Variants of point fixed theorem - MathOverflow

Variants of point fixed theorem

2010-10-26T09:07:22Z

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 $g\in G$.

A fixed point for $G$ is an element $x$ of $C$ such that $g(x)=x$ for any $g\in G$.

What conditions on $G$ assure the existence of a fixed point for $G$?

The only condition which I know is noncontracting (=distal), see Fixed point theory, Granas/Dugundji, page 173. I need other conditions.

Answer by Bill Johnson for Variants of point fixed theorem

2010-10-26T14:19:08Z

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.

Answer by Keivan Karai for Variants of point fixed theorem

2010-10-26T15:16:04Z

I think if $G$ is amenable, you always get a fixed point. Check the definition of amenability in Wikipedia: http://en.wikipedia.org/wiki/Amenable_group [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"

Answer by BigBill for Variants of point fixed theorem

2010-10-26T15:17:54Z

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!

Answer by TCL for Variants of point fixed theorem

2010-11-07T06:40:13Z

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: http://math.gmu.edu/~tlim/pams81.pdf -TCL