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A Schur multiplier characterization of coarse embeddability. (English) Zbl 1334.43004

The authors look for a characterization of coarse embeddability. They give a contractive Schur multiplier characterization of locally compact groups coarsely embeddable into Hilbert spaces. This characterization is an answer to the non-equivariant version of the problem “Is \(\wedge _{WH}\left( G\right) =1\) if and only if \(G\) has the Haagerup property?”. Also, they find that any locally compact group with weak Haagerup constant 1 embeds coarsely into a Hilbert space. Then, they prove that the Baum-Connes assembly map with coefficients is split-injective for all these groups.

MSC:

43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
43A35 Positive definite functions on groups, semigroups, etc.
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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