ON COARSE EMBEDDABILITY INTO lp-SPACES AND A CONJECTURE OF DRANISHNIKOV

We show that the Hilbert space is coarsely embeddable into any lp for 1 ≤ p < ∞. In particular, this yields new characterizations of embeddability of separable metric spaces into the Hilbert space. Coarse embeddings were defined by M. Gromov [Gr] to express the idea of inclusion in the large scale geometry of groups. G. Yu showed later that the case when a finitely generated group with a word length metric is being embedded into the Hilbert space is of great importance in solving the Novikov Conjecture [Yu], while recent work of G. Kasparov and G. Yu [KY] treats the case when the Hilbert space is replaced with just a uniformly convex Banach space. Due to these remarkable theorems coarse embeddings gain a great deal of attention, but still embeddability into the Hilbert, and more generally Banach spaces, is not entirely understood with many question remaining open. In this context the class of lp-spaces seems to be particularly interesting. Their embeddability into the Hilbert space is known lp admits such an embedding when 0 < p ≤ 2 but do not if p > 2 due to a recent result of W. Johnson and N. Randrianarivony [JR]. In this note we study the opposite situation, i.e. we show that the separable Hilbert space embeds into lp for any 1 ≤ p < ∞. As a consequence we obtain a new characterization of embeddability into l2, namely that embeddings into lp for 1 ≤ p ≤ 2 are all equivalent. In [GK, Section 6] the authors advertised a conjecture stated by A.N. Dranishnikov [Dr, Conjecture 4.4]: a discrete metric space has property A if and only if it admits a coarse embedding into the the space l1. The results presented in this note show, that this is the same as asking whether property A is equivalent to embeddability into the Hilbert space, and although it is a folk conjecture that such statement is not true, no example distinguishing between the two is known. Acknowledgements. I would like to thank Guoliang Yu for inspiring conversations on coarse geometry of Banach spaces. Lp-spaces and the Mazur Map In everything what follows we consider only separable Lp(μ)-spaces and we use the standard notation lp = lp(N). By S(X) we denote the unit sphere in the Banach space X . 2000 Mathematics Subject Classification. Primary 46C05; Secondary 46T99.