Direct sums and the Szlenk index

For $\alpha$ an ordinal and $1<p<\infty$, we determine a necessary and sufficient condition for an $\ell_p$-direct sum of operators to have Szlenk index not exceeding $\omega^\alpha$. It follows from our results that the Szlenk index of an $\ell_p$-direct sum of operators is determined in a natural way by the behaviour of the $\epsilon$-Szlenk indices of its summands. Our methods give similar results for $c_0$-direct sums.Comment: The proof of Proposition~2.4 has changed, with some of the arguments transferred to the proof of an added-in lemma, Lemma~2.8. Changes have been made to the Applications sectio

Approximation and Schur properties for Lipschitz free spaces over compact metric spaces

We prove that for any separable Banach space $X$, there exists a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space contains a complemented subspace isomorphic to $X$. As a consequence we give an example of a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space fails the approximation property and we prove that there exists an uncountable family of topologically equivalent distances on the Cantor space whose free spaces are pairwise non isomorphic. We also prove that the free space over a countable compact metric space has the Schur property. These results answer questions by G. Godefroy.Comment: 9 page

Rate-limiting Step Preceding Cytochrome c Release in Cells Primed for Fas-mediated Apoptosis Revealed by Analysis of Cellular Mosaicism of Respiratory Changes

In the present work, Jurkat cells undergoing anti-Fas antibody (anti-Fas)-triggered apoptosis exhibited in increasing proportion a massive release of cytochrome c from mitochondria, as revealed by double-labeling confocal immunofluorescence microscopy. The cytochrome c release was followed by a progressive reduction in the respiratory activity of the last respiratory enzyme, cytochrome c oxidase (COX), and with a little delay, by a decrease in overall endogenous respiration rate, as measured in vivo in the whole cell population. Furthermore, in vivo titration experiments showed that an ~30% excess of COX capacity over that required to support endogenous respiration, found in naive cells, was maintained in anti-Fas-treated cells having lost ~40% of their COX respiratory activity. This observation strongly suggested that only a subpopulation of anti-Fas-treated cells, which maintained the excess of COX capacity, respired. Fractionation of cells on annexin V-coated paramagnetic beads did indeed separate a subpopulation of annexin V-binding apoptotic cells with fully released cytochrome c and completely lacking respiration, and a nonbound cell subpopulation exhibiting nearly intact respiration and in their great majority preserving the mitochondrial cytochrome c localization. The above findings showed a cellular mosaicism in cytochrome c release and respiration loss, and revealed the occurrence of a rate-limiting step preceding cytochrome c release in the apoptotic cascade. Furthermore, the striking observation that controlled digitonin treatment caused a massive and very rapid release of cytochrome c and complete loss of respiration in the still respiring anti-Fas-treated cells, but not in naive cells, indicated that the cells responding to digitonin had already been primed for apoptosis, and that this treatment bypassed or accelerated the rate-limiting step most probably at the level of the mitochondrial outer membrane

Smooth extension of functions on a certain class of non-separable Banach spaces

Let us consider a Banach space $X$ with the property that every real-valued Lipschitz function $f$ can be uniformly approximated by a Lipschitz, $C^1$-smooth function $g$ with \Lip(g)\le C \Lip(f) (with $C$ depending only on the space $X$). This is the case for a Banach space $X$ bi-Lipschitz homeomorphic to a subset of $c_0(\Gamma)$, for some set $\Gamma$, such that the coordinate functions of the homeomorphism are $C^1$-smooth. Then, we prove that for every closed subspace $Y\subset X$ and every $C^1$-smooth (Lipschitz) function f:Y\to\Real, there is a $C^1$-smooth (Lipschitz, respectively) extension of $f$ to $X$. We also study $C^1$-smooth extensions of real-valued functions defined on closed subsets of $X$.Comment: 16 page