##
**Property \(T\) for von Neumann algebras.**
*(English)*
Zbl 1190.46047

From the introduction: Kazhdan’s property \(T\) for groups was first used in von Neumann algebras where it was shown that if \(F\) is a countable discrete group with property \(T\) (with infinite conjugacy classes), the fundamental group of the von Neumann algebra of \(F\) is countable. In [Proc.Symp.Pure Math.Vol.38, Part 2, 43–109 (1982; Zbl 0503.46043)], the first author defined a property \(T\) for type \(II_1\) factors and claimed that a discrete group \(F\) has property \(T\) if and only if its von Neumann algebra has this property. The rigidity problem was posed and some results mentioned.

In this paper, we define property \(T\) in a way that makes sense for any von Neumann algebra and makes clear the analogy with Kazhdan’s property. The key concept is that of a correspondence which plays the role of a representation of a group. Whereas the representation theory of a \(II_1\) factor is simple (just the coupling constant), the structure of its correspondences is very rich. There are notions of trivial correspondence and coefficients, the latter allowing one to topologize the space of correspondences. Property \(T\) means that the trivial correspondence is isolated from those that do not contain it. For \(II_1\) factors, the property is the same as that of [loc.cit.], but any type I factor has property \(T\).

In this paper, we define property \(T\) in a way that makes sense for any von Neumann algebra and makes clear the analogy with Kazhdan’s property. The key concept is that of a correspondence which plays the role of a representation of a group. Whereas the representation theory of a \(II_1\) factor is simple (just the coupling constant), the structure of its correspondences is very rich. There are notions of trivial correspondence and coefficients, the latter allowing one to topologize the space of correspondences. Property \(T\) means that the trivial correspondence is isolated from those that do not contain it. For \(II_1\) factors, the property is the same as that of [loc.cit.], but any type I factor has property \(T\).

### MSC:

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

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