This is an announcement for the paper "Slicely countably determined Banach spaces. Applications to the Daugavet and the alternative Daugavet equations" by Antonio Aviles, Vladimir Kadets, Miguel Martin, Javier Meri, and Varvara Shepelska.

Abstract: We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the RNP and all spaces without copies of $\ell_1$. We present many examples and several properties of this class. We give some applications to Banach spaces with the Daugavet and the alternative Daugavet properties, lush spaces and Banach spaces with numerical index $1$. In particular, we show that the dual of a real infinite-dimensional Banach with the alternative Daugavet property contains $\ell_1$ and that operators which do not fix copies of $\ell_1$ on a space with the alternative Daugavet property satisfy the alternative Daugavet equation.

Archive classification: math.FA

Mathematics Subject Classification: Primary 46B20. Secondary 46B03, 46B04, 46B22, 47A12

The source file(s), AvilesKadetsMartinMeriShepelska.tex: 107489 bytes, is(are) stored in gzipped form as 0809.2723.gz with size 30kb. The corresponding postcript file has gzipped size 172kb.

Submitted from: mmartins@ugr.es

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