Definitions and Basic Properties

von Neumann algebras were introduced by von Neumann in ref. 6. They are strong operator topology closed selfadjoint subalgebras of ℬ(ℋ), the algebra of all bounded operators on a Hilbert space ℋ. Factors are von Neumann algebras whose center consists of scalar multiples of the identity. von Neumann showed that von Neumann algebras are direct sums (or direct “integrals”) of factors. Factors are classified into more types by means of a relative dimension function. Finite factors are those for which this dimension function has a finite range. Otherwise, the factor is called infinite. For the generator question, there is an easy answer to all infinite factors: every infinite factor, or more generally, every properly infinite von Neumann algebra with a separable predual (or acting on a separable Hilbert space), is generated by two selfadjoint elements. A similar result holds for finite dimensional factors, i.e., finite dimensional full matrix algebras. Infinite dimensional finite factors are called factors of type II₁. They arise naturally from regular representations of (discrete) groups.

Suppose G is a countable discrete group with unit e. Let ℋ be the Hilbert space l²(G) and L_g the left translation by g⁻¹ of functions in l²(G). Then g → L_g is a unitary representation of G on ℋ. Let ℒ_G be the von Neumann algebra generated by {L_g : g ∈ G}. In general, ℒ_G is a finite von Neumann algebra. It is a factor of type II₁ if and only if each conjugacy class in G (other than that of e) is infinite (in this case, G is called an i.c.c. group). The vector state associated with the characteristic function at e (or any other group element) is a tracial state, denoted by τ. In fact, there is one and only one tracial state on each factor of type II₁. We refer to ref. 7 for basics on von Neumann algebras.

In this note, we will be concerned with von Neumann algebras arising from groups SL_n(Z) for n ≥ 3. [This class of groups have Kazhdan's property T (8). Property T for von Neumann algebras was introduced by Connes and Jones (9).] When n is even, I and −I lie in the center of SL_n(Z). We consider the group PSL_n(Z) (≅ SL_n(Z)/{I, −I}) in place of SL_n(Z) for the even case. For simplicity of notation, we shall denote SL_n(Z), when n is odd, or PSL_n(Z), when n is even, by G_n. From ref. 10, we know that G_n (n ≥ 3) is generated by g_jk = I + e_jk for j ≠ k and 1 ≤ j, k ≤ n, where I is the identity matrix and each e_jk is the matrix unit with (j, k)-entry equal to one and zero elsewhere. [Here G₂ is the subgroup of SL₂(Z) generated by g₁₂ and g₂₁. It is used only in the proof of Lemma 2.] It is easy to show that each g_jk gives rise to a Haar unitary element in ℒ_SLn(Z) (i.e., the spectral measure of L_gjk given by the trace τ is the Haar measure on the unit circle, the spectrum of L_gjk). It is also easy to verify that g_jk commutes with g_st whenever j = s or k = t.

Now we prove a general result on generators of finite factors.

Proposition 1.Suppose ℳ is a factor of type II₁and is generated by Haar unitary operators U₁, U₂, … , U_m, … . Assume that U^*_j+1U_jU_j+1belongs to the von Neumann subalgebra generated by U₁, … , U_j. Then ℳ is generated by a hyperfinite subfactor and two selfadjoint operators.

Proof: First we choose a hyperfinite subfactor ℛ of ℳ such that ℛ′ ∩ ℳ = CI (see ref. 11). Let ℛ_j be the von Neumann subalgebra of ℳ generated by ℛ and U₁, … , U_j, for j ≥ 1. Then ℛ′_j ∩ ℳ = CI and ℛ_j is a factor for every j. For any given large integer n, we can find unital embeddings M_n(C) ⊂ M_n²(C) ⊂ ⋯ ⊂ ℛ. Let {E} be a matrix unit system for M_nⁱ(C) ⊂ ℛ, i = 1, 2, … . From the assumption that U_i is a Haar unitary element, we choose projections F, … , F in the abelian von Neumann algebra generated by U_i such that τ(F) = 1/nⁱ and ∑F = I. Because ℛ_i (containing U_i) is a factor, we can find a unitary element W_i in ℛ_i such that W_iFW^*_i = E for j = 1, … , nⁱ. From our assumption, we have U^*_i+1FU_i+1 ∈ ℛ_i. Again, there is a unitary element V_i in ℛ_i such that V_iEV^*_i = U^*_i+1FU_i+1 = U^*_i+1W^*_iEW_iU_i+1 for j = 1, … , nⁱ. Thus W_iU_i+1V_i commutes with E, … , E and where i = 1, 2, … . Given any i, with respect to the matrix subalgebra M_nⁱ(C) of ℛ, W_iU_i+1V_i has at most nⁱ nonzero diagonal entries. Now we define, inductively, elements T_i in ℳ. When i = 1, choose integer m₁ so that (m₁(m₁ − 1)/2) ≥ n and m₁ ≤ + 1. Thus there is an injective map from {1, 2, … , n} to the set {(j, k) : 1 ≤ j ≤ k ≤ m₁}. We shall use (α₁(j), β₁(j)) to denote the image of j under this map. Let Then we know that T₁ is strictly upper triangular with respect to matrix units {E} and W₁U₂V₁ is in the algebra generated by T₁ and M_n(C). The support Q₁ of T₁ is majorized by the sum of the projections E, … , E. Thus τ(Q₁) ≤ m₁/n ≤ ( + 1/n). Similarly we define T₂ for W₂U₃V₂ so that its support Q₂ is majorized by the sum of some diagonal projections in M_n²(C), which are orthogonal to Q₁ and τ(Q₂) ≤ ( + 1/n²). Continuing this process, we shall have T_i, in a strictly upper triangular form with respect to M_nⁱ(C), such that W_iU_i+1V_i lies in the algebra generated by M_nⁱ(C) and T_i. The support Q_i of T_i is orthogonal to Q₁ + ⋯ + Q_i−1 and τ(Q_i) ≤ ( + 1/nⁱ). Now let T = (T₁/∥T₁∥) + (T₂/∥T₂∥) + ⋯ (or the strong operator limit of (T₁/∥T₁∥) + ⋯ + (T_i/∥T_i∥), and B = T + T*. Then it is easy to show that ℳ is generated by ℛ, U₁, and B. If Q is the support of B, then τ(Q) ≤ ∑ ( + 1/nⁱ), which can be arbitrarily small when n is large. Now ℳ is generated by ℛ, U₁, and B. Replacing U₁ by a selfadjoint element, we have our desired result.■

Remarks: (i) The factorial assumption on ℳ is essential. For example, ℒ_Fmℒ_Z satisfies the assumptions of the proposition. But we do not know whether it is generated by four selfadjoint elements (for m > 4).

(ii) It is well known that the hyperfinite II₁ factor is generated by a small projection and a selfadjoint element. If we choose this projection so that it is orthogonal to the support of B, then ℳ as given in the proposition is generated by two selfadjoint elements and another with an arbitrarily small support.

Next, we shall show that ℒ_SLn(Z), n ≥ 3 is generated by two selfadjoint elements, one of which has an arbitrarily small support.

Generators of ℒ_SLn(Z) for n ≥ 3

Recall that SL_n(Z), when n is odd, or PSL_n(Z), when n is even, is denoted by G_n. From the above proposition and Remark ii, we see that ℒ_Gn is generated by three selfadjoint elements, one of which has an arbitrarily small support. Now we are going to reduce the number three to two. We choose, again, the generators {g_jk : 1 ≤ j, k ≤ n, j ≠ k} for G_n defined above. Their corresponding unitary operators are denoted by L_gjk. We shall embed G_n into G_n+1 in a natural way. For g_jk = I + e_jk in G_n, we regard this g_jk as an element in G_n+1 by putting 1 at its (n + 1, n + 1) entry and zeros on the rest of the (n + 1)th row and (n + 1)th column. It is well known that G_n is an i.c.c. group. Here we prove that many of its subgroups are i.c.c. groups.

Lemma 2.If Gis a subgroup of G_n, for n ≥ 3, containing G_n−1, then Gis an i.c.c. group.

Proof: We shall show that, for any element g in G, if the conjugacy class of g by the subgroup G_n−1 is finite, then g is the identity matrix.

Suppose g = ∑a_jke_jk, where a_jk ∈ Z and {e_jk} is the matrix unit system. For g_st = I + e_st in G_n−1, g = I + me_st for any m in Z, where 1 ≤ s, t ≤ n − 1 and s ≠ t. Now, From our assumption that {ggg : m ∈ Z} is a finite set, we know that {m²a_ste_st : m ∈ Z} is a finite set. Thus a_st = 0 for all 1 ≤ s, t ≤ n − 1 and s ≠ t. Similarly, ∑a_tke_sk − ∑a_jse_jt = 0. Thus, from a_st = 0 for s ≠ t, we have This implies that a_ss = a_tt and a_ns = a_tn = 0. Therefore g must be the unit I in G_n.■

The following result is easy to check. We omit its proof here.

Lemma 3.The abelian group generated by g_js and g_jt, two of the generating elements of G_n, is isomorphic to Z × Z, i.e., the Haar unitary operators L_gjsand L_gjtare commuting independent elements. The same is true for g_sjand g_tj.

Before we state our main theorem, we prove another technical result.

Lemma 4.Let ℳ be a factor of type II₁with trace τ. Suppose ℳ is generated by a subfactor 𝒩 and a unitary operator U in ℳ, and there is a Haar unitary operator V in 𝒩 such that U*VU ∈ 𝒩. Assume 𝒩 is generated by a subalgebra 𝒩_m, isomorphic to M_m(C), and a selfadjoint element S whose support is majorized by a projection Pin 𝒩_mwith τ(P) ≤ ɛ. Then ℳ is generated by 𝒩_mand a selfadjoint element Twhose support is majorized by a projection Qin 𝒩_msuch that τ(Q) ≤ ɛ + ( + 1/m).

Proof: Because V is a Haar unitary element, there are projections F₁, … , F_m in the abelian von Neumann algebra generated by V such that ∑F_j = I and τ(F_j) = 1/m. Choose {E_jk} in 𝒩_m, which correspond to a matrix unit system in M_m(C) such that P is the sum of diagonal projections E₁₁, … , E_m1m1 for some m₁ (and (m₁/m) ≤ ɛ). Because 𝒩 is a factor, there is a unitary operator W in 𝒩 such that F_j = W*E_jjW, j = 1, … , m. Let Q_j = U*F_jU, for j = 1, … , m. Then, from our assumption, we know that Q_j ∈ 𝒩. Again, there is a unitary element W₁ in 𝒩 such that Q_j = W^*₁E_jjW₁. Now we have that W^*₁E_jjW₁ = U*F_jU = U*W*E_jjWU. Thus i.e., WUW^*₁ = ∑E_jjWUW^*₁E_jj. We use a trick similar to the one we used in the proof of Proposition 1 and choose m₂ so that (m₂(m₂ − 1)/2) ≥ m and m₂ ≤ ( + 1/m). Consider a matrix subalgebra of 𝒩_m with matrix units E_jkforj, k = m₁ + 1, … , m₁ + m₂. Now there are more than m entries in the upper triangular subalgebra of this matrix subalgebra. We can define an injective map σ : {1, … , m} → {(j, k) : j < k, j, k = m₁ + 1, … , m₁ + m₂}. We write σ(j) = (σ₁(j), σ₂(j)). Now let and T′ = T₁ + T^*₁. Then T′ is selfadjoint and its support is majorized by E_jj. Let T = S + λP + T′. By choosing an appropriate real constant λ, we have that both T′ and S + λP lie in the von Neumann algebra generated by T. It is easy to see that 𝒩_m and T generate ℳ. It is easy to check the estimate for the trace of the support of T. This completes the proof.■

The following is the main result of this article.

Theorem 5.The factor ℒ_Gn, for n ≥ 3, is generated by two selfadjoint elements, one of which can have an arbitrarily small support.

Proof: We shall use the generators g_jk, 1 ≤ j, k ≤ n and j ≠ k, for the group G_n. Let H be the subgroup of G_n generated by g₂₁, g₃₁, g₃₂ and g₁₂. The H is a subgroup of G₃ containing G₂. From Lemma 2, we know that ℒ_H is a subfactor of ℒ_Gn. For any large integer m₀, choose projections P₁, … , P_m0 in the abelian von Neumann algebra generated by L_g31 and Q₁, … , Q_m0 in the one generated by L_g32 such that τ(P_j) = τ(Q_j) = 1/m₀ and ∑P_j = ∑Q_j = I. From Lemma 3, we know that τ(P_jQ_k) = 1/m and {P_jQ_k : j, k = 1, … , m₀} is a set of m mutually orthogonal projections in ℒ_H. Let m = m and choose a full matrix subalgebra 𝒩_m in ℒ_H and a matrix unit system {E_jj} so that the diagonal projections E₁₁, … , E_mm coincide with the projections P₁Q₁, … , P₁Q_m0, … , P_m0Q₁, … , P_m0Q_m0. Because g₂₁ commutes with g₃₁ and g₁₂ commutes with g₃₂, we have Thus with respect to the matrix unit system {E_jk}, the elements L_g21, L_g31, L_g32 and L_g12 have only 2m + 2m nonzero entries. Choose m₁ so that m₁(m₁ − 1)/2 ≥ 2m + 2m and m₁ ≤ 2m₀ + 1. By the same matrix trick used in the proof of Lemma 4, there is a selfadjoint element T in ℒ_H whose support is majorized by E₁₁ + ⋯ + E_m1m1, such that ℒ_H is generated by 𝒩_m and T. When m₀ is large, we have that the trace of the support of T is less than or equal to m₁/m, which can be arbitrarily small. Next, we may consider the subfactor of ℒ_Gn generated by ℒ_H and L_g23. Continuing this process and by repeated use of Lemmas 2 and 4, we have that ℒ_Gn is generated by a matrix subalgebra M_mn(C) and a selfadjoint element T_n with an arbitrarily small support P, which is dominated by some diagonal projections in M_mn(C). It is easy to see that M_mn(C) is generated by a rank one projection E₁ and a (shift) unitary matrix U_n (of finite order). One may choose E₁ so that it is orthogonal to P. Replacing E₁ and T_n by one selfadjoint element S, we have that ℒ_Gn is generated by S and U_n, where S can have an arbitrarily small support and U_n is a unitary element with a finite order. This completes the proof of our theorem.■

Note that when n ≥ 4 even, ℒ_SLn(Z) ≅ ℒ_PSLn(Z) ⊕ ℒ_PSLn(Z). Thus the same result holds for ℒ_SLn(Z) for all n ≥ 3.

Final Comments

As pointed out by Voiculescu (4), free entropy dimension should be related to the number of generators for a factor. The “number” should be measured by its free entropy dimension, not by the length of the support of the generator. To illustrate this point, we shall introduce a notion of “generating length” and show that a large class of factors have generating length equal to one.

Definition 6: Suppose ℳ is a factor of type II₁ with trace τ or, in general, a von Neumann algebra with a state, and 𝒩 a subalgebra (or subset) of ℳ. The generating length κ_𝒩(ℳ, τ) of ℳ over 𝒩 is given as follows: When 𝒩 is CI or empty set, we shall use κ(ℳ, τ) or simply κ(ℳ) instead of κ_𝒩(ℳ, τ). The advantage of this generating length is that it tells more information, in some sense, than the number of selfadjoint generators of a von Neumann algebra. The detailed study of this invariant will appear elsewhere. Using a matrix trick, one easily proves the following result.

Proposition 7.Suppose ℳ is a factor of type II₁and P is a projection of trace ½. If PℳPis generated by two selfadjoint elements, then κ(ℳ) = 1.