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I've been looking at Folner's Condition recently, and I'm struggling to find a proof for why the existence of a Folner sequence on a locally compact group implies that it is amenable (and the converse to this).

[EDIT: So far I've looked at amenability in terms of invariant means, so my question should be:

"How does to existence of a Folner sequence imply the existence of a left invariant mean on a locally compact group?"]

If anyone could give a proof for this or point me in the direction of one on the internet that I can read that would be really helpful - have struggled to find anything myself.

Thanks in advance, Jo

Jo Williams
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  • Many people (myself included) would DEFINE a group to be amenable if it has a Folner sequence. There are about 500 equivalent definitions of amenability, so you should probably include in your question which one you're using. – Paul Siegel Apr 12 '12 at 19:42
  • Thanks for the response, I have edited my original post. I hope this makes my question more clear. :) – Jo Williams Apr 12 '12 at 19:49
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    Jo, you can use the Folner sequence to define an almost invariant mean, then take a limit of these along an ultrafilter to get a genuinely invariant one. See for example these notes, starting page 26, for a discussion in the case of Z. http://www.dpmms.cam.ac.uk/~bjg23/ATG/Chapter3.pdf I'm speaking here of the case when G is a discrete group; in the locally compact case matters are a little more complicated. There are many better sources in the literature - recent blog notes of Tao, to give just one example. – Ben Green Apr 12 '12 at 21:00
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    Google

    folner sequences imply amenable

     – Bill Johnson Apr 12 '12 at 21:01
  • I've found Tao's notes (the ones Ben Green is referring to above), http://terrytao.wordpress.com/2009/04/14/some-notes-on-amenability/, to be very useful. – Henry Towsner Apr 12 '12 at 22:05
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    If you have library access to a copy of Paterson's book "Amenability" then I think there is an explanation there - although I don't remember if parts are left "as exercises" – Yemon Choi Apr 13 '12 at 04:23
  • There is also a proof in Brown and Ozawa's book $C^{\ast}$-algebras and Finite dimensional approximations – Sabrina Gemsa Sep 22 '16 at 12:33

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