## 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)

### Keywords:

coarse embedding; Schur multipliers; Baum-Connes conjecture
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