Generator problem for certain property T factors
See allHide authors and affiliations

Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, and approved November 8, 2001 (received for review November 5, 2001)
Abstract
We show that the property T factors associated with groups SL_{n}(Z), n ≥ 3 are generated by two selfadjoint elements, one of which has an arbitrarily small support. This answers a question of Dan Voiculescu.
The generator problem for von Neumann algebras is one of the longstanding open questions in Operator Algebras. One of the first results on the problem is due to von Neumann, who showed that abelian von Neumann algebras acting on a separable Hilbert space is generated by a selfadjoint element. Let ℋ denote a separable Hilbert space (finite or infinite dimensional) and ℬ(ℋ) the algebra of all bounded linear operators on ℋ. It is well known that ℬ(ℋ) can be generated by a unitary operator and a rank one projection, by two selfadjoint operators (equivalently, one nonselfadjoint element), or by two unitary operators with order 2 and 3, respectively. Similar results hold for a large class of subalgebras of ℬ(ℋ). For example, all properly infinite von Neumann subalgebras of ℬ(ℋ) have generators with similar properties (see, e.g., ref. 1). Whether all von Neumann subalgebras of ℬ(ℋ) are generated by two selfadjoint elements remains unsolved. The question has been reduced to a special class of von Neumann algebras, i.e., the factors of type II_{1}. Partial results were obtained by several authors. Among them is the result by Popa (2), who shows that factors with a Cartan subalgebra are singly generated. More recently, Popa and Ge (3) show that the same is true for property Γ factors. In his recently developed theory of free probability and free entropy, Voiculescu introduces a notion of free entropy dimension. It is believed that free entropy dimension of the generators of a factor of type II_{1} is closely related to the minimal number of generators of the factor. After showing that certain property T factors have generators with free entropy dimension less than or equal to one, Voiculescu asks in ref. 4 whether the property T factors associated with groups SL_{n}(Z), n ≥ 3 are generated by two selfadjoint elements, one of which can have an arbitrarily small support (in the sense that its range projection has an arbitrarily small trace). In this note, we give a positive answer to this question. Our analysis and results on these property T factors may be helpful in the study of other questions that remain unsolved for this class of factors (for example, Kadison's similarity problem, vanishing higher cohomology groups, etc.).
In the following, we shall introduce some definitions concerning von Neumann algebras and prove that the factors considered in ref. 5 with “cyclic normalizing” generators can in fact be generated by two selfadjoint elements and another selfadjoint element with an arbitrarily small support. By proving some technical results on groups SL_{n}(Z), we show that their associated group von Neumann algebras can be generated by any Haar unitary operator and another selfadjoint element with an arbitrarily small support. Finally, we introduce a new notion, “generating length,” in connection with the generator problem, and obtain a basic result.
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 II_{1}. 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 l^{2}(G) and L_{g} the left translation by g^{−1} of functions in l^{2}(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 II_{1} 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 II_{1}. 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 G_{2} is the subgroup of SL_{2}(Z) generated by g_{12} and g_{21}. 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 II_{1}and is generated by Haar unitary operators U_{1}, U_{2}, … , U_{m}, … . Assume that U^{*}_{j+1}U_{j}U_{j+1}belongs to the von Neumann subalgebra generated by U_{1}, … , 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 U_{1}, … , 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/n^{i} 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_{i}FW^{*}_{i} = E for j = 1, … , n^{i}. From our assumption, we have U^{*}_{i+1}FU_{i+1} ∈ ℛ_{i}. Again, there is a unitary element V_{i} in ℛ_{i} such that V_{i}EV^{*}_{i} = U^{*}_{i+1}FU_{i+1} = U^{*}_{i+1}W^{*}_{i}EW_{i}U_{i+1} for j = 1, … , n^{i}. Thus W_{i}U_{i+1}V_{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_{i}U_{i+1}V_{i} has at most n^{i} nonzero diagonal entries. Now we define, inductively, elements T_{i} in ℳ. When i = 1, choose integer m_{1} so that (m_{1}(m_{1} − 1)/2) ≥ n and m_{1} ≤ + 1. Thus there is an injective map from {1, 2, … , n} to the set {(j, k) : 1 ≤ j ≤ k ≤ m_{1}}. We shall use (α_{1}(j), β_{1}(j)) to denote the image of j under this map. Let Then we know that T_{1} is strictly upper triangular with respect to matrix units {E} and W_{1}U_{2}V_{1} is in the algebra generated by T_{1} and M_{n}(C). The support Q_{1} of T_{1} is majorized by the sum of the projections E, … , E. Thus τ(Q_{1}) ≤ m_{1}/n ≤ ( + 1/n). Similarly we define T_{2} for W_{2}U_{3}V_{2} so that its support Q_{2} is majorized by the sum of some diagonal projections in M_{n2}(C), which are orthogonal to Q_{1} and τ(Q_{2}) ≤ ( + 1/n^{2}). Continuing this process, we shall have T_{i}, in a strictly upper triangular form with respect to M_{ni}(C), such that W_{i}U_{i+1}V_{i} lies in the algebra generated by M_{ni}(C) and T_{i}. The support Q_{i} of T_{i} is orthogonal to Q_{1} + ⋯ + Q_{i−1} and τ(Q_{i}) ≤ ( + 1/n^{i}). Now let T = (T_{1}/∥T_{1}∥) + (T_{2}/∥T_{2}∥) + ⋯ (or the strong operator limit of (T_{1}/∥T_{1}∥) + ⋯ + (T_{i}/∥T_{i}∥), and B = T + T*. Then it is easy to show that ℳ is generated by ℛ, U_{1}, and B. If Q is the support of B, then τ(Q) ≤ ∑ ( + 1/n^{i}), which can be arbitrarily small when n is large. Now ℳ is generated by ℛ, U_{1}, and B. Replacing U_{1} by a selfadjoint element, we have our desired result.■
Remarks: (i) The factorial assumption on ℳ is essential. For example, ℒ_{Fm}ℒ_{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 II_{1} 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_{jk}e_{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 {m^{2}a_{st}e_{st} : m ∈ Z} is a finite set. Thus a_{st} = 0 for all 1 ≤ s, t ≤ n − 1 and s ≠ t. Similarly, ∑a_{tk}e_{sk} − ∑a_{js}e_{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_{gjs}and L_{gjt}are commuting independent elements. The same is true for g_{sj}and g_{tj}.
Before we state our main theorem, we prove another technical result.
Lemma 4.Let ℳ be a factor of type II_{1}with 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_{m}with τ(P) ≤ ɛ. Then ℳ is generated by 𝒩_{m}and a selfadjoint element Twhose support is majorized by a projection Qin 𝒩_{m}such that τ(Q) ≤ ɛ + ( + 1/m).
Proof: Because V is a Haar unitary element, there are projections F_{1}, … , 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 E_{11}, … , E_{m1}_{m1} for some m_{1} (and (m_{1}/m) ≤ ɛ). Because 𝒩 is a factor, there is a unitary operator W in 𝒩 such that F_{j} = W*E_{jj}W, j = 1, … , m. Let Q_{j} = U*F_{j}U, for j = 1, … , m. Then, from our assumption, we know that Q_{j} ∈ 𝒩. Again, there is a unitary element W_{1} in 𝒩 such that Q_{j} = W^{*}_{1}E_{jj}W_{1}. Now we have that W^{*}_{1}E_{jj}W_{1} = U*F_{j}U = U*W*E_{jj}WU. Thus i.e., WUW^{*}_{1} = ∑E_{jj}WUW^{*}_{1}E_{jj}. We use a trick similar to the one we used in the proof of Proposition 1 and choose m_{2} so that (m_{2}(m_{2} − 1)/2) ≥ m and m_{2} ≤ ( + 1/m). Consider a matrix subalgebra of 𝒩_{m} with matrix units E_{jk}forj, k = m_{1} + 1, … , m_{1} + m_{2}. 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 = m_{1} + 1, … , m_{1} + m_{2}}. We write σ(j) = (σ_{1}(j), σ_{2}(j)). Now let and T′ = T_{1} + 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 g_{21}, g_{31}, g_{32} and g_{12}. The H is a subgroup of G_{3} containing G_{2}. From Lemma 2, we know that ℒ_{H} is a subfactor of ℒ_{Gn}. For any large integer m_{0}, choose projections P_{1}, … , P_{m0} in the abelian von Neumann algebra generated by L_{g31} and Q_{1}, … , Q_{m0} in the one generated by L_{g32} such that τ(P_{j}) = τ(Q_{j}) = 1/m_{0} and ∑P_{j} = ∑Q_{j} = I. From Lemma 3, we know that τ(P_{j}Q_{k}) = 1/m and {P_{j}Q_{k} : j, k = 1, … , m_{0}} 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 E_{11}, … , E_{mm} coincide with the projections P_{1}Q_{1}, … , P_{1}Q_{m0}, … , P_{m0}Q_{1}, … , P_{m0}Q_{m0}. Because g_{21} commutes with g_{31} and g_{12} commutes with g_{32}, 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 m_{1} so that m_{1}(m_{1} − 1)/2 ≥ 2m + 2m and m_{1} ≤ 2m_{0} + 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 E_{11} + ⋯ + E_{m1}_{m1}, such that ℒ_{H} is generated by 𝒩_{m} and T. When m_{0} is large, we have that the trace of the support of T is less than or equal to m_{1}/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 E_{1} and a (shift) unitary matrix U_{n} (of finite order). One may choose E_{1} so that it is orthogonal to P. Replacing E_{1} 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 II_{1} 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 II_{1}and P is a projection of trace ½. If PℳPis generated by two selfadjoint elements, then κ(ℳ) = 1.
Acknowledgments
We thank Prof. Bingren Li for many helpful discussions. This research was supported in part by a National Science Foundation (USA) CAREER Award.
Footnotes

↵§ To whom reprint requests should be addressed. Email: liming{at}math.unh.edu.

This paper was submitted directly (Track II) to the PNAS office.
 Received November 5, 2001.
 Copyright © 2002, The National Academy of Sciences
References
 ↵
 Wogen W
 ↵
 Popa S
 ↵
 Ge L,
 Popa S
 ↵
 Voiculescu D
 ↵
 Ge L,
 Shen J
 ↵
 ↵
 Kadison R,
 Ringrose J
 ↵
 Kazhdan D
 ↵
 Connes A,
 Jones V
 ↵
 Gaboriau D
 ↵
 Popa S