Free entropy and property T factors
Liming Ge*† and Junhao Shen‡§
*Department of Mathematics, University of New Hampshire, Durham, NH 03824-3591; and ‡Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395
Edited by Richard V. Kadison, University of Pennsylvania, PA, and approved May 19, 2000 (received for review April 26, 2000) We show that a large class of finite factors has free entropy di- generators with free entropy dimension not exceeding 1. Thus mension less than or equal to one. This class includes all prime with respect to free entropy dimension and certain generators, factors and many property T factors. property T factors behave differently from free group factors. Of course, the first question to ask after this is whether there n the early 1980s, D. Voiculescu (1) developed a noncommu- are generators in those same property T factors whose free en- I tative probability theory over von Neumann algebras, the alge- tropy dimension is greater than 1. We answer that question by bras introduced and studied by F. J. Murray and J. von Neumann showing that every set of generators of a large class of factors (2–6). von Neumann algebras are quantized measure spaces, of type II1, including those considered in ref. 13 and all non- which are the basis for the study of noncommutative analysis prime factors (11), have free entropy dimension not exceeding and geometry (7). 1 if there is at least one set of generators that satisfies a “cyclic” A von Neumann algebra is a strong-operator closed self- commuting relation. adjoint subalgebra of the algebra of all bounded linear transfor- We describe below, briefly, the construction of finite factors by mations on a Hilbert space. Factors are von Neumann algebras using regular representations of discrete groups. Then we define whose centers consist of scalar multiples of the identity. They free entropy and free entropy dimension and list some of their are the building blocks from which all the von Neumann basic properties. Finally, we state our main results and outline algebras are constructed. the proofs. F. J. Murray and J. von Neumann (2) classified factors by means of a relative dimension function. Finite factors are those Definitions for which this dimension function has a finite range. For finite There are two main classes of examples of von Neumann alge- MATHEMATICS factors, this dimension function gives rise to a unique tracial bras introduced by Murray and von Neumann (3, 5). One is ob- state. In general, a von Neumann algebra admitting a faithful tained from the “group-measure space construction,” the other normal trace is said to be finite. Infinite-dimensional finite fac- is based on the regular representation of a (discrete) group G tors are called factors of type II . The trace is the analogue of 1 (with unit e). The second class is the one needed in this note. classical integration and finite factors are basic noncommutative The Hilbert space ( is l2G. We assume that G is count- measure spaces. able so that ( is separable. For each g in G, let L denote the With classical independence replaced by “free independence” g left translation of functions in l2G by g−1. Then g → L is a on noncommutative spaces, Voiculescu develops the theory g faithful unitary representation of G on (.Let, be the von of free probability. The free central limit theorem states that G Neumann algebra generated by L x g G. Similarly, let R the analogous averaging of free independent random variables g g 2 obeys a Semicircular Law instead of a Gaussian Law in the be the right translation by g on l G and 2G be the von Neu- classical case. Voiculescu and others use this theory to answer mann algebra generated by Rg x g G. Then the commutant 0 0 several important old questions in the theory of von Neumann ,G of ,G is equal to 2G and 2G = ,G. The function ug that algebras. is 1 at the group element g and 0 elsewhere is a cyclic trace Recently, Voiculescu introduced a notion of “free entropy” vector for ,G (and 2G). In general, ,G and 2G are finite von (8), an analogue of classical entropy and Fisher’s informa- Neumann algebras. They are factors (of type II1) precisely when tion measure. Associated with free entropy, he defined a free each conjugacy class in G (other than that of e) is infinite. In entropy dimension which, in some sense, measures the “non- this case we say that G is an infinite conjugacy class (i.c.c.) group. commutative dimension" of a space (or, the minimal number of Specific examples of such II1 factors result from choosing for generators needed in the case of factors). Free entropy has be- G any of the free groups Fn on n generators (n 2),orthe come a very powerful tool in the study of von Neumann algebras permutation group 5 of integers Z (consisting of those permu- since its recent introduction (9–11). Of course, some basic ques- tations that leave fixed all but a finite subset of Z). Murray and von Neumann (4) prove that , and , are not * isomorphic to tions concerning free entropy and free entropy dimension have Fn 5 appeared. For example, it is not yet known whether the free each other. A factor is hyperfinite if it is the ultraweak closure of entropy dimension of a set of generators for a von Neumann al- the ascending union of a family of finite-dimensional self-adjoint gebra depends on the set. Is this dimension an invariant of the subalgebras. In fact, ,5 is the unique hyperfinite factor of type algebra? That is currently the major problem of the theory. II1; it is contained in any factor of type II1. When G = SLnZ, n 3, , is a von Neumann algebra with property T (12) (it For finite injective (or hyperfinite) von Neumann algebras, SLnZ Voiculescu shows that his free entropy dimension is an in- is a factor when n is odd). Property T factors are not isomorphic variant. He also shows that it is an invariant for self-adjoint to factors arising from free groups or 5 (ref. 14 or 15). The clas- operator algebras with a faithful trace (as applied to sets of sification of all von Neumann algebras has been reduced to the algebraic generators). It seems quite plausible that it is an in- case of factors of type II1 in a certain sense. Some basic prob- variant for C*-algebras with a faithful trace, as well. For von lems concerning the free group factors remain unsolved. Among Neumann algebras, this becomes somewhat more speculative, them is the one which asks whether the factors arising from free the most severe test involving its application to factors with the groups on different number of generators are isomorphic. This Connes–Jones property T (12). For some years, now, Voiculescu has set, as a primary ob- jective for study, the determination of free entropy dimension This paper was submitted directly (track II) to the PNAS office. for sets of generators in property T factors. He has proved the †E-mail [email protected]. first result in ref. 13, showing that some property T factors have §E-mail [email protected].
PNAS | August 29, 2000 | vol. 97 | no. 18 | 9881–9885 Downloaded by guest on September 30, 2021 is one of the motivating questions that led to the development (defined above) onto its first n components. Then of free probability theory. Recall that a W*-probability space is a pair -;τ consisting χX1;:::;Xn x Y1;:::;Yp of a von Neumann algebra - and a faithful normal state τ. The most important case is when the state is a trace (τAB= −2 = sup inf lim sup k log 30RX1;:::;Xn x τBA) and, hence, - is a finite von Neumann algebra. Ele- R,0 m;ε,0 k→: ments in - are called random variables. With A in -, A2 Y1;:::;Ypy m; k; ε denotes the L2-norm (or trace-norm) of A with respect to τ (that is, A2 = τA∗A). A family - , ι I, of von Neu- n 2 ι + log k : mann subalgebras of - are free (or free independent) with re- 2 spect to the trace τ if τA A ···A =0 whenever A - , 1 2 n j ιj ι 1 6= · · · 6= ιn and τAj=0 for 1 j n and every n in N.A When Y1;:::;Yp are non-self-adjoint elements, χX1;:::;Xn x family of subsets (or elements) of - are said to be free if the Y1;:::;Yp may be identified with χX1;:::;Xn x A1;:::;Ap; von Neumann subalgebras they generate are free. ∗ ∗ B1;:::;Bp, where Aj = Yj + Yj and Bj =−iYj − Yj for Voiculescu proves the free central limit theorem: Suppose each j. A , j = 1; 2;:::, are free random variables such that τA =0. The (modified) free entropy dimension δX ;:::;X x Y ; j P j 1 n 1 Denote by r2 the limit lim 1 n τA2. Assume further that :::;Y is defined by 4 n→: n j=1 j p k √ supj;k τAj +:. Then the distribution of A1 +···+An/ n converges pointwise to the semicircular law with center at 0 and δX1;:::;Xn xY1;:::;Yp radius r. He also generalizes Wigner’s semicircular law for χX +εS ;:::;X +εS xS ;:::;S ;Y ;:::;Y large random matrices to n-tuples of large random matrices = n + lim sup 1 1 n n 1 n 1 p ; and proves the asymptotic freeness in limit as the degree of ε→0 log ε the matrices tends to infinity. By using techniques developed in proving these results, he shows that the free group factor where S1;:::;Sn is a free semicircular family and the two sets on two generators is isomorphic to the 2 3 2 matrix algebra X1;:::;Xn;Y1;:::;Yp and S1;:::;Sn are free. with entries in the free group factor on five generators. This is We list some basic properties of free entropy and free entropy the first major break through in the study of the isomorphism dimension. (i) χX ;:::;X n log2πeC2n−1, where C = τX2 + problem of free group factors. 1 n 2 1 2 1/2 Now we describe briefly the definition of free entropy. Let ···+Xn . RR (ii) χX= log s − tdµsdµt+ 3 + 1 log 2π, where MkC, or simply Mk,bethek 3 k matrix algebra with entries 4 2 −1 in C, and τk be the normalized trace on Mk, i.e., τk = k Tr , µ is the (measure on the spectrum of X corresponding to the) sa where Tr is the usual trace. Let Mk denote the set of all self- distribution of X. sa n adjoint matrices in Mk and Mk denote the (vector-space) (iii) χX1;:::;Xn=χX1+···+χXn when X1;:::;Xn sa direct sum of n copies of Mk . We denote by e the eu- are free. sa n 2 2 clidean norm on Mk given by A1;:::;Ane = TrA1 + (iv) δS1;:::;Sn=n when S1;:::;Sn are free semi-circular 2 2 2 sa n ···+An=kτkA1 +···+An, for A1;:::;An in Mk . elements. (Note that the free group factor on n generators is We use 2 to denote the trace norm induced by τk.Let generated by n free semicircular elements.) sa n 3 be Lebesgue measure on Mk induced by the euclidean norm e. Main Results Suppose X ;:::;X are self-adjoint elements in a W*-prob- 1 n The assumptions of the following theorem are satisfied when ability space -;τ. The free entropy of X1;:::;Xn, denoted the algebra - is a tensor product of two factors of type II1,or by χX1;:::;Xn, is defined in the following. the group von Neumann algebra arising from SLnZ, n 3, or For any positive ε and R, and any k; m in N, let 0RX1;:::;Xny sa n certain amalgamated free product. One of the consequences of m; k; ε be the subset of Mk consisting of all A1;:::;An in Msan such that A R,1 j n, and our theorem says that the free group factors are not isomorphic k j to any factors that satisfy the assumptions of the theorem. Main Theorem. Let -;τ be a W*-probability space. Sup- τkAι ···Aι −τXι ···Xι + ε 1 q 1 q pose - is generated by its subalgebras .1;:::;.r , and, for each .j, 1 j r, there is a non-atomic self-adjoint subalgebra !j of for all 1 ι ;:::;ι n and all q with 0 q m. Then 1 q .j such that !j commutes with .j+1 (here we let .r+1 be .1). Assume that X1;X2;:::;Xn are self-adjoint elements in - that χX1;:::;Xn generate - as a von Neumann algebra. Then δX1;:::;Xn 1. The above theorem is a consequence of the following result −2 = sup inf lim sup k log 30RX1;:::;Xny m; k; ε on free entropy. R,0 m;ε,0 k→: Proposition. Suppose -;τ is a W*-probability space. We n assume that - contains subalgebras .1;:::;.r , and, for each + log k : 0 2 .j, 1 j r, there is a Haar unitary element Uj in .j such that 0 Uj commutes with .j+1 (again, .r+1 is .1). Let X1;X2;:::;Xn be self-adjoint elements in -. Suppose there are unitary elements The modified free entropy (or simply, free entropy) χX1; Vj1;Vj2;:::;Vjl in .j, for j = 1; 2;:::;r, lj N, and polyno- :::;Xn x Y1;:::;Yp of X1;:::;Xn in the presence of Y1;:::; j Y is given in the following. mials ϕjx1;x2;:::;xl in Cx1;x2;:::;xl, where l = l1 + l2 + p :::+ l , such that Let X1;:::;Xn;Y1;:::;Yp, n 1;p 0, be self-adjoint ran- r dom variables in -;τ, 0RX1;:::;Xn x Y1;:::;Ypy m; k; ε be the image of the projection of 1 X − ϕ V ;:::;V ;:::;V ;:::;V ω; j j 11 1l1 r1 rlr 2 2
0RX1;:::;Xn;Y1;:::;Ypy m; k; ε j = 1; 2;:::;n;
9882 | www.pnas.org Ge and Shen Downloaded by guest on September 30, 2021 for some positive number ω. Then Let χX ;:::;X x V ;:::;V ;U0 ;:::;U0 "W ;δ =G M CG 1; 1 n 11 rlr 1 r 1 1 k 2 n n W G − GW 3δ log4aC+ lognπ+n log 3 + 1 1 2 1 2 2 W1;δ1;δ2=H MkCG − H2 δ2; +n − 1 − ω log ω; for some G "W1;δ1: 3π where C ( e ) is a constant and a = max1jn Xj2 + 1. In the following, we outline the proof of the proposition. Note Then√ G2 is in "W1;δ1 and the covering number µ√W1;δ1;δ2; 2 2δ for W ;δ ;δ with balls of radius 2 2δ satisfies that the free entropy of X1;:::;Xn is the limit of the logarithm 2 1√ 1 2 2 2k2/p of the volume of 0RX1; ···;Xn x ···yk; m; ε. We will get a µW1;δ1;δ2; 2 2δ2 3/δ2 , where the metric on good upper bound for the volume by showing that the covering W1;δ1;δ2 is given by the normalized trace norm 2 on number of 0RX1;:::;Xn x···ym; k; ε (by small balls) is small. MkC. When Xj is approximated by ϕjV11;:::;V1l ;:::;Vr1;:::;Vrl , We will use T W1;δ1;δ2 to denote the set that has the 1 r minimal cardinality of covering balls for W ;δ ;δ with ra- the elements in 0RX1;:::;Xn x···ym; k; ε are approximated √ 1 1 2 by ϕ G ;:::;G where G values are matrices approxi- dius 2 2δ . So the cardinality of T W ;δ ;δ is bounded by j 11 rlr st 2 1 1 2 mating V values in moments. First we reduce the covering 3 2k2/p. Denote by T W ;δ ;δ , the subset of T W ;δ ;δ , st δ 1 1 1 2 1 1 2 of 0 X ;:::;X x ···ym; k; ε to the covering on the ranges 2 ∗ R 1 n consisting of balls that contain a conjugating matrix U W1U of ϕj values, which then depend on the domains of ϕj values. of W for some U in 5k. We shall enlarge the balls in When passing the commuting relations on U 0 and V values 1 j j+1;t T1W1;δ1;δ2 so that each original ball is contained in the ball to their corresponding matricial approximants, we show that the centered at some U ∗W U with radius 8δ . Then each of the G values lie in a lower dimensional submanifold of M Cl 1 2 st k balls in T1W1;δ1;δ2 contains the original ball in T W1;δ1;δ2 ∗ (l = l1 +···;lr ), so the covering number for this submanifold is containing U W U. Define, also, T W ;δ ;δ to be the subset small. The following lemmas describe the details of these steps. 1 2 1 1 2 of T W1;δ1;δ2 consisting of balls that contain some unitary Before stating them, we introduce some notation. matrix. We, again, enlarge the balls in T W ;δ ;δ the same Let p be a large prime number, k a large integer such that k 2 1 1 2 p way as for T1W1;δ1;δ2 so that each of its balls is centered is an integer. Let W be the matrix 0 at a unitary matrix with radius 8δ2.WeuseBV; a to denote the (closed) ball centered at V (in M C) with radius a (with MATHEMATICS 10::: 0 k respect to the normalized trace norm ). 0 e2πi/p ::: 0 2 : : : : Now, for any given ball in T1W1;δ1;δ2, let W2 be its center : : : : ∗ : : : : and W2 = U W1U for some U in 5k. Replacing W1;δ1; 3δ1, 2πip−1/p 00::: e and δ2 in the above lemma by W2; 8δ2; 17δ2, and δ3, respectively, we have the following. in MpC, W1 the matrix Lemma 2. Assume that G2;G3 in MkC satisfy the following I k 0 ::: 0 p G2; G3 1yW2 − G22 8δ2; 2πi/p 0 e I k ::: 0 p G2G3 − G3G22 ε1; 0 + ε1 δ1 + 8δ2: : : :: : : : : : Let 2πip−1/p 00::: e I k p "W2;δ2=G MkCG2 1; in MkC (here I k denotes the identity matrix in M k and Mk = p p W2G − GW22 17δ2 Mp ⊗ M k ). Let c0 be the distance between two adjacent pth p W ;δ ;δ =H M CG − H δ ; 2πi/p 2πis/p 2πit/p 2 2 3 k 2 3 roots of unity, i.e., c0 =1 − e . Clearly, e − e for some G "W ;δ : c0 π/p when s; t Z;s 6= t and 0 s; t p −√1. For any 2 2 given D , 0, ω , 0;r;l N, let δ be ω/8D l, define, √ r+1 Then G is in "W ;δ and µW ;δ ;δ ; 2 2δ successively, δr ;:::;δ1 by the following equations 3 2 2 2 2 3 3 3/δ 2k2/p. 17pδ 3pδ 3 r 1 Similarly, we will have corresponding T1W2;δ2;δ3 and δr+1 = ; :::; δ2 = : π π T2W2;δ2;δ3 as described preceding Lemma 2. Continuing this That is, process, we obtain, for each j 1, ω πω "W ;δ ;W ;δ ;δ ;TW ;δ ;δ ; δ = √ ;δ= √ ; :::; j j j j j+1 j j j+1 r+1 8D l r 17p · 8D l and corresponding subsets T W ;δ ;δ and T W ;δ ;δ πr ω 1 j j j+1 2 j j j+1 δ = √ of T Wj;δj;δj+1 that will satisfy conditions similar to those 1 17pr−1 · 3p · 8D l of the case when j = 2. In particular, the cardinality of 3 2k2/p T Wj;δj;δj+1 is bounded by , for j = 1;:::;r. for the given positive numbers D; l, ω and r. δj We use ε0, again, to denote minε0;δ1;ε1. Now ε0 satisfies For the given D; l and ω, δ1 is defined by the above equation. the assumptions in Lemmas 1 and 2 (in place of ε1). Then ε0 will be the positive number in 2.2 and 2.3 in ref. 16 Lemma 3. Suppose G1;:::;Gr ;S11;:::;S1l ;:::;Sr1;:::;Srl determined by this δ1 (here we only use cases when N = 2in 1 r ref. 16). are unitary matrices in MkC satisfying the following: 1GjGj+1 − Gj+1Gj2 ε0; We choose an arbitrary ε1 with 0 + ε1 + δ1. Then we have the following. 2G1 − W12 δ1; Lemma 1. Assume that G ;G in M C satisfy the following 3 for each j , 1, there is a unitary matrix Uj such that Gj − 1 2 k ∗ UjW1Uj 2 δj; G1; G2 1yW1 − G12 δ1; G1G2 − G2G12 ε1: 4Gj+1Sjl − SjlGj+12 ε0 and G1Srl − SrlG12 ε0;
Ge and Shen PNAS | August 29, 2000 | vol. 97 | no. 18 | 9883 Downloaded by guest on September 30, 2021 5 there is a unitary matrix Vjl such that Sjl − Vjl2 δ1y and an element Yrt (for 1 t lr ), the center of a ball in for j = 1;:::;r and l = 1;:::;lj. Then there is a family of balls T2W1;δ1;δ2, such that BY ; 8δ for j = 1;:::;r − 1 from T W ;δ ;δ jl j+2 2 j+1 j+1 j+2 ∗ U Srt U − Yrt 2 8δ2: such that Sjl − Yjl2 8δj+2 and a family of balls BYrl; 8δ2 from T2W1;δ1;δ2, such that Srl − Yrl2 8δ2. Choose a γ-net U in 5k with respect to the oper- Since, for each j,1 j r, Y x l = 1;:::;l are centers α αTk jl j k2 of balls in the set T W ;δ ;δ , we know that the to- ator norm such that Tk C/γ for each k in N, where 2 j+1 j+1 j+2 C is a constant. We choose γ to be ω , where a is given in the tal number of choices for Yjl x j = 1;:::;ry l = 1;:::;lj is 2a 2 2 theorem. 2k l1/p 2k lr /p bounded by 3/δ2 ···3/δr+1 . Hence there is an α in Tk such that U − U γ. Then Some of the key ideas of the proof of the proposition are α similar to those used in the proof of Theorem 2.1 in ref. 11. ∗ ∗ UαAjUα − U AjU ω; j = 1;:::;n: First we choose a non-atomic self-adjoint subalgebra !j of .j (generated by U 0). Since ! , for 1 j r, is non-atomic, there Let 8 be the map defined in Lemma 1.2 in ref. 11 and D the j j constant D8. Then we have that is a unitary Uj in !j so that its distribution is the same as that s s of W , i.e., τU =τ W ;s= 1; 2;:::. ∗ ∗ ∗ 1 j k 1 U A U −ϕ Y ;:::;Y U A U −U A U From Proposition 6.3 in ref. 9, we know that α j α j 11 rlr e α j α j e +U ∗A U −ϕ U ∗S U;:::;U∗S U;:::; χX ;:::;X x U 0 ;:::;U0;V ;:::;V j j 11 1l1 1 n 1 r 11 rlr U ∗S U;:::;U∗S U χX ;:::;X x U ;:::;U ;V ;:::;V : r1 rlr e 1 n 1 r 11 rlr +ϕ U ∗S U;:::;U∗S U;:::;U∗S U;:::;U∗S U j 11 1l1 r1 rlr By the definition of χX ;:::;X x U ;:::;U ;V ;:::;V , 1 n 1 r 11 rlr −ϕ Y ;:::;Y we know that it is the limit of the logarithm of the volume of j 11 rlr e sX 0 X ;:::;X xU r ; V y k; m; ε: 2k1/2ω+D U ∗S U −Y 2 R 1 n j j=1 jt j=1;:::;ryt=1;:::;lj jt jt e jt Our goal is to estimate the covering numbers of 0RX1;:::;Xn x sa n 2k1/2ω Uj; Vjt y k; m; ε in the euclidean space Mk . First, we study elements in 0 X ;:::;X ; U ; V y k; m; ε, then we q R 1 n j jt 1/2 2 2 2 2 project down to its first n components. From ref. 8, we know +Dk l18δ3 +l28δ4 +:::+lr−18δr+1 +lr 8δ2 √ that it is enough to choose R so that R , maxjXj+1. 1/2 1/2 Let 2k ω+8Dk lδr+1 3k1/2ω A ;:::;A ;G0 ;:::;G0 ;S0 ;:::;S0 ;:::;S0 ;:::;S0 1 n 1 r 11 1l1 r1 rlr √ where l = l + l + :::+ ł and δ = ω/8D l. be an arbitrary element in 0 X ;:::;X ; U ; V y k; m; ε. 1 2 n r+1 R 1 n j jt Thus When k; m are large and ε small enough, we have that ∗ ∗ UαA1Uα;:::;UαAnUα 0 0 0 0 2 Aj − ϕjS11;:::;S1l ;:::;Sr1;:::;Srl 2 ω; −ϕ Y ;:::;Y ;:::;ϕ Y ;:::;Y 1 r 3 1 11 rlr n 11 rlr e √ for j = 1; 2;:::;n. 3ω nk: Since the U values are unitary elements of finite order and √ j Let B8Y ;:::;Y ; 3ω nk be the ball in Msan of radius Vjt values are unitary, we may choose unitary matrices Gj (with √ 11 rlr k the same distribution as that of U ) and unitary matrices S in 3ω nk with center ϕ Y ;:::;Y ;:::;ϕ Y ;:::;Y . j jt 1 11 rlr √ n 11 rlr 1 nk2 5k such that Then the volume of B8Y ;:::;Y ; 3ω nk is π 2 01 + √ 11 rlr 1 nk2−13ω nknk2 , and Aj − ϕjS11;:::;S1l ;:::;Sr1;:::;Srl 2 ω; 2 1 r √ for j = 1; 2;:::;n. Moreover, with m large and ε small enough, A ;:::;A U nB8Y ;:::;Y ; 3ω nkU ∗n; 1 n α 11 rlr α we have n n where Uα is Uα;:::;Uα in Mk and 0· is the 0- GjGj+1 − Gj+1Gj2 ε0; Gj+1Sjl − SjlGj+12 ε0; function. By counting the number of all such balls U nB8Y ;:::; G1Srl − SrlG12 ε0: √ α 11 Y ; 3ω nkU ∗n, we know that it is bounded by rlr α Except for (2) in Lemma 3, Gj values and Sjt values satisfy all 2k2l +1/p 2k2l +1/p 2k2l +1/p other assumptions. Choose a unitary matrix U in MkC such 3 r 3 1 3 r−1 ∗ Tk ··· : that U G1U = W1. Thus, with respect to any given δ1 , 0, when ∗ δ2 δ3 δr+1 k and m are large enough and ε ε0 sufficiently small, U GjU ∗ and U Sjt U satisfy all the assumptions in Lemma 3.3. Taking Thus conjugation by U on Aj values we also have that 30RX1;:::;Xn x U1;:::;Ur ;V11;:::;Vrl y m; k; ε ∗ ∗ ∗ ∗ ∗ r U AjU −ϕjU S11U;:::;U S1l U;:::;U Sr1U;:::;U Srl U2 2 2 1 r 3 2k lr +1/p 3 2k l1+1/p ω; Tk ··· δ2 δ3 for j = 1; 2;:::;n. 2 2k lr−1+1/p −1 √ 3 1 nk2 1 2 nk2 By Lemma 3, for each pair j; t,1 j r − 1 and 3 π 2 · 0 1 + nk 3ω nk 1 t lj, there is an element Yjt , the center of a ball in δr+1 2 T W ;δ ;δ , such that 2 2 2 j+1 j+1 j+2 2k l+r/p k −1 √ 3 C 1 nk2 1 2 nk2 π 2 0 1 + nk 3ω nk U ∗S U − Y 8δ ; jt jt 2 j+2 δ2 γ 2
9884 | www.pnas.org Ge and Shen Downloaded by guest on September 30, 2021 √ 2 2 r−1 2k l+r/p k For the given ω; D; l and r, the above inequalities hold for 217p D l 2aC 1 nk2 π 2 any p. We may choose p large enough so that πr−1ω ω 1 −1 √ √ 3 0 1 + nk2 3ω nknk2 : 2l + r 217pr−1D l 2 log log 2 p πr−1ω Hence (since log p → 0 when p → :). χX ;:::;X x U ;:::;U ;V ;:::;V p 1 n 1 r 11 rlr Thus we have lim sup k−2 log 30 X ;:::;X x 0 0 R 1 n χX1;:::;Xn x U1;:::;Ur ;V11;:::;Vrl k→: r n n n G ;:::;G ;V ;:::;V y m; k; ε + log k log4aC+ lognπ+n log 3 + 1 r 11 rlr 2 √ 2 2 2l + r 217pr−1D l 2aC n log + log + lognπ +n − 1 − ω log ω: p πr−1ω ω 2 + n log 3 + n log ω This finishes the proof of the proposition. The proofs of the three lemmas are straight forward. We omit the details here. nk2 + lim sup n log k − k−2 log 0 1 + : k→: 2 This work was supported by the National Science Foundation.
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Ge and Shen PNAS | August 29, 2000 | vol. 97 | no. 18 | 9885 Downloaded by guest on September 30, 2021