Amenable correspondences and approximation properties for von Neumann algebras.

*(English)*Zbl 0892.22004Correspondences between two von Neumann algebras \(M\) and \(N\) were introduced by A. Connes (see the article by A. Connes and V. Jones [Bull. Lond. Math. Soc. 17, 57–62 (1985; Zbl 1190.46047)], as well as the two unpublished manuscripts “Correspondences” by A. Connes (1980) and “Correspondences” by S. Popa (1986)). A correspondence from \(M\) to \(N\) is simply a Hilbert space endowed with a (normal) \(M-N\)-bimodule structure. Correspondences include the concept of group representations and that of completely positive maps as well. They are useful in both the study of a single von Neumann algebra and the study of the relationships between two von Neumann algebras.

The author introduces the notion of (left) amenable correspondences and of (left) injective correspondences. She relates these notions to Popa’s definition [loc. cit.] of amenability of \(M\) relative to \(N\) and to the reviewer’s definition of amenable group representations [Inv. Math. 100, 383–403 (1990; Zbl 0702.22010)]. It is then shown that the existence of an amenable correspondence from \(M\) to \(N\) is equivalent to the existence of an injective one. In this case \(M\) is said to be amenably dominated by \(N\). This defines the notion of amenably equivalent von Neumann algebras. As an example, let \(G\) be a group acting amenably on \(M\). Then the crossed product of \(M\) by \(G\) is amenably equivalent to \(M\). Another example is the following. Let \(H\) be a subgroup of the locally compact group \(G\) and suppose that the homogeneous space \(G/H\) is amenable. Then the von Neumann algebras generated by the regular representations of \(G\) and \(H\) are amenably equivalent.

Of particular interest are the notions which are preserved by this equivalence. The author shows that, for instance, the Haagerup’s constant \(\Lambda(M)\) associated with \(M\) is such a property.

The author introduces the notion of (left) amenable correspondences and of (left) injective correspondences. She relates these notions to Popa’s definition [loc. cit.] of amenability of \(M\) relative to \(N\) and to the reviewer’s definition of amenable group representations [Inv. Math. 100, 383–403 (1990; Zbl 0702.22010)]. It is then shown that the existence of an amenable correspondence from \(M\) to \(N\) is equivalent to the existence of an injective one. In this case \(M\) is said to be amenably dominated by \(N\). This defines the notion of amenably equivalent von Neumann algebras. As an example, let \(G\) be a group acting amenably on \(M\). Then the crossed product of \(M\) by \(G\) is amenably equivalent to \(M\). Another example is the following. Let \(H\) be a subgroup of the locally compact group \(G\) and suppose that the homogeneous space \(G/H\) is amenable. Then the von Neumann algebras generated by the regular representations of \(G\) and \(H\) are amenably equivalent.

Of particular interest are the notions which are preserved by this equivalence. The author shows that, for instance, the Haagerup’s constant \(\Lambda(M)\) associated with \(M\) is such a property.

Reviewer: M.B.Bekka (Metz)

##### MSC:

22D10 | Unitary representations of locally compact groups |

43A07 | Means on groups, semigroups, etc.; amenable groups |

43A65 | Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) |

46L55 | Noncommutative dynamical systems |

22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |