Timeline for Property (T) for pseudogroups
Current License: CC BY-SA 3.0
6 events
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| Apr 15, 2011 at 21:08 | comment | added | Jesse Peterson | @Mark cont.: $I_n(\mathbb Z)$ is not defined as partial transformations on a set but rather partial transformations on a quantum set (= Hilbert space), my answer is a possible way to extend the framework of Ceccherini-Silberstein, Grigorchuk, and de la Harpe to this setting. There very well may be another way. Do you know if amenability has been defined for inverse semigroups which are not pseudogroups of transformations on a set? | |
| Apr 15, 2011 at 21:08 | comment | added | Jesse Peterson | @Mark: My answer is only relevant to pseudogroups of transformations on a set which are closed under restricting to subsets, this is the definition which is taken in the paper of Ceccherini-Silberstein, Grigorchuk, and de la Harpe. For the case when it is not closed under restriction one can still define amenability and property (T), but the proposition I wrote above will not be true in that case. | |
| Apr 15, 2011 at 16:50 | comment | added | user6976 | @Jesse: I certainly do not understand some parts of your answer. But $I_n(\mathbb{Z})$ should not be considered finite, it is a "lattice" in $I_n(\mathbb{R})$, and the proof of (T) for it cannot be muxch simpler than for $SL_n(\mathbb{Z})$. By definition a pseudogroup (inverse semigroup) of transformations of any set $X$ is any set of partial injective maps $X\to X$ closed under composition and taking inverses. We do not necessarily require that it is closed under restrictions to subsets. | |
| Apr 14, 2011 at 16:43 | history | edited | Jesse Peterson | CC BY-SA 3.0 |
Corrected proof and added more detail.
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| Apr 12, 2011 at 0:56 | history | edited | Jesse Peterson | CC BY-SA 3.0 |
Fixed typo.
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| Apr 12, 2011 at 0:48 | history | answered | Jesse Peterson | CC BY-SA 3.0 |