AN EXAMPLE OF A PROPERTY Γ FACTOR

WITH COUNTABLE FUNDAMENTAL GROUP

SORIN POPA

University of California, Los Angeles

0. Introduction.

The first examples of type II_{1} factors M having countable fundamental group F(M) have been constructed by Connes in ([C1]), as group algebras M = L(G) for G discrete ICC groups with the property T of Kazhdan. Since then, many more examples of such factors have been constructed (see e.g., [Po1,3] or [Gol]). But all such constructions are using property T groups, in a way or another. For instance, it is proved in ([Po1]) that if a type II_{1} factor M contains a group von Neumann algebra L(G) for some ICC property T group G then F(M) is countable.

All the factors M having F(M) countable that have been constructed so far do not have non-trivial asymptotically central sequences, i.e., they do not have the property Γ of Murray and von Neumann (equivalently, they are full factors, in the sense of [C]). In this respect, note that if a factor M has non-commuting such central sequences then by a result of McDuff ([McD]) they split-off the hyperfinite type II_{1} factor, thus having fundamental group equal to ℝ_{+}^{∗}. Thus, if it is for a property Γ factor M to have fundamental group ≠ ℝ_{+}^{∗}, then its central sequences must commute.

In this paper we construct a class of examples of factors which do have the property Γ yet have countable fundamental group. The construction does not use property T groups, but instead uses the rigidity properties of the inclusion ℤ^{2} ⊂ ℤ^{2} ⋊ SL(2, ℤ), also due to Kazhdan ([Kaz]). We will also use perturbation results, separability arguments and the recent striking results of Gaboriau on the cost of equivalence relations.

Typeset by AMS-TEX

Supported in part by a NSF Grant 9801324.

The paper is organised as follows: In Section 1 we prove some perturbation result for Cartan subalgebras in type II_{1} factor, in the spirit of some well known results of Eric Christensen. In Section 2 we prove the main result of the paper (Theorem 2.3), providing some classes of factors with countable fundamental group, both with and without property Γ. These factors do not contain type II_{1} factors with the property T of ([CJ1]), nor can be embedded into free group factors, yet they seem to be closer to the latter class. In Section 3 we make some comments, and sketch another construction of property Γ factors with countable fundamental group, this time by using property T groups (Remark 3.1). We also construct a class of examples of property Γ factors with fundamental group ℝ_{+}^{∗} which however are not McDuff (Remark 3.2).

I am most greatful to George Skandalis for his kind help with property (3.1.1) and to Dima Shlyakhtenko for a most useful discussion related to Remark 3.2.

1. Some perturbation results.

In this Section we prove some perturbation results for subalgebras in type II_{1} factors.

The first such perturbation result concerns maximal abelian ∗-subalgebras (abreviated as m.a.s.a. hereafter) of type II_{1} factors. Besides the technique from [Ch], the proof uses the “pull down” lemma in the basic construction ([PiPo]) and some considerations on the geometry of projections, notably a result of Kadison ([K]).

1.1. Theorem. Let N be a type II_{1} factor and B_{0}, B_{1} ⊂ N be m.a.s.a.’s of N such that sup{∥u − E_{B_{1}}(u)∥_{2} | u ∈ U_{0}} < 1, for some unitary subgroup U_{0} ⊂ B_{0} satisfying U′′_{0} = B_{0}. Then there exists a non-zero partial isometry v ∈ N such that v^{∗}v ∈ B_{0}, vv^{∗} ∈ B_{1} and vB_{0}v^{∗} = B_{1}vv^{∗}.

Proof. Consider first the basic construction for the inclusion B_{1} ⊂ N: Thus, we let e be the orthogonal projection of L^{2}(N, τ) onto L^{2}(B_{1}, τ), and which is known to satisfy exe = E_{B_{1}}(x)e, ∀x ∈ N. Then we let N_{1} = ⟨N, e⟩ be the von Neumann algebra generated inside B(L^{2}(N, τ)) by N and e. Note that eN_{1}e = B_{1}e. We endow N_{1} with the unique normal semifinite faithful trace T r which satisfies T r(xey) = τ(xy), ∀x, y ∈ N. Note that there exists a unique N − N bilinear map Φ from spNeN ⊂ N_{1} into N satisfying Φ(xey) = xy, ∀x, y ∈ N. This bilinear map satisfies the “pull down” identity eX = eΦ(eX), ∀X ∈ N_{1} from ([PiPo]).

Let now K_{e} = \overline{co}^{w}\{u_{0}eu_{0}^{∗} | u_{0} ∈ U_{0}\}. We clearly have 0 ≤ a ≤ 1 and T r(a) ≤ 1, ∀a ∈ K_{e}. Moreover, K_{e} is contained in the Hilbert space L^{2}(N_{1}, T r), where it is still weakly closed. Let h ∈ K_{e} be the unique element of minimal norm ∥ ∥_{2,T r} in K_{e}. Since uhu^{∗} ∈ K_{e} and ∥uhu^{∗}∥_{2,T r} = ∥h∥_{2,T r}, ∀u ∈ U_{0}, by the uniqueness of h it follows that uhu^{∗} = h, ∀u ∈ U_{0}. Thus h ∈ U′_{0} ∩ N_{1} = B′_{0} ∩ N_{1}.

On the other hand, since T r(eueu^{∗}) = ∥E_{B_{1}}(u)∥_{2}^{2} = 1 − ∥u − E_{B_{1}}(u)∥_{2}^{2}, if we denote δ = 1−sup{∥u − E_{B_{1}}(u)∥_{2} | u ∈ U_{0}}, it follows that T r(eueu^{∗}) ≥ δ, ∀u ∈ U_{0}. Taking appropriate linear combinations and weak limits, it follows that T r(eh) ≥ δ. Since by hypothesis we have δ > 0, it follows that h ≠ 0.

Let e_{0} be a non-zero spectral projection of h. Thus, e_{0} is a finite projection in N_{1} and e_{0} commutes with B_{0} (since h does). Since B_{0}e_{0} is abelian, it is contained in a maximal abelian subalgebra B of e_{0}N_{1}e_{0}. (Note that any element in B commutes with B_{0}.) By a result of Kadison ([K]), B contains a non-zero abelian projection e_{1} of N_{1} (i.e., e_{1}N_{1}e_{1} is abelian). Since e has central valued (semifinite) trace equal to 1, it follows that e majorizes e_{1}.

Let V ∈ N_{1} be a partial isometry such that V^{∗}V = e_{1} ≤ e_{0} and V V^{∗} ≤ e. Moreover V Be_{1}V^{∗} is a subalgebra of eN_{1}e = B_{1}e. Since e_{1} commutes with B_{0}, it follows that if we denote by f′ the maximal projection in B_{0} such that f′e_{1} = 0 and let f_{0} = 1 − f′, then there exists a unique isomorphism φ from B_{0}f_{0} into B_{1} such that φ(b)e = V bV^{∗}, ∀b ∈ B_{0}f. Let f_{1} = φ(f_{0}) ∈ B_{1}.

It follows that φ(b)eV = eV b, ∀b ∈ B_{0}f_{0}. By applying Φ to both sides and denoting a = Φ(eV) ∈ N, it follows that φ(b)a = ab, ∀b ∈ B_{0}. Since ea = eV = V, it follows that a ≠ 0.

By the usual trick, if we denote by v_{0} ∈ N the unique partial isometry in the polar decomposition of a such that the right supports of a and v_{0} coincide, then p_{0} = v_{0}^{∗}v_{0} ∈ B′_{0} ∩ N = B_{0}, p_{1} = v_{0}v_{0}^{∗} ∈ φ(B_{0})f′_{1} ∩ f_{1}Nf_{1} and φ(b)v_{0} = v_{0}b, ∀b ∈ B_{0}f_{0}.

But B_{0}f_{0} maximal abelian in f_{0}Nf_{0} implies that vB_{0}v^{∗} = φ(B_{0})p_{1} is maximal abelian in v_{0}v_{0}^{∗}Nv_{0}v_{0}^{∗}. Since any element in p_{1}B_{1}p_{1} commutes with φ(B_{0})p_{1}, which is maximal abelian, it follows that p_{1}B_{1}p_{1} = φ(B_{0})p_{1}. Thus, if P_{1} denotes the von Neumann algebra generated by p_{1} and B_{1}f_{1} inside f_{1}Nf_{1}, then P_{1} is like a basic construction for the inclusion φ(B_{0})f_{1} ⊂ B_{1}f_{1}, with p_{1} playing the role of the “Jones projection”. In particualar, p_{1}P_{1}p_{1} = φ(B_{0})p_{1} is abelian.

Thus, p_{1} is an abelian projection in P_{1}. Since P_{1} is a finite von Neumann algebra, there exists a central projection z_{1} of P_{1}, under the central support of p_{1} in P_{1}, such that p_{1}z_{1} ≠ 0 and such that P_{1}z_{1} is homogeneous of type n, for some n ≥ 1. Note that the center of P_{1} is included in B_{1}f_{1} the latter being maximal abelian in f_{1}Nf_{1}, thus in P_{1} ⊂ f_{1}Nf_{1}. Thus, z_{1} ∈ B_{1}f_{1}.

Now, since p_{1}z_{1} has central trace equal to 1/n in P_{1}z_{1} and B_{1}z_{1} is maximal abelian in P_{1}z_{1}, it follows that there exists a projection f_{11} ∈ B_{1}f_{1} such that f_{11} is equivalent to p_{1}z_{1} in P_{1}z_{1} (see [K]). Let v_{1} ∈ P_{1}z_{1} be such that v_{1}v_{1}^{∗} = f_{11}, v_{1}^{∗}v_{1} = p_{1}z_{1}. Since p_{1}z_{1} is abelian in P_{1}, f_{11} is also abelian, thus f_{11}P_{1}f_{11} = B_{1}f_{11}. This implies that v_{1}(φ(B_{0})p_{1}z_{1})v_{1}^{∗} = B_{1}f_{11}.

Finally, since B_{0}f_{0} ∋ b ↦ φ(b) ∈ φ(B_{0})f_{1} and p_{1}z_{1} belongs to φ(B_{0})p_{1}, it follows that there exists a projection f_{00} in B_{0}f_{0} such that φ(f_{00})p_{1} = z_{1}p_{1}. In particular, v_{1}^{∗}v_{1} = v_{0}f_{00}v_{0}^{∗}.

Altogether, this shows that if we denote v = v_{1}v_{0}f_{00}, then v is a partial isometry satsifying v^{∗}v = f_{00} ∈ B_{0}, vv^{∗} = f_{11} ∈ B_{1} and vB_{0}v^{∗} = B_{1}vv^{∗}. Q.E.D.

In Section 2 we will in fact need a consequence of Theorem 1.1. To state it, recall from ([D1]) that a m.a.s.a. A of a von Neumann factor N is called semiregular if the set of unitaries of N that normalize A generate a factor. Also, A is called regular if this normalizer generetes all the ambient factor N. Such regular m.a.s.a.’s were later called Cartan subalgebras in ([FM]), a terminology that seems to have prevailed and which we will therefore adopt.

We will also use the following notations from ([Ch]):

1.2. Notation. Let B_{0}, B_{1} be von Neumann subalgebras of a type II_{1} factor N. If sup{∥x_{0} − E_{B_{1}}(x_{0})∥_{2} | x_{0} ∈ B_{0}, ∥x_{0}∥ ≤ 1} ≤ ε then we write B_{0} ⊂_{ε} B_{1}. Also, we denote by d(B_{0}, B_{1}) the maximum between sup{∥x_{0} − E_{B_{1}}(x_{0})∥_{2} | x_{0} ∈ B_{0}, ∥x_{0}∥ ≤ 1} and sup{∥x_{0} − E_{B_{1}}(x_{0})∥_{2} | x_{0} ∈ B_{0}, ∥x_{0}∥ ≤ 1}).

1.3. Corollary. Let N be an arbitrary type II_{1} factor and A_{0}, A_{1} ⊂ N be semiregular m.a.s.a.’s of N. If A_{0} ⊂_{1−δ} A_{1} for some δ > 0 then there exists a unitary element u ∈ N such that uA_{0}u^{∗} = A_{1}. In particular, this is the case if A_{0}, A_{1} are Cartan subalgebras of N.

Proof. The condition implies that sup{∥u − E_{A_{1}}(u)∥_{2} | u ∈ U(A_{0})} < 1. Thus, by Theorem 1.1 there exists a non-zero partial isometry v ∈ N such that v^{∗}v ∈ A_{0}, vv^{∗} ∈ A_{1}, vA_{0}v^{∗} = A_{1}vv^{∗}. Moreover, by cutting v from the right with a smaller projection in A_{0}, we may clearly assume τ(vv^{∗}) = 1/n for some integer n.

Since A_{0}, A_{1} are semiregular, there exist partial isometries v_{1}, v_{2}, ..., v_{n}, respectively w_{1}, w_{2}, ..., w_{n} in the normalizing groupoids of A_{0} respectively A_{1} such that Σ_{i}v_{i}v_{i}^{∗} = 1, Σ_{j}w_{j}w_{j}^{∗} = 1 and v_{i}^{∗}v_{i} = v^{∗}v, w_{j}^{∗}w_{j} = vv^{∗}, ∀i, j. But then u = Σ_{i}w_{i}vv_{i}^{∗} is a unitary element and vA_{0}v^{∗} = A_{1}. Q.E.D.

Let us also mention another application of Theorem 1.1 which, although not needed later in this paper, has some independent interest. Thus, recall from ([D1]) that a m.a.s.a. A of a von Neumann algebra N is singular if the only unitaries in N that normalize A are the unitaries of A. When N is a type II_{1} factor, in ([Po3]) a numerical invariant δ(A) was associated to m.a.s.a.’s A of N, as a “measure of singularity”, as follows:

For each non-zero partial isometry v ∈ N with vv^{∗}, v^{∗}v mutually orthogonal projections in A, denote δ(vAv^{∗}, A) = sup{∥x − E_{A}(x)∥_{2} | x ∈ vAv^{∗}, ∥x∥ ≤ 1}.

Then define δ(A) to be the infimum of δ(vAv^{∗}, A)/∥vv^{∗}∥_{2}, as v runs over the set of all such partial isometries. In ([Po3]) it was noted that δ(A) > 0 implies A is singular and examples of m.a.s.a.’s with δ(A) > 0 were constructed in any type II_{1} factor with separable predual. But it was left as an open question to calculate the possible values the constants δ(A) can take. This problem was recently revived in ([SiSm]). From 1.1 we get:

1.4. Corollary. Let N be an arbitrary type II_{1} factor and A ⊂ N a singular m.a.s.a. Then δ(A) = 1.

Proof. If for some partial isometry v ∈ N with vv^{∗}, v^{∗}v ∈ A, v ≠ 0, v^{2} = 0 we have sup{∥x − E_{Avv^{∗}}(x)∥_{2}/∥vv^{∗}∥_{2} | x ∈ vAv^{∗}, ∥x∥ ≤ 1} < 1, then it follows that the m.a.s.a.’s B_{0} = vAv^{∗}, B_{1} = Avv^{∗} in the factor vv^{∗}Nvv^{∗} (with its normalized trace) verify the condition in Theorem 1.1. Thus, there exists a non-zero partial isometry v_{0} ∈ vv^{∗}Nvv^{∗} such that v_{0}^{∗}v_{0} ∈ vAv^{∗}, v_{0}v_{0}^{∗} ∈ Avv^{∗} and v_{0}vAv^{∗}v_{0}^{∗} ⊂ A. But this contradicts the singularity of A. Q.E.D.

We end this section by mentioning two well known perturbation results from ([Ch]), needed in the sequel. We have included a proof for the sake of completeness.

1.5. Lemma. [Ch]. Let N be a type II_{1} factor and B_{0}, B_{1} ⊂ N be von Neumann subalgebras of N. Assume there exists a subgroup U_{0} of the unitary group of B_{0} such that U′′_{0} = B_{0} and ∥u_{0} − E_{B_{1}}(u_{0})∥_{2} ≤ ε, ∀u_{0} ∈ U_{0}. Then B′_{1} ∩ N ⊂_{2ε} B′_{0} ∩ N.

Proof. Let x ∈ B′_{1} ∩ N, ∥x∥ ≤ 1. Since E_{B′_{0}∩N}(x) is the element of minimal norm-2 in the weakly compact convex set \overline{co}^{w}\{u_{0}xu_{0}^{∗} | u_{0} ∈ U_{0}\}, it follows that ∀δ > 0, ∃u_{1}, u_{2}, ..., u_{n} ∈ U_{0} such that ∥1/nΣ_{i}u_{i}xu_{i}^{∗} − E_{B′_{0}∩N}(x)∥_{2} < δ.

But by hypothesis, for all i = 1, 2, .., n we have the estimates:

∥x − u_{i}xu_{i}^{∗}∥_{2} ≤ ∥x − E_{B_{1}}(u_{i})xu_{i}^{∗}∥_{2} + ∥u_{i} − E_{B_{1}}(u_{i})∥_{2}
= ∥x − xE_{B_{1}}(u_{i})u_{i}^{∗}∥_{2} + ∥u_{i} − E_{B_{1}}(u_{i})∥_{2} ≤ 2∥u_{i} − E_{B_{1}}(u_{i})∥_{2} ≤ 2ε.

Altogether, this implies that ∥x−E_{B′_{0}∩N}(x)∥_{2} < 2ε+δ, with δ arbitrary, showing that x ∈_{2ε} B′_{0} ∩ N. Since x was taken arbitrary in the unit ball of B′_{1} ∩ N, we have shown that B′_{1} ∩ N ⊂_{2ε} B′_{0} ∩ N. Q.E.D.

1.6. Lemma. [Ch]. Let N be a type II_{1}. Assume there exists a subgroup U_{0} of the unitary group of N and a group morphism ρ : U_{0} → U(N) such that ∥ρ(u_{0})−u_{0}∥_{2} ≤ ε, ∀u_{0} ∈ U_{0}. Then there exists a partial isometry v ∈ N such that v^{∗}v ∈ U′_{0} ∩ N, vv^{∗} ∈ ρ(U_{0})′ ∩ N, ∥1 − v∥_{2} ≤ 2ε and Ad(v)(u_{0}) = ρ(u_{0})vv^{∗}, ∀u_{0} ∈ U_{0}.

Proof. Let K_{ρ} = \overline{co}^{w}\{ρ(u_{0})u_{0}^{∗} | u_{0} ∈ U_{0}\}. Let k ∈ K_{ρ} be the unique element of minimal norm-2. Since ρ(u_{0})K_{ρ}u_{0}^{∗} ⊂ K_{ρ} and ∥ρ(u_{0})k′u_{0}^{∗}∥_{2} = ∥k′∥_{2}, forall u_{0} ∈ U_{0}, k′ ∈ K_{ρ}, it follows that ρ(u_{0})ku_{0}^{∗} = k, ∀u_{0} ∈ U_{0}. Thus, ρ(u_{0})k = ku_{0}, ∀u_{0} ∈ U_{0}. By the usual trick, it follows that if v denotes the partial isometry in the polar decomposition of k with right support equal to the right support of k then v is an intertwiner between ρ and id on U_{0}. Also, since any element k′ in K_{ρ} satisfies ∥k′ − 1∥_{2} ≤ ε, by standard estimates (see e.g., [C2] or [Ch]) one gets ∥v − 1∥_{2} ≤ 2ε. Q.E.D.

2. Examples of factors M with countable F(M).

In this section we construct a class of examples of type II_{1} factors with countable fundamental group. Some of them have the property Γ of Murray and von Neumann while others don’t. The construction relies on the rigidity properties of the embedding of the group G_{0} = ℤ^{2} inside the group G = ℤ^{2} ⋊ SL(2, ℤ^{2}), discovered by Kazhdan in the late 60’s ([Kaz]). However, as we will prove in the next section, the factors that we construct in this section do not contain any subfactor with the property T, other than the finite dimensional ones.

2.1. The construction. Let (X_{0}, μ_{0}) be the 2-dimensional thorus, regarded as the dual group of ℤ^{2}, endowed with the Haar measure μ_{0}. Note that μ_{0} is the same as the Lebesgue measure on X_{0} = 𝕋^{2}. Let σ_{0} be the action of SL(2, ℤ) on X_{0} implemented by the action of SL(2, ℤ) on ℤ^{2}. Let (X_{1}, μ_{1}) be a probability space with a measure preserving ergodic transformation σ_{1} of SL(2, ℤ) on it. Let σ = σ_{0} × σ_{1} be the product action on the probability space (X, μ) = (X_{0} × X_{1}, μ_{0} × μ_{1}). Denote A = L^{∞}(X, μ) and M = A ⋊_{σ} SL(2, ℤ). Also, denote A_{0} = L^{∞}(X_{0}, μ_{0}) and M_{0} = A_{0} ⋊_{σ_{0}} SL(2, ℤ).

Note that we can regard A_{0} as a subalgebra A, in which case the canonical unitaries u_{g}, g ∈ SL(2, ℤ) ⊂ M implementing the action σ on A also implement the action σ_{0} on A_{0}. Thus, M_{0} can be viewed as a subfactor of M. Moreover, we can view A_{0} as L(ℤ^{2}), in which case M_{0} is identified with L(ℤ^{2} ⋊ SL(2, ℤ)).

Given any arbitrary ergodic action σ_{1} as in 2.1, the action σ and the algebras defined above have the following properties:

2.2. Lemma. 1^{∘}. The action σ is a free, ergodic action of SL(2, ℤ) on the probability space (X, μ).

2^{∘}. The action σ_{0} is strongly ergodic. The action σ = σ_{0} × σ_{1} is strongly ergodic if and only if σ_{1} is strongly ergodic.

3^{∘}. There exist ergodic actions σ_{1} of SL(2, ℤ) that are not strongly ergodic. 4^{∘}. M_{0} is a non-Γ type II_{1} factor and A_{0} ⊂ M_{0} is a Cartan subalgebra of M_{0}.

More generally, M is a type II_{1} factor, A ⊂ M is a Cartan subalgebra in M and M is non-Γ if and only if σ_{1} is strongly ergodic.

5^{∘}. When regarded as a subalgebra in M, A_{0} satisfies A′_{0} ∩ M = A.

Proof. 1^{∘}. It is well known that a non-inner automorphism of a group G implements a properly outer automorphism on L(G) that preserves the canonical trace of L(G). Thus, since each non-trivial element g in the group SL(2, ℤ) implements a non-inner automorphism of ℤ^{2} it follows that σ_{0}(g) is properly outer ∀g ∈ SL(2, ℤ), g ≠ e.

Since the tensor product of any properly outer automorphism with an arbitrary automorphism is properly outer, it follows that σ(g) is properly outer ∀g ∈ SL(2, ℤ), g ≠ e, as well.

Furthermore, the action σ_{0} is well known to be mixing. More precisely, it is easy to see that given any finite set F ⊂ SL(2, ℤ), e ∉ F, there exists g ∈ SL(2, ℤ) such that gF ∩ F = ∅. But this implies not only that σ_{0} is ergodic, but also that its tensor product with any ergodic action is still ergodic.

2^{∘}. The first part is a well known result of Klaus Schmidt ([S1]). The second part is an immediate consequence of the proof of this result in ([S1]).

3^{∘}. This is a consequence of a theorem of Connes and Weiss ([CW]), showing that any discrete group G_{1} which doesn’t have the property T has a free, ergodic but not strongly ergodic action σ_{1} on a probability space. Thus, one simply applies this result to G_{1} = SL(2, ℤ), which doesn’t have the property T.

One can in fact avoid using the general result in ([CW]), by noticing that since SL(2, ℤ) has an infinite amenable group H as a quotient (see e.g., [dHV]), any ergodic action of H on a non-atomic probability space (e.g., a Bernoulli shift action of H) composed with the quotient map G_{1} → H gives an ergodic but not strongly ergodic action of SL(2, ℤ) (note that the resulting action of SL(2, ℤ) is not free though, in fact not even faithful, but freeness is not necessary in the construction 1.1).

4^{∘}. Since SL(2, ℤ) is close to be a free group (see e.g., [dHV]), it is easy to see that any central sequences in a type II_{1} factor of the form B ⋊_{σ} SL(2, ℤ) obtained as the cross product of a finite von Neumann algebra (B, τ) by a free, τ-preserving action σ of SL(2, ℤ) on it must be supported on B.

5^{∘}. Note that if b = Σ_{g}a_{g}u_{g} ∈ M = A ⋊_{σ} SL(2, ℤ) commutes with all a ∈ A_{0} and a_{g} ≠ 0 for some g ≠ e then aa_{g}u_{g} = a_{g}u_{g}a, ∀a ∈ A_{0}, implying that a_{g}a = a_{g}σ_{g}(a), ∀a ∈ A_{0}. This in turn contradicts the fact that σ_{g} is properly outer on A_{0}. Thus, all a_{g} with g ≠ 0 must be equal to 0, implying that b lies in A. Q.E.D.

2.3. Theorem. Given any ergodic action σ_{1} of SL(2, ℤ^{2}) on a probability space (X_{1}, μ_{1}), the type II_{1} factor M constructed in 2.1 has countable fundamental group.

To prove the Theorem, we’ll use first a “separability argument”, then the result of Kazhdan on the rigidity of the embedding of the group ℤ^{2} inside ℤ^{2} ⋊ SL(2, ℤ) ([Kaz]), then the perturbation results from the previous section, and finally the recent rigidity result of the free ergodic actions of the free groups on probability spaces of Gaboriau ([G]).

We split the corresponding arguments in a series of lemmas.

2.4. Lemma. Let N be a separable type II_{1} factor. If F(N) is uncountable then there exist projections p_{n} ∈ P(N), with p_{n} ≠ 1, p_{n} ↗ 1, and isomorphisms θ_{n} : N ≃ p_{n}Np_{n}, such that lim_{n→∞}∥θ_{n}(x) − x∥_{2} = 0, ∀x ∈ N. Moreover, the projections {p_{n}}_{n} can be taken to lie in any given diffuse abelian von Neumann subalgebra A_{0} of N.

Proof. Let {q_{t} | 0 ≤ t ≤ 1} ⊂ N be a totally ordered set of projections with τ(q_{t}) = t, ∀0 ≤ t ≤ 1. For each t ∈ S = (1/2, 1) ∩ F(N) choose an isomorphism θ′_{t} : N ≃ q_{t}Nq_{t} and denote I = {θ′_{t} | t ∈ S}. Note that, since F(N) is uncountable, S is uncountable.

Let {x_{n}}_{n≥1} be a sequence of elements in the unit ball N_{1} of N, such that as a set it is dense in N_{1} in the norm ∥ ∥_{2} and such that each element appears infinitely many times in the sequence.

Since {θ′_{t}(x_{1})}_{t∈S} is a subset of N_{1}, by covering N_{1} with countably many open balls of ∥ ∥_{2}-radius 1/2 and using the separability of (N_{1}, ∥ ∥_{2}) it follows that there exists an uncountable subset S_{1} ⊂ S such that ∥θ_{t}(x_{1}) − θ′_{t′}(x_{1})∥_{2} < 1, ∀t, t′ ∈ S_{1}. Similarily, by repeatedly using the separability of N_{1} to cover it by countably many balls of radius 1/2n, we construct recursively a decreasing sequence of uncountable sets S_{n} ⊂ S, n ≥ 1, such that

∥θ′_{t}(x_{n}) − θ′_{t′}(x_{n})∥_{2} < 1/n, ∀t, t′ ∈ S_{n}.

For each n ≥ 1 choose now two distinct t_{n}, t′_{n} ∈ S_{n}, say with t_{n} > t′_{n}. Since θ′_{t_{n}}(1) ≥ θ′_{t′_{n}}(1),

θ_{n}(x) = θ′_{t_{n}}^{−1}(θ′_{t′_{n}}(x)), x ∈ N,

gives a well defined isomorphism of N onto p_{n}Np_{n}, where

p_{n} = θ′_{t_{n}}^{−1}(θ′_{t′_{n}}(1)) = θ_{t_{n}}^{−1}(q_{t′_{n}}) ∈ P(N).

Moreover, since t_{n}, t′_{n} ∈ S_{j}, ∀j ≤ n, we have for each n ≥ 1 and 1 ≤ j ≤ n the estimates:

∥θ_{n}(x_{j}) − x_{j}∥_{2}^{2} = τ(q_{t_{n}})^{−1}∥θ′_{t′_{n}}(x_{j}) − θ′_{t_{n}}(x_{j})∥_{2}^{2} ≤ 2/j.

Since each x_{j} appears infinitely many times, it follows that lim_{n→∞}∥θ_{n}(x_{j}) − x_{j}∥_{2} = 0, ∀j ≥ 1. By the density of {x_{n}}_{n} in N_{1}, it follows that lim_{n→∞}∥θ_{n}(x)−x∥_{2} = 0, ∀x ∈ N_{1}, and thus for all x ∈ N.

Finally, in case we want the projections {p_{n}}_{n} to lie in a given diffuse abelian von Neumann subalgebra A_{0}, then we first choose projections {p′_{n}}_{n} ⊂ A_{0} with τ(p′_{n}) = τ(p_{n}), ∀n, and note that ∥p′_{n} − p_{n}∥_{2} → 0 (because both sequences tend to 1). Thus, there exist partial isometries v_{n} ∈ N such that v_{n}v_{n}^{∗} = p_{n}, v_{n}^{∗}v_{n} = p′_{n}, ∀n and ∥v_{n} − 1∥_{2} → 0 (see e.g., [C2] or [Ch]). But then, by replacing n p′_{n} for p_{n} and Ad(v_{n})θ_{n} for θ_{n}, ∀n ≥ 1, the last condition will also be satisfied. Q.E.D.

We are now going to use the rigidity of the embedding of the group ℤ^{2} inside ℤ^{2} ⋊ SL(2, ℤ) (cf. [Kaz]). We recall that if G_{0} is a subgroup of a discrete group G then we say that G_{0} has the property T inside G if there exist g_{1}, g_{2}, ..., g_{n} ∈ G and ε > 0 such that if π is a unitary representation of G on a Hilbert space H such that for some unit vector ξ ∈ H one has ∥π(g_{i})ξ − ξ∥ ≤ ε, ∀i then there exists a non-zero vector ξ_{0} ∈ H such that π(h)ξ_{0} = ξ_{0}, ∀h ∈ G_{0}. Note that if this is the case, then by a well known argument it follows that ∃K ≥ 1 such that ∀δ < ε, if ∥π(g_{i})ξ − ξ∥ ≤ δ, ∀i then ∥π(h)ξ − ξ∥ ≤ Kδ, ∀h ∈ G_{0} (see e.g., [De-Ki]).

2.5. Lemma. Let G be a discrete group and G_{0} ⊂ G a property T embedding of a group G_{0} into G (e.g., G_{0} = ℤ^{2}, G = ℤ^{2} ⋊ SL(2, ℤ^{2})). Let (A_{0} ⊂ M_{0}) = (L(G_{0}) ⊂ L(G)). Denote by u_{g}, g ∈ G, the canonical unitaries in M_{0}. Let N be a type II_{1} factor that contains M_{0}. Assume {θ_{n}}_{n} is a sequence of non-necessarily unital endomorphisms of N such that lim_{t→0}∥θ_{n}(u_{g}) − u_{g}∥_{2} = 0, ∀g ∈ G. Then for

any ε > 0 there exists n_{ε} such that if n ≥ n_{ε} then ∥θ_{n}(u_{h}) − u_{h}∥_{2} ≤ ε, ∀h ∈ G_{0}.

Proof. Let π_{n} : ℤ^{2} ⋊ SL(2, ℤ^{2}) → U(H_{n}) be the unitary representation on the Hilbert space H_{n} = θ_{n}(1)L^{2}(M, τ)) defined by π_{n}(g)(ξ) = θ_{n}(u_{g})ξu_{g}^{∗}, g ∈ ℤ^{2} ⋊ SL(2, ℤ^{2}). If we let ξ_{n} ∈ H_{n} be the projection p_{n} ∈ M regarded as a vector in H_{n} then

∥π_{n}(g)ξ_{n} − ξ_{n}∥ = ∥θ_{n}(u_{g})u_{g}^{∗} − p_{n}∥_{2}
= ∥θ_{n}(u_{g}) − p_{n}u_{g}∥_{2} ≤ ∥θ_{n}(u_{g}) − u_{g}∥_{2} + ∥1 − p_{n}∥_{2}, ∀n ≥ 1, g ∈ G.

Similarily,

∥π_{n}(g)ξ_{n} − ξ_{n}∥ ≥ ∥θ_{n}(u_{g}) − u_{g}∥_{2} − ∥1 − p_{n}∥_{2}, ∀n ≥ 1, g ∈ G.

By the hypothesis and the first set of estimates it follows that lim_{n→∞}∥π_{n}(g)ξ_{n} −

ξ_{n}∥ = 0, ∀g ∈ ℤ^{2} ⋊ SL(2, ℤ^{2}). Since ∥ξ_{n}∥^{2} = τ(p_{n}) → 1 as n → ∞, by the property T of ℤ^{2} inside ℤ^{2} ⋊ SL(2, ℤ^{2}) and the second set of estimates, it follows that

lim_{n→∞} sup_{h∈ℤ^{2}}∥θ_{n}(u_{h}) − u_{h}∥_{2} = 0.

Q.E.D.

2.6. Lemma. With the same notations as in the statement of Lemma 2.5, if in addition we assume the projections p_{n} = θ_{n}(1) lie in A_{0}, ∀n, then for any n ≥ n_{ε} we have d(θ_{n}(A_{0})′ ∩ N, A_{0}p′_{n} ∩ N) ≤ 2ε(1 − ε)^{−1}.

Proof. By using the fact that p_{n} ∈ A_{0} = {u_{h}}′′_{h∈ℤ^{2}} and applying Lemma 1.5, first to U_{0} = θ_{n}({u_{h}}_{h∈ℤ^{2}}) then to U_{0} = {u_{h}p_{n}}_{h∈ℤ^{2}}, regarded as unitary subgroups in the type II_{1} factor p_{n}Np_{n}, endowed with the normalized trace τ(p_{n})^{−1}τ, it follows that

d(A_{0}p′_{n} ∩ p_{n}Np_{n}, θ_{n}(A_{0})′ ∩ p_{n}Np_{n}) ≤ 2ετ(p_{n})^{−1/2} ≤ 2ε(1 − ε)^{−1}.

Q.E.D.

Proof of Theorem 2.3. If F(M) is uncountable then by Lemma 2.4 there exists a sequence of projections p_{n} ∈ A_{0}, p_{n} ≠ 1, and isomorphisms θ_{n} of M onto p_{n}Mp_{n} such that lim_{n→∞}∥θ_{n}(x) − x∥_{2} = 0, ∀x ∈ M. In particular, aplying this to x = u_{g} ∈

M_{0} ⊂ M, g ∈ ℤ^{2} ⋊ SL(2, ℤ), we have

lim_{n→∞}∥θ_{n}(u_{g}) − u_{g}∥_{2} = 0, ∀g ∈ ℤ^{2} ⋊ SL(2, ℤ).

By 2.5 and 2.6, it follows that for n large enough, if we denote p = p_{n} and θ = θ_{n} then we have d(A_{0}p′ ∩ pMp, θ(A_{0})p′ ∩ pMp) < 1. But by Lemma 2.1, A_{0}p′ ∩ pMp = Ap is a Cartan subalgebra of pMp. Similarily,

θ(A_{0})p′ ∩ pMp = θ(A′_{0} ∩ M) = θ(A)

is a Cartan subalgebra of pMp as well. By Corollary 1.4 it follows that the Cartan subalgebras Ap and θ(A) of pMp are conjugate by a unitary element u of pMp. Thus, θ′ = Ad(u) ∘ θ is an isomorphism of M onto pMp carying A onto Ap. But by the results of Gaboriau the cost c(A ⊂ M) of the Cartan subalgebra A of M is equal to 13/12 while the cost c(Ap ⊂ pMp) of its restriction to p is equal to 1/12τ(p) + 1, which is larger than 13/12. Since the cost is invariant to the isomorphism θ′, we get a contradiction. Thus, F(M) is uncountable. Q.E.D.

2.7. Corollary. 1^{∘}. The type II_{1} factor M_{0} = L(ℤ^{2} ⋊ SL(2, ℤ)) is non-Γ and has countable fundamental group F(M_{0}).

2^{∘}. If σ_{1} is chosen to be ergodic but not strongly ergodic (cf 2.2.3^{∘}), then the type II_{1} factor M = A ⋊_{σ} SL(2, ℤ), where A and σ are defined as in 2.1, then M has the property Γ and countable fundamental group F(M).

Proof. This is now an immediate consequence of Lemma 2.2 and Theorem 2.3. Q.E.D.

3. Further remarks.

3.1. More examples of factors M with countable F(M). Recall that the first examples of factors M with countable fundamental group were constructed by Connes in ([C1]), as group von Neumann factors associated to dicrete ICC groups G having the property T of Kazhdan. More examples of factors with countable fundamental group were constructed in ([Po1,3,5], [Gol]). All these examples had relied in a way or another on constructions involving property T groups and they were non-Γ factors. The examples in the previous section are still using some rigidity phenomena, but the milder one involving the inclusion ℤ^{2} ⊂ ℤ^{2} ⋊ SL(2, ℤ). Although these examples seem “closer” to free group factors, they cannot be embedded in any free group factor, by the same argument as the one in [CJ1] (see [Po]). But they cannot contain property T groups either, because of the “relative Haagerup property” that the inclusion L(ℤ^{2}) ⊂ L(ℤ^{2} ⋊ SL(2, ℤ)) has (see [Po]). .

One can actually construct another class of examples of factors having the property Γ but countable fundamental group, by using property T groups and a strategy reminiscent of ([C1], [CJ1]) and ([Po1,3]), as follows:

Let G be an ICC group with the property T of Kazhdan and satisfying the following condition:

(3.1.1). Given any finite set F ⊂ G − {e} there exists g ∈ G such that gF g^{−1}F ∩ F gF g^{−1} = ∅.

Note that this implies in particular gF g^{−1} ∩ F = ∅. (We are greatful to George Skandalis for confirming to us that the groups SL(n, ℤ) with n odd, n ≥ 3 do satisfy this condition, besides having the property T by Kazhdan’s classical result).

Then we let (𝕋_{g}, μ_{g}), g ∈ G, be copies of the thorus 𝕋 with its Lebesgue measure and (X, μ) be the product of these spaces, with the corresponding product measure. We let σ be the action of G on (X, μ) by Bernoulli shifts. We let M_{0} be the corresponding type II_{1} factor obtained by the group-measure space construction, i.e., M_{0} = L^{∞}(X, μ) ⋊_{σ} G. Finally, we consider the action α of ℤ on M_{0}, implemented by the “transversal” action of ℤ on (X, μ) given by rotation by the same irrational number on each copy of the thorus (𝕋_{g}, μ_{g}), ∀g ∈ G. We denote M = M_{0} ⋊_{σ_{0}} ℤ.

It is easy to see that M has the property Γ. Using Lemmas 2.4 and 2.5 (the latter for G_{0} = G) and adding some more work, one can show that if F(M) is uncountable then there exists a projection p ∈ A_{0} = {u_{h}}′′_{h∈ℤ} ⊂ M, with p ≠ 1, and an isomorphism θ : M → pMp, with θ(u_{g}) = u_{g}p, ∀g ∈ G. This implies θ(A_{0}) = A_{0}p and then the condition (3.1.1) easily implies that θ(L^{∞}(X, μ) ⋊ ℤ) = p(L^{∞}(X, μ) ⋊ ℤ)p. This forces each Haar unitary v_{h} generating L^{∞}(𝕋_{h}) to be so that θ(v_{h}) has a certain specific form. But then, by using again (3.1.1), it follows that for appropriate g ∈ G the unitaries θ(v_{h}) and σ_{g}(θ(v_{h})) are “supported” on almost disjoint finite segments of the product ⊗_{g}L^{∞}(𝕋_{g}) cross product with ℤ. But then it is easy to see that θ(v_{h}) and σ_{g}(θ(v_{h})) cannot commute, a contradiction.

3.2. Examples of factors M with F(M) = ℝ_{+}^{∗}. Recall that in ([V], [Ra]) the free group factor L(𝔽_{∞}) was proved to have fundamental group equal to ℝ_{+}^{∗}. This is the most striking example, so far, of a non-Γ type II_{1} factor with fundamental group same as the hyperfinite factor.

The following construction provides a class of examples of type II_{1} factors which have the property Γ, are not McDuff, and yet have fundamental group ℝ_{+}^{∗}:

Let Q be an arbitrary non-atomic finite von Neumann algebra. Let A ⊂ R be the hyperfinite type II_{1} factor with its Cartan subalgebra. Let M = Q\overline{⊗}A ∗_{A} R be the amalgamated free product with respect to the trace preserving conditional expectations, as in ([Po3]). By the proof of (4.1 and 7.1 in [Po3]), we see that any central sequence of M must lie in A. On the other hand, any central sequence of R that lies in A is also a central sequence of M. Thus, M has the property Γ, without being McDuff.

Note that M can be alternatively described as follows: Denote by σ the irrational rotation on the thorus, viewed as an automorphism of L(ℤ). We still denote by σ the action id ∗ σ of ℤ on Q ∗ L(ℤ). Then M ≃ Q ∗ L(ℤ) ⋊_{σ} ℤ, with Q (respectively A ⊂ R) in the first construction corresponding to Q (resp. λ(ℤ)′′ ⊂ L(ℤ) ⋊_{σ} ℤ) in this construction. The fact that M has Γ but is not McDuff can then also be deduced from ([Po4]).

Using the first representation of M, we see that if p is a non-zero projection in A then p(Q\overline{⊗}A∗_{A} R)p is trivially isomorphic to Q\overline{⊗}Ap∗_{Ap} pRp. But by Dye’s theorem (Ap ⊂ pRp) is isomorphic to (A ⊂ R), while Q\overline{⊗}Ap is trivially isomorphic to Q\overline{⊗}A. Thus, pMp ≃ M for any non-zero projection in M. Thus F(M) = ℝ_{+}^{∗}.

Due to the resemblence of the above construction of M with the amalgamated free product construction in ([Po3]), it is likely that in the case Q = L(𝔽_{∞}) the factor M has the same universality property ([PoSh]) as the free group factor L(𝔽_{∞}), namely that any standard lattice G can “act” on it.

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Math.Dept., UCLA, LA, CA 90095-155505

E-mail address: popa@math.ucla.edu