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On the fundamental group of type II_{1} factors

Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, and approved July 10, 2003 (received for review June 25, 2003)
Abstract
We present here a shorter version of the proof of our earlier work, showing that the von Neumann factor associated with the group has trivial fundamental group.
The fundamental group of a type II_{1} factor M is the set of numbers t > 0 for which the “amplification” of M by t is isomorphic to M, .
This invariant for II_{1} factors was considered by Murray and von Neumann in connection with their notion of continuous dimension (1). They noticed that when M is isomorphic to the approximately finite dimensional (or hyperfinite) type II_{1} factor, and more generally when it splits off such a factor. In ref. 2, Connes showed that pairs (M, t) with , t ≠ 1, are at the “core” of the structure and classification of factors of type III. He also showed (3) that reflects the rigidity properties of M, being countable whenever M ≃ L(G _{0}) for G _{0} a group with the property T of Kazhdan. A key application in Voiculescu's free probability theory was to show that (4, 5), whereas the problem of calculating for , remained its central open problem (5–7).
On the other hand, in the ergodic theory of groups, Gaboriau (8, 9) defined and calculated several invariants for orbit equivalence relations coming from measurepreserving actions σ of groups Γ_{0} on the probability space (X, μ), such as cost, Betti numbers, etc. This allowed him to deduce that for a large class of groups Γ_{0}, including the free groups with finitely many generators , the fundamental group of , is trivial. Equivalently, if , is the II_{1} factor with its Cartan subalgebra associated with the action of (10, 11), then the fundamental group of the inclusion A ⊂ M (12), i.e., , is trivial.
In ref. 13, we introduced and studied the class of type II_{1} factors M having Cartan subalgebras A ⊂ M, with respect to which M satisfies a combination of rigidity and weakamenability properties that we called HT. The prototype examples of HT factors in ref. 13 are the crossproduct algebras , with subgroups of finite index (e.g., ) and σ_{0} the restriction to Γ_{0} of the action of on . The key result in ref. 13 was a “unique crossproduct decomposition” for HT factors, which implied that all invariants for A ⊂ M (notably the Betti numbers in ref. 9) are invariants for the HT factors M. As a consequence, we obtained many examples of type II_{1} factors M with trivial fundamental group, , including . In particular, this result solved a longstanding problem of Kadison's by showing that there exist factors M with M_{n} _{×} _{n} (M) M for any n ≥ 2‡ (see also ref. 14).
In this article we give a short proof of the fact that , for many of the concrete examples of HT factors in ref. 13. The proof is in fact not new; it is just the adaption to these particular cases of the proof in ref. 13. Thus, by considering only algebras and subalgebras associated with a group–subgroup situation, we bypass the lengthy technicalities involved in developing the abstract property HT.
The type II_{1} factors M that we consider here are defined as follows:
Let σ_{0} be the free, ergodic action of on , implemented by its action on (thus σ_{0} preserves the Lebesgue measure μ_{0} on ). Let be any subgroup with finite index. Let σ_{1} be any ergodic, measurepreserving action of Γ_{0} on some probability space (X _{1}, μ_{1}). Let and σ = σ_{0} × σ_{1} the corresponding product action of Γ_{0} on (X, μ). We define M to be the crossproduct (or groupmeasure space) von Neumann algebras M = L ^{∞} (X, μ) _{σ} Γ_{0}. The action σ_{0} is well known to be mixing on (, μ_{0}). Because σ_{1} is ergodic, σ follows ergodic and, thus, M follows type II_{1} factors (see, for example, ref. 14, 15, or 16).
Theorem. The type II _{1} factors M constructed above have trivial fundamental group, .
Corollary. For each finite n ≥ 2 there exist free, ergodic, measurepreserving actions σ of on the probability space (X, μ) such that has trivial fundamental group, . Moreover, the action σ can be taken strongly ergodic, thus getting M to be nonΓ, or nonstrongly ergodic, thus getting M to have the property Γ.
There are two crucial properties that enable us to prove the Theorem: on the one hand, Kazhdan's rigidity of the inclusion (17), which makes the entire subalgebra A = L ^{∞}(X, μ) sit “somewhat rigidly” inside M = L ^{∞}(X, μ) _{σ} Γ_{0}, and, on the other hand, Haagerup's compact approximation property of the groups Γ_{0} (18), which make the factors pMp be in some sense “weakly amenable” relative to Ap, for any nonzero projection .
As a consequence of these properties, if there would exist an isomorphism θ of pMp onto M, for some , p ≠ 0, 1, then M would have maximal abelian subalgebras A and θ(Ap) with A ⊂ M rigid and M weakly amenable relative to θ(Ap). The heart of the argument consists in proving that if a type II_{1} factor M has two such maximal abelian subalgebras, A and θ(Ap), then these subalgebras are conjugate by a unitary element in M (this is what we earlier called “unique decomposition result” for M). However, if this is the case, then the inclusions A ⊂ M and Ap ⊂ pMp are isomorphic (via Adu ° θ), thus implementing isomorphic measurable equivalence relations on the standard probability space (11, 12). When combined with the results in ref. 8 or 9, this argument gives a contradiction.
We split the details of this argument into several lemmas.
Lemma 1. Let A = L ^{∞}(X, μ) ⊂ L ^{∞} (X, μ) _{σ} Γ_{0} = M be as defined above and let {u_{g} } _{g} _{∈Γ0} ⊂ M be the canonical unitaries implementing the action σ. Let p ∈ A be a nonzero projection and θ: pMp → M an isomorphism. Denote B = θ(Ap) and v_{g} = θ(pu_{g}p), g ∈ Γ_{0}. Then we have:

v_{g} are partial isometries in the normalizing groupoid of B in M; i.e., , , and , ∀g ∈ Γ_{0}. Moreover, the spaces v_{g}B, g ∈ Γ_{0}, are mutually orthogonal in L ^{2}(M), and we have .

If ϕ_{0} is a positive definite function on Γ_{0}, then the application Σ _{g}v_{g}b_{g} ↦ Σ _{g} ϕ_{0}(g)v_{g}b_{g}, defined on finite sums with b_{g} ∈ B, extends to a unique unital, tracepreserving, completely positive, Bbilinear map φ_{ϕ0}: M → M. Similarly, Σ_{g} v _{g} b̂ _{g} ↦ Σ_{g}ϕ_{0}(g)v _{g} b̂ _{g} extends to a unique bounded linear operator T _{ϕ0} on L ^{2}(M). Moreover, we have , where e_{B} = proj _{L2(B)}. Also, T _{ϕ0}(x̂) = φ_{ϕ0}(x̂), ∀x ∈ M.
Proof: (i) Since , we have . Also, because Σ _{g}u_{g}A is dense in M, Σ _{g}pu_{g}pAp = p(Σ _{g}u_{g}A)p is dense in pMp, so Σ _{g}v_{g}B = θ(Σ _{g}pu_{g}pAp) is dense in M. Since p(u_{g}A)p = u_{g} σ _{g} _{–1}(p)pA are mutually orthogonal, v_{g}B, g ∈ Γ_{0}, follow mutually orthogonal as well.
(ii) The same argument as 1.1 in ref. 18 shows that φ_{0}(Σu_{g}a_{g} ) = Σ _{g} ϕ_{0}(g)u_{g}a_{g} , with a_{g} ∈ A all but finitely many equal to 0, extends to a unique unital, tracepreserving, completely positive, Abilinear map φ_{0} on M. Thus, is a unital, tracepreserving, completely positive, Apbilinear map on pMp and is a unital, tracepreserving, completely positive, Bbilinear map on M.
Because the vector spaces v_{g}B, g ∈ Γ_{0}, are mutually orthogonal in L ^{2}(M), it follows that is a well defined positive operator on L ^{2}(M), and, by the definitions, it is trivial to see that it coincides with T _{ϕ0}. q.e.d.
For the next lemma, recall that by a well known result of Kazhdan (17) the inclusion of groups satisfies the following rigidity property: If a unitary representation of G _{0} weakly contains the trivial representation of G _{0} then it contains the trivial representation of G. It is immediate to see that if we take a subgroup of finite index, then still satisfies the above rigidity condition. Also, the same proof as in ref. 19 shows that for inclusions of groups G ⊂ G _{0} with G normal in G _{0} this rigidity condition is equivalent to the following condition: If ϕ _{n} are positive definite functions on G _{0} with lim _{n} ϕ _{n} (g) = 1, ∀g ∈ Γ_{0}, then ϕ _{n} converge to 1 uniformly on G. This condition, in turn, is clearly equivalent to the following condition: ∀ε > 0, ∃F _{0}(ε) ⊂ G _{0} finite, δ_{0}(ε) > 0 such that if ϕ is a positive definite function on G _{0} with ϕ(g _{0}) – 1 ≤ δ_{0}(ε), ∀g _{0} ∈ F _{0}(ε) then ϕ(h) – 1 ≤ ε, ∀h ∈ G. (See refs. 20 and 21 for more on this “relative property T” condition, which for arbitrary inclusions G ⊂ G _{0} was first emphasized by Margulis.)
Lemma 2 uses group rigidity in the same spirit as Connes (3) and Connes and Jones (22).
Lemma 2. Let M and be as before. For any ε > 0 there exists a finite set F(ε) ⊂ G _{0} and δ(ε) > 0 such that if φ: M → M is a unital, completely positive map with verbar;φ(u_{g} _{0}) – u_{g} _{0}_{2} ≤ δ(ε), ∀g _{0} ∈ F(ε), then φ(u_{h} ) – u_{h} _{2} ≤ ε, ∀h ∈ .
Proof: Let φ be a unital, completely positive map satisfying φ(u_{g} _{0}) – u_{g} _{0}_{2} ≤ δ_{0} (ε^{2}/2), ∀_{g0} ∈ F _{0}(ε^{2}/2). Define , g ∈ G _{0}. Then ϕ is positive definite, and by the Cauchy–Schwartz inequality we have: for any g _{0} ∈ F _{0}(ε^{2}/2). Since by Kadison's inequality we have , it follows that for all g ∈ G _{0} we have . Thus, if h ∈ G, we get:
Thus, if we define F(ε) = F _{0}(ε^{2}/2), δ(ε) = δ_{0}(ε^{2}/2), then we are done. q.e.d.
Lemma 3. Let A, B ⊂ M be as before. There exists a finite subset F ⊂ Γ_{0} such that the projection satisfies f(û_{h} ) – û_{h} _{2} ≤ 1/4, .
Proof: Because contains free groups as subgroups with finite index, by Haagerup's work in ref. 18 it has the Haagerup approximation property and so do all its subgroups of finite index Γ_{0}. Let be positive definite functions with ϕ _{n} ∈ c _{0}(Γ_{0}), ϕ _{n} (e) = 1, ∀n, and lim _{n} ϕ _{n} (g) = 1, ∀g ∈ Γ_{0}.
By the definition of φ_{ϕ} _{n} = φ _{n} in Lemma 1, it follows that φ _{n} tend to id_{M} in the pointnorm  _{2}topology. Let n be large enough so that φ _{n} (u_{g} _{0}) – u_{g} _{0}_{2} ≤ δ(1/16), ∀g _{0} ∈ F(1/16), where F(1/16) ⊂ G _{0}, δ(1/16) are as given by Lemma 2. It follows that φ _{n} (u_{h} ) – u_{h} _{2} ≤ 1/16, . Thus, if we let T = T _{ϕ} _{n} , then 0 ≤ T ≤ 1 and T(û_{h} ) – û_{h} _{2} ≤ 1/16, . Let f be the spectral projection of T corresponding to [1/16, 1]. Thus, if F is the set of all g ∈ Γ_{0} with ϕ _{n} (g) ≥ 1/16, then F is finite and . Moreover, we have f(û_{h} ) – u_{h} _{2} ≤ 1/4, . Indeed, for if we would have (1 – f)(û_{h} )_{2} > 1/4 for some , then a contradiction. q.e.d.
Lemma 4. Let A _{0} ⊂ M be the von Neumann algebra generated by . There exists ξ ∈ L (M), ξ ≠ 0, such that A _{0} L ^{2}(ξB) ⊂ L ^{2}(ξB).
Proof: To obtain from the projection f in Lemma 3 a left A _{0} module of the form L ^{2}(ξB), we first use a trick inspired by ref. 23. Thus, let 〈M, B〉 be the von Neumann algebra generated in by M (regarded as the algebra left multiplication operators by elements in M) and by the orthogonal projection e_{B} of L ^{2}(M) onto L ^{2}(B). Note that if x ∈ M then e_{B}xe_{B} = E_{B} (x)e_{B} and that 〈M, B〉 is the weak closure of the *algebra sp{xe_{B}y  x, y ∈ M}. Also, 〈M, B〉 coincides with the commutant in of the operators of right multiplication by elements in B, i.e., with J_{M}B′J_{M} , where J_{M} is the canonical conjugation on L ^{2}(M), defined by J_{M} (x̂) = x̂*, x ∈ M. (See ref. 23 or 24 for details on 〈M, B〉 and on all this construction, called the “basic construction.”)
Because e_{B} has central support 1 in 〈M, B〉 and e_{B} 〈M, B〉e_{B} = Be_{B} , it follows that 〈M, B〉 is of type I. Moreover, Tr(xe_{B}y) = τ(xy), x, y ∈ M defines a unique normal faithful semifinite trace on 〈M, B〉.
Let be the projection given by Lemma 3. Note that f ∈ 〈M, B〉 and . We denote by . Because K is weakly closed and 0 ≤ a ≤ 1, ∀a ∈ K, K is weakly compact. Moreover, Tr(a) ≤ Tr(f), ∀a ∈ K, so K is also contained in the Hilbert space L ^{2}(〈N, B〉, Tr), where it is still weakly compact.
Let k ∈ K be the unique element of minimal norm  _{2,} _{Tr} in K. Since and , , by the uniqueness of k it follows that , . Thus, , 0 ≤ k ≤ 1 and Tr(k) ≤ 1.
Since it follows that Tr(e_{B}a) ≥ 9/16, ∀a ∈ K. In particular, Tr(e_{B}k) ≥ 9/16, so k ≠ 0. Thus, if e is a spectral projection of k corresponding to (c, ∞) for some appropriate c > 0, then , e ≠ 0 and Tr(e) < ∞. Thus, e〈M, B〉e is a finite type I von Neumann algebra. Let A _{1} ⊂ e〈M, B〉e be a maximal abelian subalgebra containing A _{0} e. By Kadison's work in ref. 25, A _{1} contains an abelian projection e _{0} ≠ 0 of e〈M, B〉e. Thus, e _{0} is an abelian projection in 〈M, B〉. By von Neumann's theorem, there exists ξ ∈ L ^{2}(M) such that e _{0} is the orthogonal projection onto . Since e _{0} ∈ A _{1} commutes with A _{0}, we have A_{0}L ^{2}(ξB) ⊂ L ^{2}(ξB). q.e.d.
Lemma 5. There exists a nonzero partial isometry w ∈ M such that w*w ∈ A, ww* ∈ B and wAw* ⊂ Bww*.
Proof: Let A _{0} be as in Lemma 4. By construction, we readily see that . Let ξ ∈ L ^{2}(M), ξ ≠ 0, be so that A _{0}ξ ⊂ L ^{2}(ξB) (conform Lemma 4).
Since ∀x ∈ M, we have . It follows that E_{B} extends by continuity to a completely positive map from L ^{1}(M, τ) to L ^{1}(B, τ), still denoted E_{B} .
By regarding ξ as a square summable operator affiliated with M and by replacing it by ξE _{B} (ξ*ξ)^{–1/2} ∈ L ^{2} (M, τ), we may assume q _{0} = E_{B} (ξ*ξ) is a projection in B. Note that we then have L ^{2}(ξB) = ξL ^{2}(B). Let p _{0} ∈ A _{0} be the minimal projection in A _{0} with the property that (1 – p _{0})ξ = 0. We denote ψ: A _{0} p _{0} → L ^{1}(q _{0} Bq _{0}, τ) the map given by ψ(a _{0}) = E_{B} (ξ*a _{0}ξ), a _{0} ∈ A _{0} p _{0}, and note that ψ is completely positive with ψ(p _{0}) = q _{0}. Thus, by positivity, it follows that ψ(A _{0} p _{0}) ⊂ q _{0} Bq _{0}. By the above remarks and by construction, it follows that as a map from A _{0} p _{0} into q _{0} Bq _{0}, ψ is normal, faithful, unital, and completely positive.
Also, since a _{0} ξ ∈ L ^{2} (ξB) = ξL ^{2}(Bq _{0}), it follows that a _{0}ξ = ξψ(a _{0}), ∀a _{0} ∈ A _{0}. Indeed, for if a _{0}ξ = ξb, for some b ∈ L ^{2}(B)q _{0}, then ξ*a _{0}ξ = ξ*ξb and so Thus, for a _{1}, a _{2} ∈ A _{0} we get a _{1} a _{2}ξ = a _{1}ξψ(a _{2}) = ξψ(a _{1})ψ(a _{2}). Because we also have (a _{1} a _{2})ξ = ξψ(a _{1} a _{2}), this equality shows that ψ(a _{1} a _{2}) = ψ(a _{1})ψ(a _{2}). Thus, ψ is a *morphism of A _{0} into A.
Thus, since a _{0}ξ = ξψ(a _{0}), ∀a _{0} ∈ A _{0}, and ψ is a *isomorphism, by a standard trick, if v ∈ M is the partial isometry in the polar decomposition of ξ with the property that the right supports of ξ and v coincide, then , q = v*v ∈ ψ(A _{0})′ ∩ q _{0} Mq _{0} and a _{0} v = vψ(a _{0}), ∀a _{0} ∈ A _{0}. Since it follows that q is an abelian projection in ψ(A _{0})′ ∩ q _{0} Mq _{0}. However, Bq _{0} is maximal abelian in q _{0} Mq _{0}, so by the work of Kadison (25) it follows that there exists a projection q′ ∈ Bq _{0} such that q and q′ are equivalent in ψ(A _{0})′ ∩ q _{0} Mq _{0}, say via a partial isometry v′ ∈ ψ(A _{0})′ ∩ q _{0} Mq _{0}. Thus, if we put w = (vv′)*, then w*w ∈ A, ww* ∈ B, and wAw* ⊂ B. q.e.d.
Lemma 6. There exists a unitary element u ∈ M such that uAu* = B.
Proof: Note first that both A and B are regular maximal abelian subalgebras in M in the sense of ref. 26 (N.B.: Such algebras are called Cartan subalgebras in refs. 10 and 11). Indeed, this is because Σ _{g} u_{g}A (resp. Σ _{g} v_{g}B) is dense in M, implying that (resp. ) generates the von Neumann algebra M, so the work of Dye (27, 28) applies.
Let w ∈ M be the nonzero partial isometry provided by Lemma 5. By cutting w from the right with a smaller projection in A, we may clearly assume τ(ww*) = 1/n for some integer n.
Because A, B are regular in M, by the work of Dye (27, 28) 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 , and , , ∀i, j. However, then is a unitary element, and vA _{0} v* = B. q.e.d.
We summarize Lemmas 1–6 as the following unique decomposition result.
Proposition. If p ∈ A is a nonzero projection and θ: pMp → M is an isomorphism, then there exists a unitary element u ∈ M such that Adu(θ(Ap)) = A. Thus, the inclusion Ap ⊂ pMp and A ⊂ M follow isomorphic.
Proof of the Theorem: If and t < 1, then let p ∈ M be a projection with trace τ(p) = t and θ: pMp ≃ M an isomorphism. Without loss of generality, we may take p ∈ A. By the Proposition, Ap ⊂ pMp and A ⊂ M follow isomorphic. By the work of Feldman and Moore (11, 12), this implies that the countable, measurepreserving equivalence relations and are isomorphic. By the work of Gaboriau (8, 9), this is a contradiction. q.e.d.
Proof of the Corollary: Because any free group with finitely many generators , n ≥ 2, can be embedded into with finite index, we can take in the construction of M, proving the first part of the statement.
If , then any central sequence for M is contained in L ^{∞} (X, μ) (see, for example, ref. 29). Thus, if we take σ_{1} to be trivial, i.e., with σ = σ_{0}, then by taking into account that σ_{0} is strongly ergodic (cf. ref. 30), it follows that M has no nontrivial central sequences; i.e., it is nonΓ.
On the other hand, if we take σ_{1} to be any free ergodic action of on a probability space (X _{1}, μ_{1}) (e.g., a Bernoulli shift action of ) composed with the quotient map , then σ = σ_{0} × σ_{1} gives an ergodic, but not strongly ergodic, action of on (X, μ), thus getting M to have property Γ. q.e.d.
Remark: Note that the arguments in Lemmas 1–6 prove a more general unique decomposition result than the Proposition, namely let G^{i} be commutative groups and some groups satisfying Haagerup's compact approximation property, i = 1, 2. Assume acts on G^{i} mixingly and outerly, and so that the inclusion has the relative property T, for each i = 1, 2, in the sense of Margulis (ref. 20; see ref. 31 for examples). For each i = 1, 2 let be the corresponding tracepreserving action of on L(G_{i} ) and let be an ergodic measurepreserving action of on a probability space () such that the action of on is ergodic. Let . If there exists some isomorphism θ from p _{1} M _{1} p _{1} onto p _{2} M _{2} p _{2} for some , then there exists a unitary element u ∈ p _{2} M _{2} p _{2} such that Adu ° θ(A _{1} p _{1}) = A _{2} p _{2}. Note that this statement covers most of the concrete examples of HT factors in ref. 13, but that it only proves the uniqueness of the “concrete” HT Cartan subalgebras.
The proof of the general unique decomposition result for arbitrary HT factors in ref. 13 requires a full discussion of the operator algebraic concept of (relative) properties H and T. As a consequence, although going along the same lines, the proof there is unavoidably lengthier. The full generality of that result is needed in order to show that the class of abstract HT factors is well behaved to elementary operations (amplification, tensor product, finite index extension/restriction), to make the definition of Betti numbers for factors in the class HT be more conceptual (while relying on the work of Gaboriau in ref. 9) and to show that they check the appropriate formulas. It is also needed in order to investigate the structure of finite Connes's correspondences and of subfactors of finite Jones index of HT factors (see section 7 in ref. 13), which in turn is of interest because it relates a number theory framework to the theory of type II_{1} factors.
Acknowledgments
S.P. was supported by National Science Foundation Grant 0100883.
Footnotes
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