Property Γ factors and the Hochschild cohomology problem

Abstract

The main result of this article is that the kth continuous Hochschild cohomology groups H^k(ℳ, ℳ) and H^k(ℳ, B(H)) of a von Neumann factor ℳ ⊆ B(H) of type II₁ with property Γ are 0 for all positive integers k. The method of proof involves the construction of hyperfinite subfactors with special properties and a new inequality of Grothendieck type for multilinear maps. We prove joint continuity in the ∥⋅∥₂ norm of separately ultraweakly continuous multilinear maps and combine these results to reduce to the case of completely bounded cohomology, which is already solved.

1. Introduction

The study of the continuous Hochschild cohomology groups Hⁿ(ℳ, ℳ), n ≥ 1, of a von Neumann algebra ℳ with coefficients in itself was begun in a series of papers by Johnson, Kadison, and Ringrose (1–4). Their work was an outgrowth of the Kadison–Sakai theorem on derivations (5, 6), which proved, in an equivalent formulation, that H¹(ℳ, ℳ) = 0 for all von Neumann algebras. It was natural to conjecture that the higher cohomology groups Hⁿ(ℳ, ℳ) also vanish, and this was settled affirmatively for hyperfinite von Neumann algebras in ref. 4. These authors established many general results on cohomology, some of which are reviewed below. One particular consequence is that it suffices to consider separately the cases when ℳ is type I, II₁, II_∞, or III in the Murray–von Neumann classification scheme; the general von Neumann algebra is a sum of these four types. Because type I von Neumann algebras are hyperfinite (but by no means exhaust this class), attention has been focused on the remaining three types. Considerable progress on the problem has been made recently by the introduction of the completely bounded cohomology groups H(ℳ, ℳ). E.C. and A.M.S. (7) used the structure theory of completely bounded multilinear maps (8) to show that H(ℳ, ℳ) = 0 for all von Neumann algebras (see chapter 4 in ref. 9). These authors and Effros (10) also proved that the continuous and completely bounded cohomology groups coincide when ℳ is type II_∞, III, or II₁ and stable under tensoring with the hyperfinite type II₁ factor, showing that Hⁿ(ℳ, ℳ) = 0 in these cases. Thus the conjecture remains open only for type II₁ von Neumann algebras.

Within the class of type II₁ factors the main results are that Hⁿ(ℳ, ℳ) = 0, for separable factors with a Cartan subalgebra and n ≥ 1 (11, 12), and for factors with property Γ when n = 2. The latter result is due to E.C. and may be found in chapter 6 of ref. 9 (see ref. 13 for a later proof). The main result of this article, which builds on the techniques of ref. 13 is that Hⁿ(ℳ, ℳ) = 0, n ≥ 1, for type II₁ von Neumann factors that have property Γ. This was introduced by Murray and von Neumann (14) to distinguish certain nonisomorphic factors from one another. They defined this property by requiring that for each finite subset {x_i} ∈ ℳ and each ɛ > 0, there should exist a unitary u ∈ ℳ of zero trace satisfying

We will work with a later characterization due to Dixmier (15), which we describe in Section 5. The class of Γ factors contains all that are stable under tensoring with the hyperfinite factor but also contains many factors that are not of this type.

Recent approaches to cohomology (9–13) have focused on proving that the relevant cocycles are completely bounded as multilinear maps, and that is the method that we use here. It has often been the case that complete boundedness of maps of a factor into itself could be proved but only by techniques that were ineffective for larger ranges. For Γ factors, the situation is different, and our results also apply to the cohomology groups with coefficients in any containing B(H). For n = 1 and n = 2, see refs. 16 and 17, respectively.

In Section 2 of the article we review the basic definitions of cohomology theory and we include a brief discussion of completely bounded maps. Section 3 presents a new version (Theorem 3.1) of the Pisier–Haagerup–Grothendieck inequality, which has been important in earlier work on cohomology, and also discusses some results on the joint continuity in the ∥⋅∥₂ norm for multilinear maps (Theorem 3.3). Section 4 expands on a theorem of Popa (17), which states that hyperfinite subfactors of trivial relative commutant can be found in any type II₁ factor with separable predual. When property Γ is assumed, we show that the hyperfinite subfactor can be chosen to contain certain auxiliary projections (Theorem 4.1). In Section 5 we describe how the results of the previous two sections can be combined to prove the complete boundedness of certain multilinear maps (Theorem 5.1), and this leads to our main result on the cohomology groups of Γ factors (Theorem 5.2). Complete details will appear elsewhere.

We refer the reader to ref. 18 for an early survey of cohomology theory and to a later account in ref. 9, which contains all the necessary background material for this article, as well as a discussion of applications.

2. Preliminaries

The matrix algebras 𝕄_n(𝒜), n ≥ 1, over a C*-algebra 𝒜 ⊆ B(H) carry natural norms, defined by viewing 𝕄_n(𝒜) as a subalgebra of 𝕄_n(B(H)) and identifying the latter algebra with B(H ⊕ ⋯ ⊕ H) (n-fold direct sum). Thus a bounded linear map φ: 𝒜 → B(H) induces a family φ⁽ⁿ⁾: 𝕄_n(𝒜) → 𝕄_n(B(H)), n ≥ 1, of bounded maps on the matrix algebras by applying φ in each entry, and φ is said to be completely bounded if

This supremum then defines the completely bounded norm ∥φ∥_cb. There is substantial literature on completely bounded maps (see refs. 9 and 19 and references therein). Now let 𝒜^k denote the k-fold Cartesian product of copies of 𝒜. A k-linear map φ: 𝒜^k → B(H) may be lifted to a k-linear map φ⁽ⁿ⁾: 𝕄_n(𝒜)^k → 𝕄_n(B(H)), n ≥ 1. For clarity we take k = 2, because this case contains the essential ideas. For matrices X = (x_ij), Y = (y_ij) ∈ 𝕄_n(𝒜), the (i, j) entry of φ⁽ⁿ⁾(X, Y) ∈ 𝕄_n(B(H)) is defined to be ∑ φ(x_ir, y_rj). Following the linear case, the completely bounded norm is also defined by Eq. 2.1. Such maps have important applications in cohomology theory (9).

We will require the notion of multimodular maps below. If ℛ ⊆ ℳ ⊆ B(H) is an inclusion of algebras then ℛ-multimodularity of φ: ℳ^k → B(H) is defined by the equations

where r ∈ ℛ and x_i ∈ ℳ for 1 ≤ i ≤ k. A simple but important consequence of the definitions is that φ⁽ⁿ⁾ is 𝕄_n(ℛ)-multimodular, for all n ≥ 1, when φ is ℛ-multimodular.

We recall from refs. 1 and 9 the basic definitions of Hochschild cohomology theory. Let ℳ be a von Neumann algebra and denote by ℒ^k(ℳ, 𝒳) the space of k-linear bounded maps φ: ℳ^k → 𝒳 into a Banach ℳ-bimodule 𝒳. The coboundary operator ∂: ℒ^k(ℳ, 𝒳) → ℒ^k+1(ℳ, 𝒳) is defined by

for x₁, … , x_k+1 ∈ ℳ. Then φ is a k cocycle if ∂φ = 0, whereas φ is a k coboundary if φ = ∂ψ for some ψ ∈ ℒ^k−1(ℳ, 𝒳). A short algebraic calculation shows that ∂∂ = 0, and thus coboundaries are cocycles. The cohomology group H^k(ℳ, 𝒳) then is defined to be the space of k cocycles modulo the space of k coboundaries (for k ≥ 2). For k = 1, H¹(ℳ, 𝒳) is defined to be the space of bounded derivations modulo the space of inner derivations. The definition gives rise to a related family of cohomology groups by imposing further restrictions on the bounded maps. We might require ultraweak continuity (H(ℳ, 𝒳)), ℛ-multimodularity (H^k(ℳ, 𝒳:/ℛ)), complete boundedness (H(ℳ, 𝒳)), or any combination of these. The interplay between these various cohomology theories gives an important tool for the determination of H^k(ℳ, 𝒳). The following theorem summarizes the work of several authors (2–4, 7, 10), and the various parts are described in ref. 9.

Theorem 2.1.

Let ℳ ⊆ B(H) be a von Neumann algebra with a hyperfinite subalgebra ℛ, and let 𝒳 be a dual normal ℳ-bimodule.

1. (i)  For k ≥ 1,

   
2. (ii)  For k ≥ 1 and either 𝒳 = ℳ or 𝒳 = B(H),

   

   The second part of this theorem shows that vanishing of cohomology for these two bimodules can be established by proving that each cocycle is cohomologous to one that is completely bounded, whereas the first part shows that attention can be restricted to those cocycles that are separately normal and ℛ-multimodular for a suitably chosen ℛ. This is now the standard approach to such problems, and we adopt it below.

3. Grothendieck's Inequality and Joint Continuity

One of the most useful tools for cohomology theory has been the noncommutative generalization of Grothendieck's inequality for bilinear forms. This was proved first by Pisier (20) under mild restrictions and then by Haagerup (21) in full generality. For a bilinear form θ: ℳ × ℳ → ℂ on a von Neumann algebra ℳ, separately normal in each variable, the appropriate formulation is as follows. There exist four normal states f₁, f₂, g₁, g₂ ∈ ℳ_* such that

for x, y ∈ ℳ. This inequality is valid for all von Neumann algebras; we assume for the rest of the section that ℳ is a type II₁ factor, and we describe some more specialized inequalities that then are available. If θ above has the additional property of being inner ℛ-modular, in the sense that

for r in a hyperfinite subfactor ℛ with trivial relative commutant ℛ′ ∩ ℳ = ℂ1, then Eq. 3.1 takes the form

where F and G are normal states. This is accomplished by replacing x and y in Eq. 3.1 by xu and u*y, respectively, where u lies in an amenable group 𝒢 of unitaries, which generates ℛ, and then averaging the right-hand side of the inequality over 𝒢. The x*x and yy* terms in Eq. 3.1 become tr(x*x)1 and tr(yy*)1 after averaging and are then replaced by tr(xx*)1 and tr(y*y)1, explaining why only two types of terms appear on the right-hand side of Eq. 3.3. Further replacement of x by tx and y by t⁻¹y for t ∈ ℝ⁺ in Eq. 3.3, followed by minimization over t, leads to

There is no direct generalization of Eq. 3.4 to multilinear forms, but nevertheless certain inequalities can be obtained, based on the bilinear equalities above, by fixing all but two of the variables. We illustrate this for three variables. We observe that a bilinear map φ: ℳ × ℳ → B(H) may be studied by applying a vector functional 〈⋅ξ, η〉 and examining the bilinear form

For a trilinear map φ: ℳ³ → B(H), various auxiliary bilinear maps are used to focus on each variable separately. For the outer variables we use maps of the type

and

whereas the inner variable is handled by

The x_i values in these equations are taken to be arbitrary but fixed, and an assumption of ℛ-multimodularity on φ ensures that each ψ_i, 1 ≤ i ≤ 3, is inner ℛ-modular. This approach applies also to the nth amplification φ⁽ⁿ⁾ of φ to 𝕄_n(ℳ), and we now state an inequality for k-linear maps that results from applying the bilinear Grothendieck inequality repeatedly to bilinear maps of the above types. We have indicated that the trace is implicit in Eqs. 3.3 and 3.4; the new inequality will contain it explicitly. We let tr_n denote the unique normalized normal trace on 𝕄_n(ℳ), and we introduce a new norm on 𝕄_n(ℳ) by

ρn(X)=(∥X∥²+n tr_n(X*X))^1/2, X, Y∈𝕄_n(ℳ).

Theorem 3.1.

Let ℳ ⊆ B(H) be a typeII₁ factor containing a hyperfinite subfactor ℛ with trivial relative commutant. Let φ: ℳ^k → B(H) be a bounded ℛ-multimodular k-linear map that is separately normal in each variable. Then

for all X₁, … , X_k ∈ 𝕄_n(ℳ) and n ∈ ℕ.

If the tracial term in Eq. 3.9, and thus in Eq. 3.10, could be removed, then Theorem 3.1 would give complete boundedness of maps that satisfy the hypotheses. The following corollary is a step in this direction.

Corollary 3.2.

In addition to the hypotheses of Theorem 3.1, letP ∈ 𝕄_n(ℳ) be a projection of trace n⁻¹. Then

for X₁, … , X_k ∈ 𝕄_n(ℳ).

This inequality is a simple consequence of Eq. 3.10 by estimating that

for X ∈ 𝕄_n(ℳ).

We now turn to joint continuity of multilinear separately normal maps. We will need the topology of ∥⋅∥₂-norm convergence, where ∥⋅∥₂ is defined by

To see the necessity for this, consider the free group 𝔽₂ with generators a and b and its associated type II₁ factor V N(𝔽₂). The group element a becomes a unitary in this factor and has the property that lim_n→∞ a^±n = 0 ultraweakly. For the separately normal bilinear multiplication map m(x, y) = xy, defined on any factor, the equation lim_n→∞ m(aⁿ, a⁻ⁿ) = 1 precludes joint ultraweak continuity. The simple estimates

when restricted to the unit ball of ℳ, lead to

and this inequality generalizes to

This suggests joint ∥⋅∥₂-norm continuity on bounded balls in any type II₁ factor. The example of a sequence of projections {p_n} with tr(p_n) = n⁻⁴ and thus satisfying

shows that joint continuity on the whole factor cannot be expected. These remarks indicate that the following result is optimal.

Theorem 3.3.

Let ℳ ⊆ B(H) be a typeII₁ factor, let k ∈ ℕ, and let φ: ℳ^k → B(H) be a bounded separately normal k-linear map.

1. (i)  If the range of φ is contained in ℳ, then the restriction of φ to the product of closed balls is jointly ∥⋅∥₂-norm continuous.

2. (ii)  If ξ, η ∈ H, then the same conclusion holds for the k-linear form

   

   For k = 1, a normal linear map φ: ℳ → ℳ induces a separately normal bilinear form

   

   to which Grothendieck's inequality can be applied. The ∥⋅∥₂-norm continuity of φ on balls is a consequence of this (using corollary 5.4.3 of ref. 9). The proof of the theorem for k ≥ 2 is by induction. The first step is to obtain joint continuity at a single point, and this is a special case of a theorem in ref. 22, which relies on the Baire category theorem. A separate argument is required to deduce joint continuity at the origin, from which joint continuity at all points in a product of balls follows. ii is a simple consequence of i by treating the range ℂ as ℂ1 ⊆ ℳ.

4. Hyperfinite Subfactors

The purpose of this section is to combine two results, due to Popa (17) and Dixmier (15), respectively. If we assume that a type II₁ factor ℳ has a separable predual, then the first states that we may find a hyperfinite subfactor ℛ with trivial relative commutant. Any hyperfinite subfactor ℛ may be expressed as the ultraweak closure of a union of an increasing sequence of matrix subfactors 𝒜_n, n ≥ 1. Denoting their unitary groups by 𝒰_n and then integrating with respect to normalized Haar measure leads to

where 𝔼_ℛ′∩ℳ is the trace preserving conditional expectation of ℳ onto ℛ′ ∩ ℳ, and the limit is taken ultraweakly (see lemma 5.4.4 of ref. 9). Consequently, the relative commutant will be trivial precisely when the limit in Eq. 4.1 is tr(x)1 for all x ∈ ℳ, giving a criterion for ℛ′ ∩ ℳ to be ℂ1.

Dixmier's result is an equivalent formulation of property Γ: given ɛ > 0, n ∈ ℕ, and {x₁, … , x_m} ∈ ℳ, there exist n orthogonal projections {p_i}, each of trace n⁻¹, so that

where [⋅, ⋅] denotes the commutator. For Γ factors these results can be combined.

Theorem 4.1.

Let ℳ be a type II₁ factor with property Γ and a separable predual. Then ℳ has a hyperfinite subfactor ℛ with trivial relative commutant, and such that Eq.4.2 can always be satisfied by projections from ℛ.

This subfactor is built inductively from matrix subfactors 𝒜_k, k ≥ 1. A ∥⋅∥₂-norm dense sequence {m_i} is fixed in the unit ball of ℳ. At the kth stage of the induction, 𝒜_k−1 is tensored with a large matrix subfactor to form ℬ_k in such a way that this algebra contains projections satisfying Eq. 4.2 for ɛ = k⁻¹, for any subset of {m₁, … , m_k}, and for any n ∈ {1, … , k}. Then 𝒜_k is formed by tensoring ℬ_k with another matrix subfactor, chosen to ensure that the limit in Eq. 4.1 is tr(x)1 for all x ∈ ℳ. This last step requires Popa's theorem on the existence of hyperfinite subfactors with trivial relative commutant.

5. Main Results

We are now able to state the main results of the article. In the first theorem we note that there is no restriction on the dimension of the Hilbert space.

Theorem 5.1.

Let ℳ ⊆ B(H) be a typeII₁ factor with property Γ and a separable predual. Let ℛ be a hyperfinite subfactor satisfying the conclusions of Theorem 4.1, and let φ: ℳ^k → B(H) be a separately normal ℛ-multimodular k-linear map. Then φ is completely bounded and ∥φ∥_cb ≤ 2^k∥φ∥.

The argument is best illustrated by k = 2. If p ∈ ℛ is a projection, then ℛ-multimodularity is used to establish the following algebraic identity, which has more complicated counterparts for k ≥ 3:

Now fix n ∈ ℕ and two unit vectors ξ, η ∈ Hⁿ. Choose sets of projections {p_i,j ∈ ℛ: 1 ≤ j ≤ n, i ≥ 1} of trace n⁻¹ such that

and let P_i,j = I_n ⊗ p_i,j ∈ 𝕄_n(ℛ). Note that for each j, the sequence {P_i,j} asymptotically commutes in the ∥⋅∥₂ norm with the elements of 𝕄_n(ℳ). Because φ⁽ⁿ⁾ is 𝕄_n(ℛ)-multimodular, we may replace φ by φ⁽ⁿ⁾ in Eq. 5.1 while also substituting X, Y ∈ 𝕄_n(ℳ) for x, y ∈ ℳ and P_i,j ∈ 𝕄_n(ℛ) for p ∈ ℛ. Then apply the vector functional 〈⋅ξ, η〉 to the resulting equation, sum over j, and let i → ∞. On the left-hand side we obtain 〈φ⁽ⁿ⁾(X, Y)ξ, η〉. On the right-hand side, the term ∑〈P_i,jφ⁽ⁿ⁾(XP_i,j, YP_i,j)P_i,jξ, η〉 is estimated by

using Corollary 3.2 and the orthogonality of {P_i,j} for i ≥ 1. The remaining terms all contain commutators and thus vanish in the limit, by Theorem 3.3, leaving

|〈φ⁽ⁿ⁾(X, Y)ξ, η〉|≤2²∥φ∥∥X∥∥Y∥, X, Y∈𝕄_n(ℳ).

The inequality ∥φ∥_cb ≤ 2²∥φ∥ follows immediately, because the right-hand side of Eq. 5.4 is independent of n.

We have already remarked at the end of Section 2 that, in determining cohomology groups of von Neumann algebras, it suffices to restrict to normal ℛ-multimodular cocycles and prove that they are completely bounded. When ℳ has a separable predual, the following theorem is a consequence of Theorem 5.1. The general case is then deduced by showing that any type II₁ factor with property Γ is a union of factors with this property, each of which has a separable predual.

Theorem 5.2.

Let ℳ ⊆ B(H) be a typeII₁ factor with property Γ. Then, for n ≥ 1,

As we have noted previously, the cases n = 1, 2 of this theorem are already known (5, 6, 13, 16).