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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
\[ \| f\|_{C(p)} = \left(\int_I \left(\frac{1}{x} \int_0^x |f(t)|\, dt\right)^p dx\right)^{1/p} \quad \text{if }1\leq p<\infty \]
\[ \| f\|_{C(\infty)}= \sup_{x\in I,\, x>0} \frac{1}{x} \int_0^x |f(t)|\, dt \quad \text{if }p=\infty \]
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 \(1<p<\infty\).

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
Full Text: DOI
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