Variants of point fixed theorem - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T12:20:36Z http://mathoverflow.net/feeds/question/43646 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43646/variants-of-point-fixed-theorem Variants of point fixed theorem BigBill 2010-10-26T09:07:22Z 2010-11-07T06:40:13Z <p>Let $E$ be a dual Banach space and $C$ a nonempty convex weak* compact subset of $E$. Let $G$ be a group of weak* continuous linear isometries on $E$. Suppose that $g(C)\subset C$ for all $g\in G$. </p> <p>A fixed point for $G$ is an element $x$ of $C$ such that $g(x)=x$ for any $g\in G$.</p> <blockquote> What conditions on $G$ assure the existence of a fixed point for $G$? </blockquote> <p>The only condition which I know is noncontracting (=distal), see Fixed point theory, Granas/Dugundji, page 173. I need other conditions. </p> http://mathoverflow.net/questions/43646/variants-of-point-fixed-theorem/43668#43668 Answer by Bill Johnson for Variants of point fixed theorem Bill Johnson 2010-10-26T14:19:08Z 2010-10-26T14:19:08Z <p>Kakutani's fixed point theorem says that it is enough for $G$ to be equicontinuous on $C$. Now equicontinuity in the weak$^*$ topology might be too restrictive, but the pre adjoints of of the elements of $G$ are equicontinuous in the normed topology and this can sometimes (always?) be used to find a fixed point of $G$ in $C$. This is what Rudin does in his book Functional Analysis to prove the existence of Haar measure on a compact group. </p> http://mathoverflow.net/questions/43646/variants-of-point-fixed-theorem/43674#43674 Answer by Keivan Karai for Variants of point fixed theorem Keivan Karai 2010-10-26T15:16:04Z 2010-10-26T15:16:04Z <p>I think if $G$ is amenable, you always get a fixed point. Check the definition of amenability in Wikipedia: <a href="http://en.wikipedia.org/wiki/Amenable_group" rel="nofollow">http://en.wikipedia.org/wiki/Amenable_group</a> [Even though the definition is given for discrete groups, it generalizes to second countable, locally compact groups: see Bob Zimmer's book semisimple groups and ergodic theory"</p> http://mathoverflow.net/questions/43646/variants-of-point-fixed-theorem/43675#43675 Answer by BigBill for Variants of point fixed theorem BigBill 2010-10-26T15:17:54Z 2010-10-26T15:36:18Z <p>Finally, Bourbaki "topological vector spaces" seems to answer completely the question if $C$ has a denumerable type. None condition is needed. A such group has a fixed point!</p> http://mathoverflow.net/questions/43646/variants-of-point-fixed-theorem/45141#45141 Answer by TCL for Variants of point fixed theorem TCL 2010-11-07T06:40:13Z 2010-11-07T06:40:13Z <p>Let X be the predual of E. If the dual of every separable subspace of X is separable, then C contains a point that is fixed by EVERY weak* continuous affine isometry of C into C. This is Theorem 2 in the following paper: <a href="http://math.gmu.edu/~tlim/pams81.pdf" rel="nofollow">http://math.gmu.edu/~tlim/pams81.pdf</a> -TCL</p>