Is there any clear definition of amenable non Hausdorff groupoids? It should be possibly non-separable nuclear C*-algebras? Please let me know if there is any existing literature talking about this.
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1$\begingroup$ Why do you think such groupoids would have anything to do with nonseparability of associated Cstar algebras? $\endgroup$– Yemon ChoiCommented Oct 27 at 2:17
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$\begingroup$ because etale hausdorff groupoids has countable bisections covering any open set $\endgroup$– PKOACommented Oct 27 at 15:32
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1$\begingroup$ I don't follow the reasoning at all. Take any discrete uncountable group, then its reduced Cstar algebra is non-separable. $\endgroup$– Yemon ChoiCommented Oct 27 at 17:01
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$\begingroup$ I took my words back, discard the non-separability, What is the exact definition of non-hausdorff amenble groupoids? $\endgroup$– PKOACommented Oct 27 at 23:38
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1$\begingroup$ You may find the definition of amenable groupoid (for non Hausdorff étale groupoids as well) in my paper with Buss arxiv.org/pdf/2302.14466 (see definition 4.23). Nevertheless, as Yemon Choi pointed out, non Hausdorffness has nothing to do with separability of the algebra. $\endgroup$– Diego MartinezCommented Oct 28 at 20:30
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Just to add to my comment, the definition of amenability is the same in the Hausdorff/non-Hausdorff settings (as long as $G$ is assumed to be étale). In particular, what we want are functions $(\xi_i)_i \subset C_c(G)$ such that $||\xi_i^* * \xi_i|| \leq 1$ and $\xi_i^* * \xi_i$ converges to $1$ uniformly over compact sets of $G$. In the paper https://arxiv.org/pdf/2302.14466 it is proven (albeit in a very roundabout way) that if $G$ is amenable then $C_{red}^*(G) \cong C_{max}^*(G)$ is a nuclear C*-algebra.
Note that by $C_c(G)$ here we mean the set of functions $a \colon G \to \mathbb{C}$ that are finite linear combinations of functions that are continuous and compactly supported on some open bisection of $G$. Thus, elements in $C_c(G)$ are not continuous as functions on $G$, but if $G$ is Hausdorff then they actually are. I hope this helps.
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$\begingroup$ What do you mean? Whether I can provide some examples of amenable groupoids? Examples 4.8 and 4.9 in the paper I referenced are somewhat enlightening. $\endgroup$ Commented Oct 30 at 18:48
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$\begingroup$ I mean examples of non-Hausdorff amenable groupoids. Does groupoids of germs give an example of such? $\endgroup$– PKOACommented Oct 31 at 22:35
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$\begingroup$ Again, look at example 4.9 in the paper. Any étale groupoid is a groupoid of germs of an inverse semigroup action (some people may disagree what 'germs' means here, but what I said is true with the notion of germs in that paper). And, thus, some groupoids of germs are amenable, some are not. It depends on the groupoid itself. $\endgroup$ Commented Nov 1 at 10:59