ANNALES MATHÉMATIQUES BLAISE PASCAL. Wolfgang Lück Estimates for spectral density functions of matrices over C[Z d ]

Transcription

1 NNLES MTHÉMTIQUES BLISE PSCL Wolfgang Lück Estimates for spectral density functions of matrices over C[Z d ] Volume 22, n o ), p < 22_1_73_0> nnales mathématiques Blaise Pascal, 2015, tous droits réservés L accès aux articles de la revue «nnales mathématiques Blaise Pascal» implique l accord avec les conditions générales d utilisation Toute utilisation commerciale ou impression systématique est constitutive d une infraction pénale Toute copie ou impression de ce fichier doit contenir la présente mention de copyright Publication éditée par le laboratoire de mathématiques de l université Blaise-Pascal, UMR 6620 du CNRS Clermont-Ferrand France cedram rticle mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques

2 nnales mathématiques Blaise Pascal 22, ) Estimates for spectral density functions of matrices over C[Z d ] Wolfgang Lück bstract We give a polynomial bound on the spectral density function of a matrix over the complex group ring of Z d It yields an explicit lower bound on the Novikov- Shubin invariant associated to this matrix showing in particular that the Novikov- Shubin invariant is larger than zero Estimation de fonctions de densité spectrale de matrices de C[Z d ] Résumé Nous donnons une estimation polynomiale pour la fonction de densité spectrale d une matrice sur l algèbre complexe du groupe Z d Ce résultat donne une borne inférieure explicite à l invariant de Novikov-Shubin associé à la matrice, montrant en particulier que l invariant de Novikov-Shubin est strictement positif 1 Introduction 11 Summary The main result of this paper is that for a m, n)-matrix over the complex group ring of Z d the Novikov-Shubin invariant of the bounded Z d - equivariant operator r 2) : L2 Z d ) m L 2 Z d ) n given by right multiplication with is larger than zero ctually rather explicit lower bounds in terms of elementary invariants of the minors of the matrix will be given This is a direct consequence of a polynomial bound of the spectral density function of r 2) which is interesting in its own right It will play a role This paper is financially supported by the Leibniz-Preis of the author granted by the Deutsche Forschungsgemeinschaft DFG The author wants to thank the referee for his useful comments Keywords: spectral density function, Novikov-Shubin invariants Math classification: 46L99, 58J50 73

3 W Lück in the forthcoming paper [8], where we will twist L 2 -torsion with finite dimensional representations and it will be crucial that we allow complex coefficients and not only integral coefficients Novikov-Shubin invariants were originally defined analytically in [10, 11] More information about them can be found for instance in [7, Chapter 2] Before we state the main result, we need the following notions 12 The width and the leading coefficient Consider a non-zero element p = pz 1 ±1,, z±1 d ) in C[Zd ] = C[z 1 ±1,, z±1 d ] for some integer d 1 There are integers n d and n + d and elements q n z 1 ±1,, z±1 d 1 ) in C[Z d 1 ] = C[z 1 ±1,, z±1 d 1 ] uniquely determined by the properties that n d n + d ; q n z 1 ±1,, z±1 d 1 ) 0; d q n + d z ±1 1,, z±1 d 1 ) 0; + nd pz 1 ±1,, z±1 d ) = q n z ±1 1,, z±1 d 1 ) zn d n=n d In the sequel denote wp) = n + d n d ; q + p) = q n + z 1 ±1,, z±1 d 1 ) d Define inductively elements p i z ±1 1,, z±1 d i ) in C[Zd i ] = C[z ±1 1,, z ±1 d i ] and integers w ip) 0 for i = 0, 1, 2,, d by p 0 z 1 ±1 d ) := pz±1 1,, z±1 d ); p 1 z 1 ±1 d 1 ) := q+ p) p i := q + p i 1 ) for i = 1, 2, d; w 0 p) := wp) w i p) := wp i ) for i = 1, 2, d 1) 74

4 Estimates for spectral density functions Define the width of p = pz 1 ±1,, z±1 ) to be d wdp) = max{w 0 p), w 1 p),, w d 1 p)}, 11) and the leading coefficient of p to be Obviously we have leadp) = p d 12) wdp) wdp 1 ) wdp 2 ) wdp d ) = 0; leadp) = leadp 1 ) = = leadp 0 ) 0 Notice that p i, wdp) and leadp) do depend on the ordering of the variables z 1,, z d Remark 11 Leading coefficient) The name leading coefficient comes from the following alternative definition Equip Z d with the lexicographical order, ie, we put m 1,, m d ) < n 1,, n d ), if m d < n d, or if m d = n d and m d 1 < n d 1, or if m d = n d, m d 1 = n d 1 and m d 2 < n d 2, or if, or if m i = n i for i = d, d 1),, 2 and m 1 < n 1 We can write p as a finite sum with complex coefficients a n1,,n d pz 1 ±,, z± d ) = a n1,,n d z1 n1 zn 2 2 zn d d n 1,,n d ) Z d Let m 1, m d ) Z d be maximal with respect to the lexicographical order among those elements n 1,, n d ) Z d for which a n1,,n d 0 Then the leading coefficient of p is a m1,,m d 13 The L 1 -norm of a matrix For an element p = g Z d g g C[Z d ] define p 1 := g G g For a matrix M m,n C[Z d ]) define 1 = max{ a i,j 1 1 i m, 1 j n} 13) The main purpose of this notion is that it gives an a priori upper bound on the norm r 2) : L2 Z d ) L 2 Z d ), namely, we get from [7, Lemma 1333 on page 466] r 2) m n 1 14) 75

5 W Lück 14 The spectral density function Given M m,n C[Z d ]), multiplication with induces a bounded Z d - equivariant operator r 2) : L2 Z d ) m L 2 Z d ) n We will denote by F r 2) ) : [0, ) [0, 5) its spectral density function in the sense of [7, Definition 21 on page 73], namely, the von Neumann dimension of the image of the operator obtained by applying the functional calculus to the characteristic function of [0, 2 ] to the operator r 2) ) r 2) In the special case m = n = 1, where is given by an element p C[Z d ], it can be computed in terms of the Haar measure µ T d of the d-torus T d see [7, Example 26 on page 75] F r 2) ) ) = µt d {z1,, z d ) T d pz 1,, z d ) } ) 16) 15 The main result Our main result is: Theorem 12 Main Theorem) Consider any natural numbers d, m, n and a non-zero matrix M m,n C[Z d ]) Let B be a quadratic submatrix of of maximal size k such that the corresponding minor p = det C[Z d ]B) is non-trivial Then: 1) If wdp) 1, the spectral density function of r 2) : L2 Z d ) m L 2 Z d ) n satisfies for all 0 F r 2) ) 2)) ) F r 0) 8 3 k d wdp) k 2k 2 B 1 ) k 1 leadp) d wdp) If wdp) = 0, then F r 2) ) ) = 0 for all < leadp) and F r 2) ) ) = 1 for all leadp) ; 2) The Novikov-Shubin invariant of r 2) is or + or a real number satisfying α r 2) d wdp), and is in particular larger than zero 76

6 Estimates for spectral density functions It is known that the Novikov-Shubin invariants of r 2) for a matrix over the integral group ring of Z d is a rational numbers larger than zero unless its value is or + This follows from Lott [4, Proposition 39] The author of [4] informed us that his proof of this statement is correct when d = 1 but has a gap when d > 1 The nature of the gap is described in [5, page 16] The proof in this case can be completed by the same basic method used in [4]) This confirms a conjecture of Lott-Lück [6, Conjecture 72] for G = Z d The case of a finitely generated free group G is taken care of by Sauer [12] Virtually finitely generated free abelian groups and virtually finitely generated free groups are the only cases of finitely generated groups, where the positivity of the Novikov-Shubin invariants for all matrices over the complex group ring is now known In this context we mention the preprints [1, 2], where examples of groups G and matrices M m,n ZG) are constructed for which the Novikov-Shubin invariant of r 2) disproving a conjecture of Lott-Lück [6, Conjecture 72] 16 Example is zero, Consider the case d = 2, m = 3 and n = 2 and the 3, 2)-matrix over C[Z 2 ] z = 1 3 ) z 1 z z 2 z 1 z 2 Let B be the 2, 2)-submatrix obtained by deleting third column Then k = 2, z B = 1 3 ) 1 2 z 1 z z 2 and we get p := det C[Z 2 ]B) = z1 3 z z 1 z Using the notation of Section 12 one easily checks p 1 z 1 ) = 2 z 1, wdp) = 2, and leadp) = 2 Obviously 1 = max{ 1, 1, , 1 } = 18 Hence Theorem 12 implies for all 0 F r 2) ) 2)92 2 ) F r 0) 1 4 α r 2) )

7 W Lück 2 The case m = n = 1 The main result of this section is the following Proposition 21 Consider an non-zero element p in C[Z d ]If wdp) = 0, then F r 2) ) 2)) ) = 0 for all < leadp) and F r ) = 1 for all leadp) If wdp) 1, we get for the spectral density function of r p 2) for all 0 F r p 2) ) 8 3 ) d wdp) d wdp) leadp) For the case d = 1 and p a monic polynomial, a similar estimate of the shape F r p 2) ) ) Ck 1 k 1 can be found in [3, Theorem 1], where the k 2 is the number of non-zero coefficients, and the sequence of real numbers C k ) k 2 is recursively defined and satisfies C k k 1 21 Degree one In this subsection we deal with Proposition 21 in the case d = 1 We get from the Taylor expansion of cosx) around 0 with the Lagrangian remainder term that for any x R there exists θx) [0, 1] such that cosx) = 1 x2 cosθx) x) + x 4 2 4! This implies for x 0 and x 1/2 2 2 cosx) x 2 1 = 2 cosθx) x) x 2 4! Hence we get for x [ 1/2, 1/2] 2 cosθx) x) 4! = 1 48 x 2 48 x2 2 2 cosx) 21) Lemma 22 For any complex number a Z we get for the spectral density function of z a) C[Z] = C[z, z 1 ] F r 2) ) 8 3 z a ) for [0, ) 78

8 Estimates for spectral density functions Proof We compute using 16), where r := a, F r 2) z a) ) = µs 1{z S 1 z a } = µ S 1{z S 1 z r } = µ S 1{φ [ 1/2, 1/2] cosφ) + i sinφ) r } = µ S 1{φ [ 1/2, 1/2] cosφ) + i sinφ) r 2 2 } = µ S 1{φ [ 1/2, 1/2] cosφ) r) 2 + sinφ) 2 2 } = µ S 1{φ [ 1/2, 1/2] r 2 2 cosφ) + r 1) 2 2 } We estimate using 21) for φ [ 1/2, 1/2] r 2 2 cosφ)) + r 1) 2 r 2 2 cosφ)) 48 φ2 This implies for 0 F r z a) 2) ) = µs 1{φ [ 1/2, 1/2] r 2 2 cosφ) + r 1) 2 2 } µ S 1{φ [ 1/2, 1/2] 48 φ2 2 } { 48 = µ S 1 φ [ 1/2, 1/2] φ 2 48 = 8 3 Lemma 23 Let pz) C[Z] = C[z, z 1 ] be a non-zero element If wdp) = 0, then F r p 2) ) 2)) ) = 0 for all < leadp) and F r p ) = 1 for all leadp) If wdp) 1, we get F r p 2) ) 8 3 ) wdp) leadp) wdp) } for [0, ) Proof If wdp) = 0, then p is of the shape C z n, and the claim follows directly from 16) Hence we can assume without loss of generality that wdp) 1 We can write pz) as a product r pz) = leadp) z k z a i ) 79 i=1

9 W Lück for an integer r 0, non-zero complex numbers a 1,, a r and an integer k Since for any polynomial p and complex number c 0 we have for all [0, ) F r c p 2) ) ) = F r 2)) ) p, c we can assume without loss of generality leadp) = 1 If r = 0, then pz) = z k for some k 0 and the claim follows by a direct inspection Hence we can assume without loss of generality r 1 Since the width, the leading coefficient and the spectral density functions of pz) and z k pz) agree, we can assume without loss of generality k = 0, or equivalently, that pz) has the form for some r 1 r pz) = z a i ) i=1 We proceed by induction over r The case r = 1 is taken care of by Lemma 22 The induction step from r 1 1 to r is done as follows Put qz) = r 1 i=1 z a i) Then pz) = qz) z a r ) The following inequality for elements q 1, q 2 C[z, z 1 ] and s 0, 1) is a special case of [7, Lemma 213 3) on page 78] F r 2) q 1 q 2 ) ) F r 2) q 1 ) 1 s ) + F r 2) q 2 ) s ) 22) We conclude from 22) applied to pz) = qz) z a r ) in the special case s = 1/r F r 2) p ) ) F r 2)) r 1 2) ) q r ) + F r z a r 1/r ) We conclude from the induction hypothesis for [0, ) F r q 2) ) 8 3 ) r 1) 1 r 1 ; F r 2) ) 8 3 z a r ) 80

10 Estimates for spectral density functions This implies for [0, ) F r 2) p ) ) F r 2)) r 1 2) q r ) + F r 8 3 r 1) r 1 r ) z a r 1/r ) r r 8 3 r 1) 1 r r = 8 3 r 1 r 22 The induction step Now we finish the proof of Proposition 21 by induction over d If wdp) = 0, then p is of the shape C z n 1 1 zn 2 2 zn d d, and the claim follows directly from 16) Hence we can assume without loss of generality that wdp) 1 The induction beginning d = 1 has been taken care of by Lemma 23, the induction step from d 1 to d 2 is done as follows Since F r p 2) ) ) 1, the claim is obviously true for leadp) 1 Hence we can assume in the sequel leadp) 1 We conclude from 16) and Fubini s Theorem applied to T d = T d 1 S 1, where χ denotes the characteristic function of a subset and p 1 z 1 ±,, z±1 d 1 ) has been defined in Subsection 12 F r p 2) ) ) 23) = µ T d {z1,, z d ) T d pz 1,, z d ) } ) = χ {z1 T d,,z d ) T d pz 1,,z d ) } dµ T n ) = χ {z1 T d 1 S 1,,z d ) T d pz 1,,z d ) } dµ S 1 dµ T d 1 81

11 W Lück = χ {z1 T d 1,,z d 1 ) T d 1 p 1 z 1,,z d 1 ) leadp) 1/d d 1)1/d } ) χ {z1 S 1,,z d ) T d pz 1,,z d ) } dµ S 1 dµ T d 1 + χ {z1 T d 1,,z d 1 ) T d 1 p 1 z 1,,z d 1 )> leadp) 1/d d 1))/d } ) χ {z1 S 1,,z d ) T d pz 1,,z d ) } dµ S 1 dµ T d 1 χ z1 T d 1,,z d 1 ) p 1 z 1,,z d 1 ) leadp) 1/d d 1)1/d } + { max χ {z1 S 1,,z d ) T d pz 1,,z d ) } dµ S 1 z 1,, z d 1 ) T d 1 with p 1 z 1,, z d 1 ) > leadp) 1/d d 1)/d } We get from the induction hypothesis applied to p 1 z 1,, z d 1 ) and 16) since leadp) 1, wdp 1) wdp) and leadp) = leadp 1 ) χ z1 T d 1,,z d 1 ) p 1 z 1,,z d 1 ) leadp) 1/d d 1)1/d } 24) = χ z1 T d 1,,z d 1 ) p 1 z 1,,z d 1 ) leadp 1 ) 1/d d 1)1/d } = F r p 2) ) 1 leadp1 ) 1/d d 1)/d) 8 3 leadp1 ) 1/d d 1)/d d 1) wdp 1 ) d 1) wdp 1 ) leadp 1 ) = 8 3 d wdp 1 ) d 1) wdp 1 ) = 8 3 d 1) wdp 1 ) 8 3 d 1) wdp) 8 3 d 1) wdp) leadp 1 ) leadp) leadp) leadp) 82 d wdp 1 ) d wdp 1 ) d wdp)

12 Estimates for spectral density functions Fix z 1,, z d 1 ) T d 1 with p 1 z 1,, z d 1 ) > leadp/d d 1)/d Consider the element fz d ±1 ) := pz 1, z d 1, z d ± ) C[z± d ] It has the shape fz ± d ) = n + n=n q n z 1,, z d 1 ) z n d The leading coefficient of fz d ±1 ) is p 1z 1, z d 1 ) = q n+ z 1,, z d 1 ) Hence we get from Lemma 23 applied to fz d ±1 ) and 16) since leadp) 1, wdf) wdp) and leadf) = p 1 z 1, z d 1 )) > leadp) 1/d d 1)/d χ {z1 S 1,,z d ) T d pz 1,,z d ) } dµ S 1 25) = χ {zd S 1 fz S 1 d ) } dµ S 1 = 8 3 wdf) wdf) leadf) 8 3 wdf) wdf) leadp/d d 1)/d = 8 3 d wdf) wdf) leadp) 8 3 d wdp) wdp) leadp) Combining 23), 24) and 25) yields for with F r p 2) ) 8 3 ) d 1) wdp) wdp) = 8 3 d wdp) This finishes the proof of Proposition leadp) leadp) leadp) leadp) 1 d wdp) d wdp) d wdp)

13 W Lück 3 Proof of the main Theorem 12 Now we can complete the proof of our Main Theorem 12 We need the following preliminary result Lemma 31 Consider B M k,k C[Z d ]) such that p := det C[Z d ]B) is non-trivial Then we get for all 0 F r 2) B ) ) k F r 2)) 2) p r B k 1 ) Proof In the sequel we will identify L 2 Z d ) and L 2 T d ) by the Fourier transformation We can choose a unitary Z d -equivariant operator U : L 2 Z d ) k L 2 Z d ) k and functions f 1, f 2,, f k : T d R such that 0 f 1 z) f 2 z) f k z) holds for all z T d and we have the following equality of bounded Z d -equivariant operators L 2 Z d ) k = L 2 T d ) k L 2 Z d ) k = L 2 T d ) k, see [9, Lemma 22] r 2) B ) r 2) B r 2) = U f r 2) f r 2) f r 2) f k r 2) f k U 31) Since p 0 holds by assumption and hence the rank of B over C[Z d ] 0) is maximal, we conclude from [7, Lemma 134 on page 35] that r 2) B and hence r 2) f i for each i = 1, 2,, k are weak isomorphisms, ie, they are injective and have dense images We conclude from [7, Lemma ) on page 77 and Lemma 213 on page 78] F r 2) B )) = F r 2) B ) r 2) ) B 2 ) = k i=1 F r 2) f i ) 2 ) For i = 1, 2,, k we have f 1 z) f i z) for all z T d and hence F r 2) ) 2)) f i ) F r f 1 ) for all 0 This implies F r 2) ) 2) B ) k F r f 1 ) 2 ) 32) 84

14 Estimates for spectral density functions Let B M k,k C[Z d ) be the matrix obtain from B by transposition and applying to each entry the involution C[Z d ] C[Z d sending g G g g to g G g g 1 Then r 2) ) 2) B = r B Since r 2) B ) r 2) B = r2) BB and det C[Z d ]BB ) = det C[Z d ]B) det C[Z d ]B ) = p p holds, we conclude from 31) the equality of functions T d [0, ] k pp = f i i=1 Since sup{ f i z) z T d } agrees with the operatornorm r 2 f i and we have r 2) B 2 = r 2) B ) r 2) B = max{ r 2) f i i = 1, 2,, k} = r 2) f k, we obtain the inequality of functions T d [0, ] k ) pp r 2) f i f 1 r 2) B 2) k 1 f1 i=2 Hence we get for all 0 F r 2) ) 2) pp r B k 1 ) 2 ) = F r 2) ) pp r 2) B 2) k 1 ) 2 F r 2) ) B 2) k 1 2) r f 1 r 2) B 2) k 1 ) 2 = F r 2) f 1 ) 2 ) This together with 32) and [7, Lemma ) on page 77] implies F r 2) ) 2) B ) k F r f 1 ) 2 ) k F r 2) ) 2) pp r B k 1 ) 2 ) k F r 2) p ) r 2) B k 1 ) Proof of the Main Theorem 12 1) In the sequel we denote by dim N G) the von Neumann dimension, see for instance [7, Subsection 113] The rank of the matrices and B over the quotient field C[Z d ] 0) is k The operator r 2) B : L2 Z d ) k L 2 Z d ) k is a weak isomorphism, and dim N Z d )imr 2) )) = k because of [7, Lemma 134 1) on page 35] In particular we have F r 2) ) B 0) = 0 85

15 W Lück Let i 2) : L 2 Z d ) k L 2 Z d ) m be the inclusion corresponding to I {1, 2,, m} and let pr 2) : L 2 Z d ) n L 2 Z d ) k be the projection corresponding to J {1, 2,, n}, where I and J are the subsets specifying the submatrix B Then r 2) B : L2 Z d ) k L 2 Z d ) k agrees with the composite r 2) B : L2 Z d ) k i 2) L 2 Z d ) m r2) L 2 Z d ) n pr2) L 2 Z d ) k Let p 2) : L 2 Z d ) m kerr 2) )) be the orthogonal projection onto the orthogonal complement kerr 2) ) L 2 G) m of the kernel of r 2) Let j 2) : imr 2) ) L2 G) n be the inclusion of the closure of the image of r 2) Let r2) ) : kerr 2) ) imr 2) ) be the Zd -equivariant bounded operator uniquely determined by r 2) = j 2) r 2) ) p 2) The operator r 2) ) is a weak isomorphism by construction We have the decomposition of the weak isomorphism r 2) B = pr2) r 2) i2) = pr 2) j 2) r 2) ) p 2) i 2) 33) This implies that the morphism p 2) i 2) : L 2 Z d ) k ) kerr 2) ) is injective and the morphism pr 2) j 2) : imr 2) ) L2 Z d ) k has dense image Since we already know dim N G) imr 2) )) = k = dim N G) L 2 Z d ) k), the operators p 2) i 2) : L 2 Z d ) k kerr 2) ) and pr 2) j 2) : imr 2) ) L 2 Z d ) are weak isomorphisms Since the operatornorm of pr 2) j 2) and of p 2) i 2) is less or equal to 1, we conclude from [7, Lemma 213 on page 78] and 33) F r 2) ) 2)) ) F r 0) = F r 2) ) ) ) F pr 2) j 2) r 2) ) p 2) i 2)) pr 2) j 2) p 2) i 2) ) = F r 2) ) B pr 2) j 2) p 2) i 2) ) F r 2) ) B ) 86

16 Estimates for spectral density functions Put p = det C[Z d ]B) If wdp) = 0, the claim follows directly from Proposition 21 It remains to treat the case wdp) 1 The last inequality together with 14) applied to B, Proposition 21 applied to p and Lemma 31 applied to B yields for 0 F r 2) ) 2)) ) F r 0) ) ) ) 2) r B k 1 ) ) k2 B 1 ) k 1 ) F r 2) B k F r 2) p k F r 2) p 8 3 k d wdp) k 2k 2 B 1 ) k 1 leadp) d wdp) This finishes the proof of assertion 1) ssertion 2) is a direct consequence of assertion 1) and the definition of the Novikov-Shubin invariant This finishes the proof of Theorem 12 References [1] L Grabowski Group ring elements with large spectral density, [2] L Grabowski & B Virág Random walks on Lamplighters via random Schrödinger operators, Preprint, 2013 [3] W M Lawton problem of Boyd concerning geometric means of polynomials, J Number Theory ), no 3, p [4] J Lott Heat kernels on covering spaces and topological invariants, J Differential Geom ), no 2, p [5], Delocalized L 2 -invariants, J Funct nal ), no 1, p 1 31 [6] J Lott & W Lück L 2 -topological invariants of 3-manifolds, Invent Math ), no 1, p [7] W Lück L 2 -Invariants: Theory and pplications to Geometry and K-Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 3 Folge Series of Modern Surveys in Mathematics [Results in 87

17 W Lück Mathematics and Related reas 3rd Series Series of Modern Surveys in Mathematics], vol 44, Springer-Verlag, Berlin, 2002 [8], Twisting L 2 -invariants with finite-dimensional representations, in preparation, 2015 [9] W Lück & M Rørdam lgebraic K-theory of von Neumann algebras, K-Theory ), no 6, p [10] S P Novikov & M Shubin Morse inequalities and von Neumann II 1 -factors, Dokl kad Nauk SSSR ), no 2, p [11], Morse inequalities and von Neumann invariants of nonsimply connected manifolds, Uspekhi Matem Nauk ), no 5, p , in Russian [12] R Sauer Power series over the group ring of a free group and applications to Novikov-Shubin invariants, in High-dimensional manifold topology, World Sci Publ, River Edge, NJ, 2003, p Wolfgang Lück Mathematicians Institut der Universität Bonn Endenicher llee Bonn, Germany tkohl@buedu wolfganglueck@himuni-bonnde 88