We introduce the notion
of amenable equivalence between von Neumann algebras, and study some
approximation properties which remain invariant by this relation. We show for
instance that the constant Λ(M) associated with a von Neumann algebra M when
considering the weak∗ completely bounded approximation property is an invariant for
this equivalence relation. As an example, let α be an amenable action of a locally
compact group G on a von Neumann algebra M; then the crossed product MG is
amenably equivalent to M. Another example is obtained by considering a
pair G1⊂ G of locally compact groups such that the homogeneous space
G∕G1 is amenable. Then the von Neumann algebras W∗(G) and W∗(G1)
generated by the left regular representations of G and G1 respectively are
amenably equivalent. Therefore, if moreover G is discrete, we get that G and G1
are simultaneously weakly amenable with the same Haagerup’s constants
ΛG= ΛG1.
