Generator problem for certain property T factors
Liming Ge†‡§ and Junhao Shen¶
†Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China; ‡Department of Mathematics, University of New Hampshire, Durham, NH 03824; and ¶Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104
Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, and approved November 8, 2001 (received for review November 5, 2001)
We show that the property T factors associated with groups SLn(Z), that von Neumann algebras are direct sums (or direct ‘‘inte- n > 3 are generated by two selfadjoint elements, one of which grals’’) of factors. Factors are classified into more types by means has an arbitrarily small support. This answers a question of Dan of a relative dimension function. Finite factors are those for Voiculescu. which this dimension function has a finite range. Otherwise, the factor is called infinite. For the generator question, there is an he generator problem for von Neumann algebras is one of easy answer to all infinite factors: every infinite factor, or more Tthe longstanding open questions in Operator Algebras. One generally, every properly infinite von Neumann algebra with a of the first results on the problem is due to von Neumann, who separable predual (or acting on a separable Hilbert space), is showed that abelian von Neumann algebras acting on a separable generated by two selfadjoint elements. A similar result holds for Hilbert space is generated by a selfadjoint element. Let H denote finite dimensional factors, i.e., finite dimensional full matrix a separable Hilbert space (finite or infinite dimensional) and algebras. Infinite dimensional finite factors are called factors of B(H) the algebra of all bounded linear operators on H. It is well type II1. They arise naturally from regular representations of known that B(H) can be generated by a unitary operator and a (discrete) groups. rank one projection, by two selfadjoint operators (equivalently, Suppose G is a countable discrete group with unit e. Let H be 2 Ϫ1 one nonselfadjoint element), or by two unitary operators with the Hilbert space l (G) and Lg the left translation by g of 2 3 order 2 and 3, respectively. Similar results hold for a large class functions in l (G). Then g Lg is a unitary representation of subalgebras of B(H). For example, all properly infinite von of G on H. Let LG be the von Neumann algebra generated by ʦ Neumann subalgebras of B(H) have generators with similar {Lg : g G}. In general, LG is a finite von Neumann algebra. properties (see, e.g., ref. 1). Whether all von Neumann subal- It is a factor of type II1 if and only if each conjugacy class in G gebras of B(H) are generated by two selfadjoint elements (other than that of e) is infinite (in this case, G is called an i.c.c. remains unsolved. The question has been reduced to a special group). The vector state associated with the characteristic class of von Neumann algebras, i.e., the factors of type II1. Partial function at e (or any other group element) is a tracial state, results were obtained by several authors. Among them is the denoted by . In fact, there is one and only one tracial state on result by Popa (2), who shows that factors with a Cartan each factor of type II1. We refer to ref. 7 for basics on von subalgebra are singly generated. More recently, Popa and Ge (3) Neumann algebras. show that the same is true for property ⌫ factors. In his recently In this note, we will be concerned with von Neumann algebras Ն developed theory of free probability and free entropy, Voi- arising from groups SLn(Z) for n 3. [This class of groups have culescu introduces a notion of free entropy dimension. It is Kazhdan’s property T (8). Property T for von Neumann algebras believed that free entropy dimension of the generators of a factor was introduced by Connes and Jones (9).] When n is even, I and Ϫ of type II1 is closely related to the minimal number of generators I lie in the center of SLn(Z). We consider the group PSLn(Z) Х ͞ Ϫ of the factor. After showing that certain property T factors have ( SLn(Z) {I, I}) in place of SLn(Z) for the even case. For generators with free entropy dimension less than or equal to one, simplicity of notation, we shall denote SLn(Z), when n is odd, or Voiculescu asks in ref. 4 whether the property T factors associ- PSLn(Z), when n is even, by Gn. From ref. 10, we know that Gn Ն Ն ϭ ϩ Յ Յ ated with groups SLn(Z), n 3 are generated by two selfadjoint (n 3) is generated by gjk I ejk for j k and 1 j, k elements, one of which can have an arbitrarily small support (in n, where I is the identity matrix and each ejk is the matrix unit the sense that its range projection has an arbitrarily small trace). with (j, k)-entry equal to one and zero elsewhere. [Here G2 is the In this note, we give a positive answer to this question. Our subgroup of SL2(Z) generated by g12 and g21. It is used only in analysis and results on these property T factors may be helpful the proof of Lemma 2.] It is easy to show that each gjk gives rise in the study of other questions that remain unsolved for this class to a Haar unitary element in LSL (Z) (i.e., the spectral measure n of factors (for example, Kadison’s similarity problem, vanishing of Lgjk given by the trace is the Haar measure on the unit circle, higher cohomology groups, etc.). the spectrum of Lg ). It is also easy to verify that gjk commutes jkϭ ϭ
In the following, we shall introduce some definitions concern- with gst whenever j s or k t. MATHEMATICS ing von Neumann algebras and prove that the factors considered Now we prove a general result on generators of finite factors. in ref. 5 with ‘‘cyclic normalizing’’ generators can in fact be PROPOSITION 1. Suppose M is a factor of type II1 and is generated by two selfadjoint elements and another selfadjoint generated by Haar unitary operators U1, U2,...,Um,....Assume element with an arbitrarily small support. By proving some that U*jϩ1UjUjϩ1 belongs to the von Neumann subalgebra generated technical results on groups SLn(Z), we show that their associated by U1,...,Uj. Then M is generated by a hyperfinite subfactor and group von Neumann algebras can be generated by any Haar two selfadjoint operators. unitary operator and another selfadjoint element with an arbi- Proof: First we choose a hyperfinite subfactor R of M such that Ј പ ϭ trarily small support. Finally, we introduce a new notion, ‘‘gen- R M CI (see ref. 11). Let R j be the von Neumann Ն erating length,’’ in connection with the generator problem, and subalgebra of M generated by R and U1,...,Uj, for j 1. Then Ј പ ϭ obtain a basic result. R j M CI and R j is a factor for every j. For any given large ʚ ʚ ⅐⅐⅐ integer n, we can find unital embeddings Mn(C) Mn2(C) Definitions and Basic Properties ʚ (i) ni ʚ ϭ R. Let {Ejk }j,kϭ1 be a matrix unit system for Mni(C) R, i von Neumann algebras were introduced by von Neumann in ref. 1, 2, . . . . From the assumption that Ui is a Haar unitary element, 6. They are strong operator topology closed selfadjoint subal- gebras of B(H), the algebra of all bounded operators on a Hilbert space H. Factors are von Neumann algebras whose center This paper was submitted directly (Track II) to the PNAS office. consists of scalar multiples of the identity. von Neumann showed §To whom reprint requests should be addressed. E-mail: [email protected].
www.pnas.org͞cgi͞doi͞10.1073͞pnas.022593699 PNAS ͉ January 22, 2002 ͉ vol. 99 ͉ no. 2 ͉ 565–567 Downloaded by guest on September 26, 2021 (i) (i) ϩ we choose projections F1 ,...,Fni in the abelian von Neumann (n 1)th column. It is well known that Gn is an i.c.c. group. Here (i) ϭ ͞ i ͚ni (i) ϭ algebra generated by Ui such that (Fj ) 1 n and jϭ1Fj we prove that many of its subgroups are i.c.c. groups. Ն I. Because R i (containing Ui) is a factor, we can find a unitary LEMMA 2. If G is a subgroup of Gn, for n 3, containing GnϪ1, (i) ϭ (i) ϭ i element Wi in R i such that WiFj W*i Ejj for j 1,...,n . then G is an i.c.c. group. (i) ʦ From our assumption, we have U*iϩ1Fj Uiϩ1 R i. Again, there Proof: We shall show that, for any element g in G, if the (i) (i) ϭ ϩ ϩ ϭ is a unitary element Vi in R i such that ViEjj V*i U*i 1Fj Ui 1 conjugacy class of g by the subgroup GnϪ1 is finite, then g is the (i) ϭ i U*iϩ1W*iEjj WiUiϩ1 for j 1,...,n . Thus WiUiϩ1Vi commutes identity matrix. (i) (i) i n i ϭ͚ ʦ with E11,...,En n and Suppose g j,kϭ1ajkejk, where ajk Z and {ejk} is the matrix ϭ ϩ m ϭ ϩ ni unit system. For gst I est in GnϪ1, gst I mest for any m Յ Յ Ϫ ϭ ͑i͒ ͑i͒ in Z, where 1 s, t n 1 and s t. Now, WiUi ϩ 1Vi Ejj WiUi ϩ 1ViEjj , ϭ j 1 n Ϫ m m ϭ ͑ Ϫ ͒ͩ ͪ͑ ϩ ͒ where i ϭ 1, 2, . . . . Given any i, with respect to the matrix gst ggst I mest ajkejk I mest i j,k ϭ 1 subalgebra Mni(C)ofR , WiUiϩ1Vi has at most n nonzero diagonal entries. Now we define, inductively, elements Ti in M. ϭ Ϫ ͞ Ն n n n When i 1, choose integer m1 so that (m1(m1 1) 2) n and Յ ͌ ϩ ϭ Ϫ ϩ m1 2n 1. Thus there is an injective map from {1, 2, . . . , ajkejk m ajkestejk m ajkejkest Յ Յ Յ ␣  ϭ ϭ ϭ n} to the set {(j, k):1 j k m1}. We shall use ( 1(j), 1(j)) j,k 1 j,k 1 j,k 1 to denote the image of j under this map. Let n n Ϫ 2 m ajkestejkest ͑ ͒ ͑ ͒ ϭ 1 1 ϭ ␣ ͑ ͒  ͑ ͒ j,k 1 T1 E 1 j jW1U2V1Ej 1 j . j ϭ 1 n n n ϭ Ϫ ϩ Then we know that T1 is strictly upper triangular with respect to ajkejk m atkesk m ajsejt (1) m1 matrix units {Ejk }j,kϭ1 and W1U2V1 is in the algebra generated j,k ϭ 1 k ϭ 1 j ϭ 1 by T1 and Mn(C). The support Q1 of T1 is majorized by the sum (1) (1) Յ ͞ Յ Ϫ 2 of the projections E11 , ..., Em1m1. Thus (Q1) m1 n m atsest. (͌2n ϩ 1͞n). Similarly we define T for W U V so that its 2 2 3 2 Ϫm m ʦ support Q2 is majorized by the sum of some diagonal projections From our assumption that {gst ggst : m Z} is a finite set, we Յ ͌ 2 ϩ 2 ʦ ϭ in Mn2(C), which are orthogonal to Q1 and (Q2) ( 2n know that {m astest : m Z} is a finite set. Thus ast 0 for all 2 Յ Յ Ϫ ͚n Ϫ͚n ϭ 1͞n ). Continuing this process, we shall have T , in a strictly 1 s, t n 1 and s t. Similarly, kϭ1atkesk jϭ1ajsejt i ϭ upper triangular form with respect to Mni(C), such that WiUiϩ1Vi 0. Thus, from ast 0 for s t, we have lies in the algebra generated by Mni(C) and Ti. The support Qi ϩ ⅐⅐⅐ϩ Յ ͌ i ϩ a e ϩ a e Ϫ a e Ϫ a e ϭ 0. of Ti is orthogonal to Q1 QiϪ1 and (Qi) ( 2n tt st tn sn ss st ns nt ͞ i ϭ ͞ʈ ʈ ϩ ͞ʈ ʈ ϩ ⅐⅐⅐ 1 n ). Now let T (T1 T1 ) (T2 T2 ) (or the strong ϭ ϭ ϭ ͞ʈ ʈ ϩ ⅐⅐⅐ϩ ͞ʈ ʈ ϭ ϩ This implies that ass att and ans atn 0. Therefore g must operator limit of (T1 T1 ) (Ti Ti ), and B T T*. ■ be the unit I in Gn. Then it is easy to show that M is generated by R, U1, and B.If Յ ͚ϱ ͌ i ϩ ͞ i Q is the support of B, then (Q) iϭ1 ( 2n 1 n ), which can be arbitrarily small when n is large. Now M is generated by The following result is easy to check. We omit its proof here. LEMMA 3. The abelian group generated by gjs and gjt, two of the R, U1, and B. Replacing U1 by a selfadjoint element, we have our ϫ desired result. ■ generating elements of Gn, is isomorphic to Z Z, i.e., the Haar unitary operators Lgjs and Lgjt are commuting independent ele- Remarks: (i) The factorial assumption on M is essential. For ments. The same is true for gsj and gtj. R Before we state our main theorem, we prove another technical example, LFm LZ satisfies the assumptions of the proposition. But we do not know whether it is generated by four selfadjoint result. elements (for m Ͼ 4). LEMMA 4. Let M be a factor of type II1 with trace . Suppose M (ii) It is well known that the hyperfinite II factor is generated is generated by a subfactor N and a unitary operator U in M, and 1 ʦ by a small projection and a selfadjoint element. If we choose this there is a Haar unitary operator V in N such that U*VU N. projection so that it is orthogonal to the support of B, then M as Assume N is generated by a subalgebra Nm, isomorphic to Mm(C), given in the proposition is generated by two selfadjoint elements and a selfadjoint element S whose support is majorized by a Յ and another with an arbitrarily small support. projection PinNm with (P) . Then M is generated by Nm and Ն Next, we shall show that LSL (Z), n 3 is generated by two a selfadjoint element T whose support is majorized by a projection n Յϩ ͌ ϩ ͞ selfadjoint elements, one of which has an arbitrarily small QinNm such that (Q) ( 2m 1 m). support. Proof: Because V is a Haar unitary element, there are pro- jections F1, ..., Fm in the abelian von Neumann algebra Generators of LSL (Z) for n > 3 ͚m ϭ ϭ ͞ n generated by V such that jϭ1Fj I and (Fj) 1 m. Choose m Recall that SLn(Z), when n is odd, or PSLn(Z), when n is even, {Ejk}j,kϭ1 in Nm, which correspond to a matrix unit system in is denoted by Gn. From the above proposition and Remark ii,we Mm(C) such that P is the sum of diagonal projections E11,..., ͞ Յ see that LGn is generated by three selfadjoint elements, one of Em m for some m1 (and (m1 m) ). Because N is a factor, 1 1 ϭ ϭ which has an arbitrarily small support. Now we are going to there is a unitary operator W in N such that Fj W*EjjW, j ϭ ϭ reduce the number three to two. We choose, again, the gener- 1,...,m. Let Qj U*FjU, for j 1,...,m. Then, from our Յ Յ ʦ ators {gjk :1 j, k n, j k} for Gn defined above. Their assumption, we know that Qj N. Again, there is a unitary ϭ corresponding unitary operators are denoted by Lgjk. We shall element W in N such that Q W*E W . Now we have that ϭ ϩ 1 j 1 jj 1 embed Gn into Gnϩ1 in a natural way. For gjk I ejk in Gn, W*E W ϭ U*F U ϭ U*W*E WU. Thus ϩ 1 jj 1 j jj we regard this gjk as an element in Gnϩ1 by putting 1 at its (n ϩ ϩ ϭ ϭ 1, n 1) entry and zeros on the rest of the (n 1)th row and EjjWUW*1 WUW*1Ejj, j 1,...,m,
566 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.022593699 Ge and Shen Downloaded by guest on September 26, 2021 ϭ͚m m i.e., WUW*1 jϭ1EjjWUW*1Ejj. We use a trick similar to the one Thus with respect to the matrix unit system {Ejk}j,kϭ1, the 3 ϩ 2 we used in the proof of Proposition 1 and choose m2 so that elements Lg , Lg , Lg and Lg have only 2m0 2m0 non- Ϫ ͞ Ն Յ ͌ ϩ ͞ 21 31 32 12 Ϫ ͞ Ն 3 ϩ 2 (m2(m2 1) 2) m and m2 ( 2m 1 m). Consider a zero entries. Choose m1 so that m1(m1 1) 2 2m0 2m0 and ϭ ϩ Յ ͌ ϩ ϩ matrix subalgebra of Nm with matrix units Ejk for j, k m1 m1 2m0 m0 1 1. By the same matrix trick used in the ϩ 1,...,m1 m2. Now there are more than m entries in the upper proof of Lemma 4, there is a selfadjoint element T in LH whose ϩ ⅐⅐⅐ ϩ L triangular subalgebra of this matrix subalgebra. We can define an support is majorized by E11 Em1m1, such that H is 3 Ͻ ϭ ϩ N injective map : {1, . . . , m} {(j, k):j k, j, k m1 1, generated by m and T. When m0 is large, we have that the trace ϩ ϭ of the support of T is less than or equal to m ͞m, which can be ...,m1 m2}. We write (j) ( 1(j), 2(j)). Now let 1 arbitrarily small. Next, we may consider the subfactor of LGn
m generated by LH and Lg23. Continuing this process and by
ϭ repeated use of Lemmas 2 and 4, we have that LGn is generated ͑ ͒ * ͑ ͒ T1 E 1 j jWUW 1Ej 2 j by a matrix subalgebra Mm (C) and a selfadjoint element Tn with j ϭ 1 n an arbitrarily small support P, which is dominated by some Јϭ ϩ Ј diagonal projections in Mm (C). It is easy to see that Mm (C)is and T T1 T*1. Then T is selfadjoint and its support is n n ϩ generated by a rank one projection E and a (shift) unitary ͚m1 m2 ϭ ϩ ϩ Ј 1 majorized by jϭm ϩ1 Ejj. Let T S P T . By choosing an matrix U (of finite order). One may choose E so that it is 1 Ј ϩ n 1 appropriate real constant , we have that both T and S P orthogonal to P. Replacing E1 and Tn by one selfadjoint element lie in the von Neumann algebra generated by T. It is easy to see S, we have that LGn is generated by S and Un, where S can have that Nm and T generate M. It is easy to check the estimate for an arbitrarily small support and Un is a unitary element with a the trace of the support of T. This completes the proof. ■ finite order. This completes the proof of our theorem. ■ Ն Х Q The following is the main result of this article. Note that when n 4 even, LSLn(Z) LPSLn(Z) LPSLn(Z). Thus Ն L Ն THEOREM 5. The factor LGn, for n 3, is generated by two the same result holds for SLn(Z) for all n 3. selfadjoint elements, one of which can have an arbitrarily small Final Comments support. Յ Յ Proof: We shall use the generators gjk,1 j, k n and j As pointed out by Voiculescu (4), free entropy dimension should be related to the number of generators for a factor. The k, for the group Gn. Let H be the subgroup of Gn generated by ‘‘number’’ should be measured by its free entropy dimension, not g21, g31, g32 and g12. The H is a subgroup of G3 containing G2. by the length of the support of the generator. To illustrate this From Lemma 2, we know that LH is a subfactor of LG . For any n point, we shall introduce a notion of ‘‘generating length’’ and large integer m0, choose projections P1,...,Pm0 in the abelian von Neumann algebra generated by L and Q ,...,Q in the show that a large class of factors have generating length equal to g31 1 m0 one. L ϭ ϭ ͞ one generated by g32 such that (Pj) (Qj) 1 m0 and ͚m0 ϭ͚m0 ϭ ϭ Definition 6: Suppose M is a factor of type II1 with trace or, jϭ1Pj jϭ1Qj I. From Lemma 3, we know that (PjQk) ͞ 2 ϭ 2 in general, a von Neumann algebra with a state, and N a 1 m0 and {PjQk : j, k 1,..., m0} is a set of m0 mutually ϭ 2 subalgebra (or subset) of M. The generating length N(M, )of orthogonal projections in LH. Let m m0 and choose a full m M over N is given as follows: matrix subalgebra Nm in LH and a matrix unit system {Ejj}jϭ1 so ͑ ͒ ϭ ͕͑ ͒ ϩ ͑ ͒ that the diagonal projections E11, ..., Emm coincide with the N M, inf P1 P2 projections P Q ,...,P Q ,...,P Q ,...,P Q . Because 1 1 1 m0 m0 1 m0 m0 ϩ g21 commutes with g31 and g12 commutes with g32, we have ···: Pj is the range projection of Aj, Aj
m0 m0 ϭ ʦ ͖ A*j M, M is generated by N and A1, A2,... . ϭ ϭ Lg PlLg Pl PlQk Lg Qk Pl, 21 21 1 21 2 When N is CI or empty set, we shall use (M, ) or simply (M) l ϭ 1 l,k1,k2 ϭ 1 instead of N(M, ). The advantage of this generating length is m0 m0 that it tells more information, in some sense, than the number of ϭ ϭ selfadjoint generators of a von Neumann algebra. The detailed Lg12 QkLg12Qk Pl1QkLg12QkPl2, k ϭ 1 k,l1,l2 ϭ 1 study of this invariant will appear elsewhere. Using a matrix trick, one easily proves the following result. m0 m0 PROPOSITION 7. Suppose M is a factor of type II1 and P is a ϭ ϭ 1 Lg31 PlLg31Pl PlQkLg31QkPl, projection of trace . If PMP is generated by two selfadjoint 2 l ϭ 1 l,k ϭ 1 elements, then (M) ϭ 1.
m0 m0 MATHEMATICS ϭ ϭ We thank Prof. Bingren Li for many helpful discussions. This research Lg32 QkLg32Qk PlQkLg32QkPl. was supported in part by a National Science Foundation (USA) k ϭ 1 k,l ϭ 1 CAREER Award.
1. Wogen, W. (1969) Bull. Am. Math. Soc. 75, 95–99. 7. Kadison, R. & Ringrose, J. (1983, 1986) Fundamentals of the Operator Algebras 2. Popa, S. (1985) Math. Scand. 57, 171–188. (Academic, Orlando, FL), Vols. 1 and 2. 3. Ge, L. & Popa, S. (1998) Duke Math. J. 94, 79–101. 8. Kazhdan, D. (1967) Funkcional. Anal. Prilozˇen. 1, 71–74. 4. Voiculescu, D. (1999) J. Reine Angew. Math. 514, 113–118. 9. Connes, A. & Jones, V. (1985) Bull. London Math. Soc. 17, 57–62. 5. Ge, L. & Shen, J. (2000) Proc. Natl. Acad. Sci. USA 97, 9881–9885. 10. Gaboriau, D. (2000) Invent. Math. 139, 41–98. 6. von Neumann, J. (1930) Math. Ann. 102, 370–427. 11. Popa, S. (1983) J. Oper. Theory 9, 253–268.
Ge and Shen PNAS ͉ January 22, 2002 ͉ vol. 99 ͉ no. 2 ͉ 567 Downloaded by guest on September 26, 2021