On Coarse Isometries between Banach Spaces

Lixin Cheng ³ÌÁ¢ÐÂ

*(Xiamen University)*

14:45-15:45, Oct 30, 2017 Science Building A1510

__Abstract:__

A mapping $f$ from a Banach space $X$ into a Banach space $Y$
is said to be a Lindenstrauss-Szankowski provided $\lim_{t\rightarrow+\infty}\frac{\varepsilon(t)}t=0$,
where $$\varepsilon(t)=\sup\{\big|\|f(x)-f(y)\|-\|x-y\|: \|x-y\|\vee\|f(x)-f(y)\|\leq t\},\;\;t\geq0.$$
Lindenstrauss and Szankowski first studied such mappings and showed the following remarkable
representation theorem for a surjective coarse isometry $f: X\rightarrow Y$ with the additional assumption.
$$\int_1^\infty\frac{\varepsilon(t)}{t^2}dt\le \infty$$
There is a linear surjective isometry $U: X\rightarrow Y$ so that
$$\|f(x)-Ux\|=o(\|x\|),\;\;{\rm as\;}\;\|x\|\rightarrow\infty.$$
Nevertheless, the research of this topic has stopped
for over 30 years, because they have given examples showing
that this theorem fails without the integral convergence assumption above, even if both $X$ and $Y$ are the real
line $\mathbb R$.
In this talk, we shall show that, in certain Banach spaces, the results analogous to the Lindenstrauss-Szankowski theorem
may hold without the integral convergence assumption£¬ even without the surjectivity on the mappings, but we substitute
$w^*$-topology and the free ultrafilter limits on $\mathbb N$ for the norm-topology and the usual limits, respectively.

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