The Hochschild cohomology problem for von Neumann algebras

Abstract

In 1967, when Kadison and Ringrose began the development of continuous cohomology theory for operator algebras, they conjectured that the cohomology groups Hⁿ(M, M), n ≥ 1, for a von Neumann algebra M, should all be zero. This conjecture, which has important structural implications for von Neumann algebras, has been solved affirmatively in the type I, II_∞, and III cases, leaving open only the type II₁ case. In this paper, we describe a positive solution when M is type II₁ and has a Cartan subalgebra and a separable predual.

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 (1–4) by Johnson, Kadison, and Ringrose. 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_cbⁿ(M, M). Christensen and Sinclair (7) used the structure theory of completely bounded multilinear maps to show that H_cbⁿ(M, M) = 0 for all von Neumann algebras (see chapter 4 of ref. 8). These authors and Effros (9) 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.

There have been some partial results in this direction, mainly concerned with the lower order groups. The vanishing of H²(M, M) was proved by Christensen and Sinclair for the type II₁ factors with property Γ (see chapter 6 of ref. 8), while the same conclusion was reached for a type II₁ von Neumann algebra with a Cartan subalgebra in ref. 10 (for n = 2) and ref. 11 (for n = 3 and a separable predual). The von Neumann algebras having Cartan subalgebras form a rich class (12), but this class does not contain the von Neumann algebra arising from the free group on two generators (13). The main result of this paper is that Hⁿ(M, M) = 0, n ≥ 1, for type II₁ von Neumann algebras with a Cartan subalgebra and separable predual, generalizing the results of refs. 10 and 11. Recent approaches to cohomology (8) have focused on proving that the relevant cocycles are completely bounded as multilinear maps, and this was successful in refs. 10 and 11 for the lower order groups. The principal idea of the present paper is to recognize that we need only establish complete boundedness in the last variable, a much weaker requirement.

In the second section of the paper, we review the basic definitions of cohomology theory, and we include a brief discussion of completely bounded maps. Various forms of averaging over amenable groups play a fundamental and continuing role in the theory, so we have taken the opportunity to recall the most important aspects in the third section. Theorem 3.1 shows the equivalence of several cohomology groups, Theorems 3.2 and 3.3 present useful inequalities based on the Pisier–Haagerup–Grothendieck inequality, and Theorem 3.4 concerns the existence of a projection of completely bounded maps onto the subspace of right module maps which subsequently produces cobounding maps. The last section gives a sketch of our main result, and we indicate how the previously quoted theorems can be combined with some important work of Popa (14, 15) to establish that Hⁿ(M, M) = 0 when M has a Cartan subalgebra and a separable predual. Complete details will appear elsewhere.

We refer the reader to ref. 16 for an early survey of cohomology theory, and to a later account in ref. 8 that contains all the necessary background material for this paper, as well as a discussion of applications.

2. Preliminaries

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

2.1

This supremum then defines the completely bounded norm ∥φ∥_cb. The spaces Row_k(A) and Col_k(A) are, respectively, rows and columns of length k with entries from A, and are obviously identified with subspaces of M_k(A). Then φ: A → B(H) is said to be row bounded if the following quantity (which then defines the row bounded norm) is finite:

2.2

There is a substantial literature on completely bounded maps (see ref. 8 and the references therein), but row bounded maps are much less studied. Nevertheless, they will play a crucial role subsequently. We note that the inequalities

2.3

are immediate from the definitions.

Now let Aⁿ denote the n-fold Cartesian product of copies of A. An n-linear map φ: Aⁿ → B(H) may be lifted to an n-linear map φ_k: M_k(A)ⁿ → M_k(B(H)), k ≥ 1. For clarity we take n = 2 because this case contains the essential ideas. For matrices X = (x_ij), Y = (y_ij) ∈ M_k(A), the (i, j)-entry of φ_k(X, Y) ∈ M_k(B(H)) is defined to be Σ_r=1^k 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 (8, 11).

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

2.4

2.5

2.6

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

We recall from refs. 1 and 8 the basic definitions of Hochschild cohomology theory. Let M be a von Neumann algebra (or C*-algebra) and denote by Lⁿ(M, X) the space of n-linear bounded maps φ: Mⁿ → X into a Banach M-bimodule X. The coboundary operator ∂: Lⁿ(M, X) → Lⁿ⁺¹ (M, X) is defined by

2.7

for x₁, … , x_n+1 ∈ M. Then φ is an n-cocycle if ∂φ = 0, while φ is an n-coboundary if φ = ∂ψ for some ψ ∈ Lⁿ⁻¹(M, X). A short algebraic calculation shows that ∂∂ = 0, and thus coboundaries are cocycles. The cohomology group Hⁿ(M, X) is then defined to be the space of n-cocycles modulo the space of n-coboundaries (for n ≥ 2). For n = 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_wⁿ(M, X)), R-multimodularity (Hⁿ(M, X, :/R)), complete boundedness (H_cbⁿ(M, X)), or any combination of these. The interplay between these various cohomology theories gives an important tool for the determination of Hⁿ(M, X) (see Theorem 3.1).

3. Averaging of Maps

One of the earliest and most fruitful techniques in cohomology theory is to replace a given cocycle with an equivalent one that has several desirable properties. This is often achieved by averaging over a suitable amenable group G of unitary operators in the von Neumann algebra M, using an invariant mean β. If φ is a bounded n-cocycle, we may define a bounded (n − 1)-linear map α by

3.1

where the action of β is denoted by integration. The invariance of β yields

3.2

whenever x₁ ∈ G, and thus whenever x₁ ∈ B, the C*-algebra generated by G. Further applications of unitary averaging lead to the conclusion that φ is equivalent to an n-cocycle ψ with the property that

3.3

when at least one of the variables is in B. As a consequence, such a ψ is B-multimodular. We use n = 2 to illustrate this point: the cocycle equation

3.4

b ∈ B, x, y ∈ M, reduces to

3.5

because the last two terms in 3.4 are 0.

Now let A be a C*-algebra acting on a Hilbert space H, and let M be its ultraweak closure. There is a central projection z ∈ A** such that A**z and M are isomorphic and, employing this isomorphism, M becomes a dual normal A**-bimodule. By using second dual techniques, it is then possible to replace a cocycle φ: Aⁿ → M by an equivalent cocycle ψ: Aⁿ → M that is separately ultraweakly-weak* continuous in each variable. Moreover, such a cocycle extends to a cocycle ψ̄: Mⁿ → M that is separately normal in each variable. In the particular case when A = M, we conclude that each cocycle is equivalent to one that is separately normal in each variable.

Now consider a hyperfinite subalgebra R of a von Neumann algebra M. Because R is the ultraweak closure of an increasing family of finite dimensional subalgebras, it is possible to find an amenable group G of unitary operators in R that generates a C*-algebra B whose ultraweak closure is R. The averaging and second dual techniques can be applied in tandem to replace a cocycle φ: Mⁿ → M by an equivalent cocycle ψ that is both B-multimodular and separately normal in each variable. Of course, ψ is then R-multimodular by ultraweak continuity.

All the results discussed above are due to refs. 2–4 and may also be found in chapter 3 of ref. 8. They are summarized by the following theorem, which is undoubtedly the most important for cohomological calculations.

Theorem 3.1. (See ref. 8.) Let M ⊆ B(H) be a von Neumann algebra with a hyperfinite von Neumann subalgebra R and an ultraweakly dense C*-subalgebra A. Then the cohomology groups

are pairwise isomorphic, for each n ≥ 1.

As will be seen subsequently, this theorem gives several options for the determination of Hⁿ(M, M).

A second application of averaging over an amenable unitary group leads to a very useful inequality for bilinear maps. Grothendieck’s inequality for abelian C*-algebras was extended to C*-algebras with the approximation property by Pisier (17), and then to all C*-algebras by Haagerup (18). The latter formulation, appropriate for von Neumann algebras, is as follows. Given a bounded bilinear form θ: M × M → C, separately normal in each variable, there exist four states f₁, f₂, g₁, g₂ ∈ M_∗, the predual of M, such that, for x, y ∈ M,

3.6

If θ has the additional property of being inner R-modular, in the sense that

3.7

for x, y ∈ M, r ∈ R, where R is a hyperfinite von Neumann subalgebra whose relative commutant R′ ∩ M is the center Z of M, then we may fix an amenable group G of unitary operators in R and average in 3.6. The resulting inequality, when M is type II₁, is

3.8

leading to the existence of two states F, G ∈ M_∗ such that

3.9

The x*x and yy* terms have disappeared from 3.6 because, for m ∈ M,

3.10

where E is the tracial conditional expectation of M onto Z. The following result from ref. 11, to which we refer for details, is a straightforward deduction from 3.9.

Theorem 3.2. Let M be a type II₁ von Neumann algebra with a hyperfinite subalgebra R satisfying R′ ∩ M = Z. If ψ: M × M → M is inner R-modular and separately normal in each variable, then

3.11

for x_i, y_i ∈ M.

This theorem applies to a normal right R-module map φ: M → M by considering the inner R-modular bilinear map

3.12

The next result follows from 3.11 by taking y_i to be x^*_i. The crucial point is the equivalence of the operator norm and the row bounded norm, which is immediate from 3.13.

Theorem 3.3. Let R ⊆ M satisfy the hypotheses of Theorem 3.2, and let φ: M → M be a bounded normal right R-module map. Then

• (i) for x_i ∈ M, 1 ≤ i ≤ n,

  3.13

• (ii) φ is row bounded, and

  3.14

  A nonhyperfinite type II₁ von Neumann algebra will not be generated by an amenable group of unitaries, but nevertheless there is a notion of averaging that works in this situation, but only on completely bounded maps. The idea is to replace an average ∫_Gφ(xu)u*dβ(u) by an ultraweak limit of maps of the form

  3.15

  where m_i ∈ M and Σ_i=1^∞ m_im^*_i = 1, to obtain right M-module maps from bounded maps. The next result, taken from ref. 19 but tailored to our needs, relies on the minimal invariant set technique pioneered by Kadison (5) and Sakai (6).

• Theorem 3.4. There exists a contractive projection ρ from L_cb(M, M) onto the subspace L_cb(M, M)_M of right M-module maps with the following properties:

• (i) There exists a net of maps ρ_α: L_cb(M, M) → L_cb(M, M), each of the form 3.15, such that

  3.16

  ultraweakly for φ ∈ L_cb(M, M) and x ∈ M.

• (ii) For all φ ∈ L_cb(M, M),

  3.17

• (iii) If a ∈ M is fixed and φ_a ∈ L_cb(M, M) is defined, for each φ ∈ L_cb(M, M), by

  3.18

  then

  3.19

4. The Main Result

Throughout this section, M is a type II₁ von Neumann algebra with a separable predual, center Z, and a faithful trace tr. We assume that M is represented on L²(M, tr), in which case there is a conjugate linear isometry J: [x] ↦ [x*] so that JMJ = M′, the commutant of M. We also assume that M has a Cartan subalgebra A. This is a maximal abelian self-adjoint subalgebra of M whose unitary normalizer

4.1

generates M as a von Neumann algebra (12).

Theorem 4.1. With the above assumptions on M,

4.2

Sketch of Proof:

From Theorem 3.1, it suffices to show that H_wⁿ(C*(U), M) = 0, n ≥ 1. Because the case n = 1 is a special case of the Kadison–Sakai derivation theorem (5, 6), we make the further restriction of n ≥ 2. By refs. 14 and 20 there is a hyperfinite subalgebra R such that A ⊆ R ⊆ M and R′ ∩ M = Z. The averaging techniques of the previous section allow us to consider a cocycle φ: Mⁿ → M which is R-multimodular and separately normal in each variable, and it is required to show that the restriction of φ to C*(U)ⁿ is a coboundary.

The first step is to prove that, for fixed u₁, … , u_n−1 ∈ U, the map

4.3

is completely bounded in the x-variable. The Cartan subalgebra and R-multimodularity hypotheses are used to establish this. Given X ∈ M_k(M), R₀ ∈ Row_k(A), C ∈ Col_k(A), all of norm 1, we may find rows R₁, … , R_n−1 ∈ Row_k(A), again of norm 1, so that

4.4

because u₁, … , u_n−1 ∈ U. Then elements of A can be passed through the variables of φ, as discovered by Rădulescu (21), giving

4.5

Thus

4.6

By ref. 15, A and JAJ generate a maximal abelian subalgebra of B(L²(M, tr)), and this is sufficient to conclude that the supremum in 4.6 is ∥μ_k(X)∥ (see theorem 2.1 of ref. 22). Thus ∥μ∥_cb ≤ ∥φ∥. It is then clear that, for any fixed y₁, … , y_n−1 ∈ Alg(U),

4.7

is completely bounded in the x-variable, and is a normal right R-module map. For y₁, … , y_n ∈ Alg U, x ∈ M, each term in the cocycle equation

4.8

is a completely bounded map in x, so the projection ρ of Theorem 3.4 may be applied. Because L_cb(M, M)_M consists of maps of the form x ↦ m₀x for a fixed m₀ ∈ M, we obtain α: Alg(U)ⁿ⁻¹ → M so that 4.8 becomes

4.9

and the estimate ∥α∥ ≤ ∥φ∥ follows from Theorem 3.3 (ii) and Theorem 3.4 (ii). Eq. 4.9, with x = 1, gives φ = ∂((−1)ⁿα) on Alg(U)ⁿ, and the same conclusion holds on C*(U)ⁿ by continuity. This establishes that the restriction of φ to C*(U) is a coboundary, as required.