We introduce a strictly weaker version of the Daugavet property as follows: a Banach space X has this alternative Daugavet property (ADP in short) if the norm identity max |ω|=1 Id + ωT = 1 + Tholds for all rank-one operators T : X → X. In such a case, all weakly compact operators on X also satisfy (aDE). We give some geometric characterizations of the alternative Daugavet property in terms of the space and its successive duals. We prove that the ADP is stable for c 0 -, l 1 -and l ∞ -sums and characterize when some vector-valued function spaces have the property. Finally, we show that a C * -algebra (or the predual of a von Neumann algebra) has the ADP if and only if its atomic projection (respectively, the atomic projection of the algebra) are central. We also establish some geometric properties of JB * -triples, and characterize JB * -triples possessing the ADP and the Daugavet property.

The final publication is available at Springer via http://dx.doi.org/10.1007/s00365-013-9209-zWe show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N-term error of approximation times a function of N which depends on the democracy functions and the quasi-greedy constant of the basis. If the basis is democratic this function is bounded by C log N. We show with two examples that this bound is attained for quasi-greedy democratic basesFirst and second authors supported by Grant MTM2010-16518 (Spain). Third author supported by a travel grant from Simons Foundation, and by a COR grant from University of California Syste

We obtain Lebesgue-type inequalities for the greedy algorithm for arbitrary complete seminormalized biorthogonal systems in Banach spaces. The bounds are given only in terms of the upper democracy functions of the basis and its dual. We also show that these estimates are equivalent to embeddings between the given Banach space and certain discrete weighted Lorentz spaces. Finally, the asymptotic optimality of these inequalities is illustrated in various examples of non necessarily quasi-greedy bases.

International audienceWe construct several examples of Hilbertian operator spaces with few completely bounded maps. In particular, we give an example of a separable $1$-Hilbertian operator space $X_0$ such that, whenever $X'$ is an infinite dimensional quotient of $X_0$, $X$ is a subspace of $X'$, and $T : X \raw X'$ is a completely bounded map, then $T = \lambda I_{X} + S$, where $S$ is compact Hilbert-Schmidt and $||S||_2/16 \leq ||S||_{cb} \leq ||S||_2$. Moreover, every infinite dimensional quotient of a subspace of $X_0$ fails the operator approximation property. We also show that every Banach space can be equipped with an operator space structure without the operator approximation property

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We construct a "universal space" for 1-exact finite dimensional operator spaces (an analogue of the Gurarii space).

We investigate various aspects of the "weighted" greedy algorithm with respect to a Schauder basis. For a weight w, we describe w-greedy, w-almost-greedy, and w-partiallygreedy bases, and examine some properties of w-semi-greedy bases. To achieve these goals, we introduce and study the w-Property (A).2000 Mathematics Subject Classification. 46B15, 41A65.

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