear is that such a cluster point will be a norm $1$ map with values in $\mathfrak A$, which is the identity on $\mathfrak A$. It is therefore a conditional expectation. Not normal by your proof. | |
| Jul 31, 2011 at 7:15 | history | answered | Dima Shlyakhtenko | CC BY-SA 3.0 |