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Cesàro function spaces fail the fixed point property. (English) Zbl 1168.46014
The Cesàro function space $$\text{Ces}_p(I)$$ consists of equivalence classes of Lebesgue measurable functions $$f$$ on $$I=[0,1]$$ or $$I=[0,\infty)$$ such that the norms
$\| f\|_{C(p)} = \left(\int_I \left(\frac{1}{x} \int_0^x |f(t)|\, dt\right)^p dx\right)^{1/p} \quad \text{if }1\leq p<\infty$
and
$\| f\|_{C(\infty)}= \sup_{x\in I,\, x>0} \frac{1}{x} \int_0^x |f(t)|\, dt \quad \text{if }p=\infty$
are finite. The authors prove that, if $$1\leq p \leq \infty$$ and $$I=[0,1]$$ or if $$1< p \leq \infty$$ and $$I=[0,\infty)$$, the Cesàro function spaces $$\text{Ces}_p(I)$$ contain asymptotically isometric copies of $$\ell^1$$. Consequently, for these $$p$$ and $$I$$, $$\text{Ces}_p(I)$$ and its dual $$\text{Ces}_p(I)^*$$ fail the fixed point property for nonexpansive maps. This contrasts with the known result [Y. Cui and H. Hudzik, Collect. Math. 50, No. 3, 277–288 (1999; Zbl 0955.46007)] that the sequential analogs of these spaces, the Cesàro sequence spaces, have the fixed point property for nonexpansive maps if $$1<p<\infty$$.

##### MSC:
 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B20 Geometry and structure of normed linear spaces 46B42 Banach lattices
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