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**Central sequences in crossed products.**
*(English)*
Zbl 0624.46044

Operator algebras and mathematical physics, Proc. Summer Conf., Iowa City/Iowa 1985, Contemp. Math. 62, 539-544 (1987).

[For the entire collection see Zbl 0602.00004.]

A. Connes announced an exact sequence which allows one to calculate Connes invariant \(\chi\) (M) when M is the crossed product of a \(II_ 1\) factor N by a finite group G of automorphisms that meets the closure of the inner automorphisms only in the identity. The main ingredient in the proof of the exact sequence is that under the above assumption on G it is easy to prove that all central sequences in M are asymptotically contained in N.

The present paper contains two results devoted to the question whether this exact sequence works for more general groups of automorphisms. The first is a counterexample showing that so called condition “G discrete modulo inner” is not enough, the second showing how the powerful property T can be used to control central sequences.

A. Connes announced an exact sequence which allows one to calculate Connes invariant \(\chi\) (M) when M is the crossed product of a \(II_ 1\) factor N by a finite group G of automorphisms that meets the closure of the inner automorphisms only in the identity. The main ingredient in the proof of the exact sequence is that under the above assumption on G it is easy to prove that all central sequences in M are asymptotically contained in N.

The present paper contains two results devoted to the question whether this exact sequence works for more general groups of automorphisms. The first is a counterexample showing that so called condition “G discrete modulo inner” is not enough, the second showing how the powerful property T can be used to control central sequences.

Reviewer: Sh.Ayupov

### MSC:

46L55 | Noncommutative dynamical systems |

46L35 | Classifications of \(C^*\)-algebras |

46L40 | Automorphisms of selfadjoint operator algebras |