## Variants of point fixed theorem

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

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.

-

## 4 Answers

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.

-

### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.

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"

-

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!

-

Is this an answer or a comment to your original question? – Yemon Choi Oct 26 2010 at 19:13
This is an answer to the initial question in the case where $C$ is second countable for the topology of the norm. – BigBill Oct 26 2010 at 19:56

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

-

Thank you very much. It is a surprising result.   Actually, I am interested in the case where $E$ is not separable (unfortunately for me). What is known about my question?  – BigBill Nov 8 2010 at 18:08
If $G$ is commutative and $E$ is uniformly convex (or uniformly smooth), then $G$ has a common fixed point. Other weaker conditions like weak* normal structure on $C$ will also suffice.  – TCL Nov 10 2010 at 21:00
Thank you.   A last question:  What do you think about the case where $E$ is the space $B(\ell_\infty)$ of bounded linear operators on $\ell_\infty$ (its predual is the space $\ell_\infty\hat{\otimes}\ell_1$ where $\hat{\otimes}$ denotes the projective tensor product) and a noncommutative discrete group $G$?   Same question with $E=B_{w*}(\ell_\infty)$ the space of weak* continuous bounded operators on $\ell_\infty$.      – BigBill Nov 11 2010 at 22:55
I don't know the answer to your last question. I will look into it. BTW commutativity is not needed in my previous comment, the group of isometries is left-reversible semigroup, so the theorem in math.gmu.edu/~tlim/pams74.pdf  applies. – TCL Nov 13 2010 at 15:31
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 36, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9249351620674133, "perplexity_flag": "head"}