A CHARACTERIZATION OF THE LEINERT PROPERTY

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 11, November 1997, Pages S (97)03966-X A CHARACTERIZATION OF THE LEINERT PROPERTY FRANZ LEHNER (Communicated by Palle E. T. Jorgensen) Abstract. Let G be a discrete grou and denote by λ G its left regular reresentation on l 2 (G). Denote further by F n thefreegrouonngenerators {g 1,g 2,...,g n} and λ its left regular reresentation. In this aer we show that a subset S = {t 1,t 2,...,t n} of G has the Leinert roerty if and only if for some real ositive coefficients α 1,α 2,...,α n the identity α i λ G (t i ) = α i λ(g i ) λ (G) λ (F n) holds. Using the same method we obtain some metric estimates about abstract unitaries U 1,U 2,...,U n satisfying the similar identity n U i U i min =2 n 1. Notation. Throughout in this aer G will denote a discrete grou with unit element e and Cλ (G) the sub-c -algebra of B(l 2 (G)) generated by its left regular reresentation λ G. This algebra is equied with the trace state τ G (X) = Xδ e,δ e. A subset {t 1,t 2,...,t n } of G is called free if it generates a coy of F n,thefree grou on n generators. We shall denote the canonical generators of the free grou by {g 1,g 2,...,g n }. When considering the free grou we shall omit the subscrit in λ Fn. We shall almost exclusively deal with finitely generated grous and reeatedly use the fact that given a generating set {h 1,h 2,...,h n } for the grou G there is a (unique) quotient maing q : F n G with q(g i )=h i i {1,2,...,n}. The following roosition collects some results from [K, Lemma 3.1 and Theorem 3]: Proosition 1. Let {h 1,h 2,...,h n } be a generating set of the grou G and denote by α 1,α 2,...,α n ositive real numbers. Then α i (λ G (h i )+λ G (h i ) ) α i (λ(g i )+λ(g i ) ) λ (G) λ (F n). Moreover in the case of equal coefficients, (λ G (h i )+λ G (h i ) ) = (λ(g i )+λ(g i ) ) = 2 2n 1 λ (G) λ (F n) if and only if {h 1,h 2,...,h n } is a free set. Received by the editors February 22, 1996 and, in revised form, May 21, Mathematics Subject Classification. Primary 22D25; Secondary 43A05, 43A15, 60J15. Key words and hrases. Norm of a convolution oerator, Leinert roerty, free grou, random walk c 1997 American Mathematical Society

2 3424 FRANZ LEHNER This result has a grah-theoretical interretation, where the oerator in consideration corresonds to the combinatorial Lalacian on the Cayley grah of the grou G; see [Pa] in articular for a generalization of the second statement. However it remained unclear whether the free grou can be characterized by the norms of non-selfadjoint oerators, and this question will be the subject of this aer. The following roosition is an extension of [K, Lemma 3.1]. Proosition 2. Let G and H be discrete grous and q : G H a grou homomorhism. Then for every finite subset {t 1,t 2,...,t m } of G and corresonding nonnegative real numbers α 1,α 2,...,α m the inequality m m (1) α i λ H (q(t i )) α i λ G (t i ) λ (H) holds. λ (G) Proof. We will use the following well known fact about the noncommutative L - norms associated to τ G : X λ (G) = su (τ G ((X X) )) 1/2. 1 < It suffices to consider the integer values of. Denote by W alt (A) thesetofall alternating words of length 2 in the letters A = {t 1,t 2,...,t m }: { } W alt (A) = t j1 t 1 t j2 t 1 i t j :,j 1,,j 2,...,i,j {1,2,...,m}. For v = t 1 t 1 t j1 t 1 t j2 t 1 i t j W alt (A) we use the following abbreviation: (2) α v = α i1 α j1 α i2 α j2 α i α j. Then we can write (note that we identify words in W alt (A) with the corresonding elements in the grou G): m m α i λ H (q(t i )) =su τ H α i α j λ H (q(t 1 i t j )) λ (H) =su su i,j=1 α v τ H (λ H (q(v))) v W alt(a) v W alt v=e (A) α v 1/2 = 1/2 = su m α i λ G (t i ) λ (G). 1/2 v W alt q(v)=e (A) α v 1/2

3 A CHARACTERIZATION OF THE LEINERT PROPERTY 3425 In [P2] the following more general statement is roved: Proosition 3. Let H be a Hilbert sace and U 1,U 2,...,U n a finite sequence of unitary oerators acting on H. Consider the reresentation π of F n which is determined by the condition π(g i )=U i for all i {1,2,...,n}. Then for every set of words {w 1,w 2,...,w m } F n and every sequence α 1,α 2,...,α m of ositive real numbers we have m m α i π(w i ) π(w i ) α i λ(w i ) min λ (F n). The set {g 1,g1 1,g 2,g2 1,...,g n,gn 1 } is only a secial case of a Leinert set, a concet which aeared first in [Le]. Definition 4 ( [A-O, Definition III B and Theorem III D]). A subset A = {t 1,t 2,...,t n } of a discrete grou G with unit element e is called a Leinert set if it satisfies one of the following equivalent conditions: Every sequence t i1,t j1,t i2,t j2,...,t im,t jm with i k,j k {1,2,...,n} such that j 1 j 2 j m satisfies t 1 t j1 t 1 t j2 t 1 t jm e. The set A can be written as y(b {e}), where B is a free subset of G and y G. The next roosition extends Kesten s formula for the norm (see also [B] for a weaker version). Simlified roofs can be found in [P-P] and [W]. For recent examles of Leinert sets in one-relator grous see [C-V]. Proosition 5 ([A-O, Theorem IV J]). Let n 2 and {t 1,...,t n } be a Leinert set in a discrete grou G. Then λ G (t i ) =2 n 1. λ (G) The next roosition is standard. Proosition 6. If u 1,u 2,...,u n are unitaries in a -algebra A for which there exist ositive real numbers β 1,β 2,...,β n satisfying β i u i = β i then there exists a state ϕ of A such that for every m N ϕ(u u j1 u u j2 u u jm )=1 i k,j k =1,2,...,n. In articular, if t 1,t 2,...,t n are some elements of the grou G then the subgrou of G generated by {t 1 i t j : i, j = 1,2,...,n} is amenable if and only if β i λ G (t i ) = β i for some ositive coefficients β 1,β 2,...,β n. Proof. By a comactness argument, there is a state on A such that ϕ( β i β j u i u j) = β i β j u i u j = β i β j It follows that ϕ(u i u j) = 1 for all i, j =1,2,...,nand by induction on m we have φ(u u j1 u u j2 u u jm ) = 1 for any choice of indices i k,j k. Indeed, φ(u u j1 u u j2 u u jm u i u ju u j1 u u j2 u u jm ) φ((i u i u j ) (I u i u j )) 1/2 =0.

4 3426 FRANZ LEHNER Now in the case where u i = λ G (t i ) this imlies that the trivial reresentation of the grou generated by t 1 i t j extends to a reresentation of the reduced grou -algebra, which imlies that it is amenable. The following lemma and its corollary will lay a key rôle in the generalization of Kesten s result. Lemma 7 ([H-R-V1, Lemma 8]). Let µ be a robability measure on R with comact suort and suose m = min su µ<max su µ = M, µ n = t n dµ(t) 0 for all n N. R Then lim su (µ n ) 1/n = M =max(m, M). n Corollary 8 ([K, Lemma 3.2]). For every subset {t 1,t 2,...,t n } of G and every sequence of ositive real numbers α 0,α 1,...,α n we have α 0I + α i (λ G (t i )+λ G (t i ) ) = α 0 + α i (λ G (t i )+λ G (t i ) ) and a similar result holds for general unitaries: α 0I + α i (U i U i + Ui U i ) = α 0 + α i (U i U i + Ui U i ). Proof. This follows from Lemma 7 alied to the robability measure µ on the sectrum of the self-adjoint oerator T = α i (λ(t i )+λ(t i ) ) which is determined by the moments µ n = τ G (T n ). For the case of general unitaries Ûi = U i U i we refer to [P1, Examle 5.6] where the following formula is roved: For any finite sequence of oerators x 1,x 2,...,x n on some Hilbert sace H we have min (3) xi x i = su tr (xi yx i z) y,z (S + 2 )1 where (S 2 + ) 1 is the intersection of the unit ball of the Hilbert-Schmidt class oerators on H with the cone of ositive oerators. In our case the oerator is self-adjoint and we can restrict (3) to the symmetric art: ( ) α 0 I + α i (Ûi + Û i ) = su tr α 0 y 2 +2 Û i yû i y min y (S + 2 )1 = α 0 + α i (Ûi + Û i ). min

5 A CHARACTERIZATION OF THE LEINERT PROPERTY 3427 Now we are ready to rove the main result. Theorem 9. Let G be a discrete grou, n 3 an integer and let t 1,t 2,...,t n be some elements in G. Then for any sequence α 1,α 2,...,α n of strictly ositive numbers we have α i λ G (t i ) = α i λ(g i ) λ (G) λ (F n) if and only if {t 1,t 2,...,t n } is a Leinert set. In articular, this is equivalent to the identity λ G (t i ) = 2 n 1. λ (G) Proof. We only have to show the only if art and we shall use the method introduced in [K, Theorem 3] (see also [B], where similar methods are used. We are grateful to A. Valette for bringing this article to our attention). Let us comare the norms of the oerators T = α i λ G (t i ) and T = α i λ(g i ). We can assume that G is generated by the subset {t 1,t 2,...,t n }. Suose that the set {t 1,t 2,...,t n } does not have the Leinert roerty. This means that there are an integer m and a sequence of indices j 1 j 2 j m, with i k,j k {1,2,...,n} and such that t 1 t j1 t 1 t j2 t 1 t jm =e. Let g 1,g 2,...,g n be the generators of the free grou F n and let q : F n G be the quotient maing associated to the generating set {t 1,t 2,...,t n }. Taking the same indices as above, set w := gi 1 1 g j1 gi 1 2 g j2 gi 1 m g jm. Since {g 1,g 2,...,g n } is a Leinert set, the word w does not reduce to the identity in F n but it is in the kernel of q. We can assume without loss of generality that w is cyclically reduced, i.e. j m. For if this is not the case, we can consider the word g 1 g i1 wg 1 g im =gi 1 m g j1 g 1 g j2 g 1 1 g jm 1 which is cyclically reduced and still in the kernel of q. Note that it cannot be reduced to the emty word in this way, since the conjugacy class of the latter in F n is trivial. So we can assume for the sake of clarity that =1andj m = 2. Now consider the elements w = g1 1 g 3 wg3 1 g 1, w = g2 1 g 3 wg3 1 g 2 which are both in the kernel of q and have infinite order, because w has, being cyclically reduced. Moreover, they form a free set, since there are no ossible

6 3428 FRANZ LEHNER cancellations in non-trivial reduced words built out of them. Thus we can aly Proosition 6 to the oerator T 1 = α w (λ(w )+λ(w ) )+α w (λ(w )+λ(w ) ) and we get (4) T 1 < 2(α w + α w ) because the grou generated by {w 1 w,w 2,w 2 } is not amenable. To comare the norms of T and T we use the identity X 2 = X X = (X X) 1/, which is a consequence of Gel fand s theorem. Thus it suffices to show that for some (T T ) < ( T T ). We consider = m +2sothat2= w = w.denotebyw alt (n) thesetofall alternating words of length 2 in n generators: W alt (n) ={g 1 g j1 g 1 g j2 g 1 i g j :,j 1,,j 2,...,i,j {1,2,...,n}}. Using the notation of (2) we can write (T T ) m+2 = α v λ(v) v Wm+2 alt (n) = T 0 + T 1 where T 0 = v W alt m+2 (n)\{w,w 1,w,w 1 } and T 1 is the same as above. Obviously, setting T 0 = we have v W alt m+2 (n)\{w,w 1,w,w 1 } α v λ(v) α v λ G (q(v)) ( T T ) m+2 = T 0 +2(α w +α w )I. Observe that T 0 (and hence T 0 ) is selfadjoint with ositive coefficients so that we can aly (1) and Corollary 8 and we get (T T ) m+2 T 0 + T 1 < T 0 +2(α w +α w ) = T 0 +2(α w +α w )I = ( T T ) m+2. Remark. Actually in the case of equal coefficients the above roof gives the following quantitative estimate (cf. [K, (4.15)] and [Pa, Theorem 3.2]): With the notation as in the roof, let m denote the minimal length of a nontrivial word in the kernel of the quotient maing q. Then T T + 2(n 1) 2 2n 3 (2m +4)n 2m+3.

7 A CHARACTERIZATION OF THE LEINERT PROPERTY 3429 Indeed, when there are n free generators, we can build a free set {w 1,w 2,...,w n 1 } in F n by setting w i = gi 1 g n wgn 1g i for i {1,2,...,n 1}.Then n 1 (λ(w i )+λ(w i ) ) =2 2n 3 and by modifying T 0 and T 1 (res. T0 and T 1 aroriately), we have the inequality T 2(m+2) T 2(m+2) +2(n 1) 2 2n 3. Now a simle convexity argument and the fact that T nyield the claim. The following corollary gives a characterization of the free grou in terms of the norms oerators in the reduced -algebra. Note that the free grou cannot be characterized by the sectrum of the sum of the generators, as shown in [H-R-V, sect.4]. Corollary 10. Let G be a discrete grou and S = {t 0 = e, t 1,...,t n } be a generating subset with n 2. Then 2 n λ G (t i ) i=0 with equality if and only if G isthefreegrouonngenerators. A similar roof yields the following result about unitaries which do not necessarily arise from a regular reresentation of a discrete grou, as for examle in [L-P-S]. Theorem 11. Let U 1,...,U n be unitary oerators acting on some Hilbert sace H and suose U i U i =2 min n 1. Then for all roducts of the form V = U 1 U j1 U 1 U j2 U 1 j 2... j m we have V V I 1. U jm with j 1 Proof. To simlify notation we use the unitary reresentation ˆπ = π π : F n B(H 2 H) which we already introduced in Proosition 3 and which mas the generators g i to the corresonding unitaries Ûi = U i U i. Now let v 0 = g 1 g j1 g 1 g j2 g 1 g jm be a word as in the claim and set ˆV =ˆπ(v 0 )=V V. We can assume v 0 to be cyclically reduced, because ˆV I = W ˆVW 1 I for any unitary W and in the case v 0 is not cyclically reduced, we can take a cyclically reduced one in its conjugacy class. We can of course assume that =1andj m =2. Nowforthe cyclically reduced word v 0 the set F k = {(g1 1 g 3) r v 0 (g1 1 g 3) r : r =1,2,...,k} is a free set and we can do the same estimate as in the above roof. Setting T = n λ(g i)asaboveand ˆT = n Ûi we can identify F k with some subset of the set

8 3430 FRANZ LEHNER of unreduced words Wm+2k alt (n), e.g. the set {(g 1 1 g 3) r v 0 (g1 1 g 3) r (g1 1 g 1) k r : r = 1, 2,...,k},andget (T T) m+2k = v W alt m+2k (n) λ(v) v Wm+2k alt (n)\(f k F 1 k v Wm+2k alt (n)\(f k F 1 k λ(v) + λ(v)+λ(v) ) v F k ˆπ(v) +2 2k 1. ) The last inequality follows from (1) and Proosition 5. We can now aly Corollary 8 to the first term on the right-hand side and we obtain (T T ) m+2k ˆπ(v)+ 2I +2 2k 1 2k v Wm+2k alt (n)\(f k F 1 k ) v F k ˆπ(v) + 2I ˆπ(v) ˆπ(v) +2 2k 1 2k v Wm+2k alt (n) v F k (ˆT ˆT) m+2k +2k ˆπ(v 0 ) I +2 2k 1 2k. This together with the assumtion T = ˆT yields ˆV 2k 1 I 1 k and since this holds for all k, the roof is comlete. Acknowledgements I would like to thank my advisor Prof. Gilles Pisier for suggesting the roblem and for his advice and encouragement. I am grateful to the referee and Alain Valette for helful remarks on a first draft of this aer. Part of this work was done during a stay at the Texas A&M University, which I thank for its hositality. Note added in roof We noticed a connection to the aer of V. Flory, Estimating norms in - algebras of discrete grous, Math. Ann. 224 (1976) Using our results Theorem 8 in the latter aer can be sharened for finite sets as follows. Let G be a discrete grou and E G a finite subset. Theorem 8 can be sharened for finite subsets as follows. Define for K G the Letin constant #(KU) ω(k) = inf U G finite #(U). Then the following are equivalent. (1) E has the Leinert roerty. (2) For every y E the set {y 1 x : x E and x y} is a free subset of G. (3) ω(e) =#(E) 1.

9 A CHARACTERIZATION OF THE LEINERT PROPERTY 3431 References [A-O] Akemann, C.A., Ostrand, P.A., Comuting norms in grou -algebras, Am. J. of Math. 98 (1976) MR 56:1079 [B] Bożejko, M., On Λ() sets with minimal constant in discrete noncommutative grous, Proc. A.M.S. 51 (1975) MR 52:11481 [C-V] Cherix, A.P., Valette, A., On sectra of simle random walks on one-relator grous, Pac. J. of Math. 175 (1996) [H-R-V1] de la Hare, P., Robertson, A.G., Valette, A., On the Sectrum of the Sum of Generators for a Finitely Generated Grou I, Isr. J. ofmath. 81 (1993) MR 94j:22007 [H-R-V] de la Hare, P., Robertson, A.G., Valette, A., On the sectrum of the sum of generators for a finitely generated grou II, Coll. Math. 65 (1993) [K] Kesten, H., Symmetric Random Walks on Grous, Trans. A.M.S. 92 (1959) MR 22:253 [Le] Leinert, M., Faltungsoeratoren auf gewissen diskreten Gruen, Studia math. 52 (1974) MR 50:7954 [L-P-S] Lubotzky, A., Phillis, R., Sarnak, P., Hecke oerators and distributing oints on the shere II, Comm. Pure and Alied Math. 40 (1987) MR 88m:11025b [Pa] Paschke, W., Lower bound for the norm of a vertex-transitive grah, Math.Z.213 (1993) MR 94j:05085 [P-P] Picardello, M.A., Pytlik, T., Norms of free oerators, Proc. A.M.S. 104 (1988) MR 90b:47016 [P1] Pisier, G., The oerator Hilbert sace OH, comlex interolation and tensor norms, Mem. A.M.S. 122(1996), 103 gs. MR 97a:46024 [P2] Pisier, G., Quadratic forms in unitary oerators, toaear [W] Woess, W., A short comutation of the norms of free convolution oerators, Proc. A.M.S. 96 (1986) MR 87e:43002 Institut für Mathematik, Johannes Keler Universität Linz, A4040 Linz, Austria address: lehner@caddo.bayou.uni-linz.ac.at Current address: IMADA, Odense Universitet, Camusvej 55, DK 5230 Odense M, Denmark address: lehner@imada.ou.dk