Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một yếu tố của II_1 loại với nhóm cơ bản đếm được. | J. OPERATOR THEORY 4 1980 151-153 Copyright by INCREST 1980 A FACTOR OF TYPE nx WITH COUNTABLE FUNDAMENTAL GROUP A. CONNES Introduction. Let M be a factor OutAf AutAT IntAf will denote the quotient of the group of automorphisms of M by the normal subgroup of inner automorphisms. For M R the hyperfinite factor of type H1 the group OutR is very large in fact it contains any locally compact separable group and is however simple as an abstract group. If M is of type IIj and has no non trivial central sequences then IntM is a closed subgroup of Aut M when AutAf is endowed with the topology of pointwise norm convergence in M its preduaf and then Out M inherits a topology which makes it into a Polish group. So in order to show that OutAf is countable it is enough to show that it is discrete. The next result is the first rigidity result pertaining to the theory of operator algebras. Theorem. Let r be a countable discrete group with infinite conjugacy classes satisfying property T of Kazhdan then M 2 F is a factor of type II such that 1 IntAf is closed in AutAf for the topology of norm pointwise convergence in AQ. 2 OutAf is a discrete group. Corollary. OutAf is countable. Proof of Corollary . Since OutAf is a Polish topological space it has a dense countable subset so being discrete it is countable. Proof of T . Let M act in l2 r and let J be the canonical antilinear involution such that JẸữ Ỉ0 JxS ữ .x j0 Vie M where j0 is the vector associated to the neutral element of r . 0 g 0 if g e and 0 e 1 if we consider 4o as a function on F . For each X eAf JxJ e M the commutant of M moreover the map X - JxJ is an antilinear isomorphism of M onto M . Let 2 be the left regular representation of r in 2 F so forg e r 2 g is the operator of left translation 152 A. CONNES by g one has Ấ g e M and the equality p g 2 g J2 g J defines a representation of F in Z2 T . By hypothesis the non-trivial conjugacy classes of r are infinite and r satisfies the Kazhdan property 3 There exists a
