Noncommutative de Leeuw theorems

Transcription

1 Noncommutative de Leeuw theorems Éric Ricard Laboratoire de Mathématiques Nicolas Oresme Université de Caen Basse-Normandie December, 2014 Joint work with M. Caspers, J. Parcet and M. Perrin

2 De Leeuw's Theorems General setting : H R d = R d subgroup, H = Z and d = 1 to simplify. Question : Relating L p -multipliers with symbols m : R d C : T m : L p ( R d ) L p ( R d ) T m f (ξ) = m(ξ) f (ξ), T m f (x) = m(ξ) f (ξ)e 2πi x,ξ dξ, R n to H R d. Recall that R d Ĥ.

3 De Leeuw's Theorems The restriction theorem (1965) H R d closed subgroup, m : R d C continuous, let m = m H then T m : L p (Ĥ) Lp(Ĥ) Tm : L p ( R d ) L p ( R d ).

4 De Leeuw's Theorems The restriction theorem (1965) H R d closed subgroup, m : R d C continuous, let m = m H then T m : L p (Ĥ) Lp(Ĥ) Tm : L p ( R d ) L p ( R d ). Basic case : Transference of multipliers on L p (R) to L p (T)

5 De Leeuw's Theorems The restriction theorem (1965) H R d closed subgroup, m : R d C continuous, let m = m H then T m : L p (Ĥ) Lp(Ĥ) Tm : L p ( R d ) L p ( R d ). Basic case : Transference of multipliers on L p (R) to L p (T) m continuous can be weakened : Riesz Transforms on L p (R) Riesz Transforms on L p (T)

6 De Leeuw's Theorems The periodization theorem H R d closed subgroup, m : R d /H C, let m p : R d C where m p = m q with q : R d R d /H then T mp : L p ( R d ) L p ( R d ) T m : L p ( R d /H) L p ( R d /H). Basic case : Transference of convolutors on l p (Z) to L p (R)

7 De Leeuw's Theorems Let R d disc be Rd with the discrete topology as a LCA group. R d disc = Rd bohr

8 De Leeuw's Theorems Let R d disc be Rd with the discrete topology as a LCA group. R d disc = Rd bohr The compactication theorem Let m : R d C continuous then T m : L p (R d bohr ) L p(r d bohr ) Tm = : L p ( R d ) L p ( R d ).

9 De Leeuw's Theorems Let R d disc be Rd with the discrete topology as a LCA group. R d disc = Rd bohr The compactication theorem Let m : R d C continuous then T m : L p (R d bohr ) L p(r d bohr ) Tm = : L p ( R d ) L p ( R d ). Compactication is somehow the strongest result. It implies the restriction thm as it is clear for discrete G. All these theorems are about transference.

10 R is commutative Ideas behind the proof of compactication

11 Ideas behind the proof of compactication R is commutative R is well approximated by discrete subgroups : R = k 0 2 k Z.

12 Ideas behind the proof of compactication R is commutative R is well approximated by discrete subgroups : R = k 0 2 k Z. R is amenable : existence of Folner sets (intervals)

13 Ideas behind the proof of compactication R is commutative R is well approximated by discrete subgroups : R = k 0 2 k Z. R is amenable : existence of Folner sets (intervals) ( ) Convolution with Gaussians γ(x) = 1 2π e x 2 2, γ ɛ = 1 ɛ γ x ɛ Good approximations on both frequency and space sides. Nice semi-group of convolution

14 Extensions Saeki 1970 Those 3 theorems hold more generally for LCA groups. Proof : De Leeuw thms + structure theory of LCA groups.

15 Extensions Saeki 1970 Those 3 theorems hold more generally for LCA groups. Proof : De Leeuw thms + structure theory of LCA groups. There are also related works by Igari There is another approach to compactication : Extension of multipliers from a closed subgroup to the whole group

16 Extension of multipliers Let F : R R, F = 1 [ 1, 1 ] 1 [ 1, 1 ] be a Fejer-type kernel Fn d = F d : R d R

17 Extension of multipliers Let F : R R, F = 1 [ 1, 1 ] 1 [ 1, 1 ] be a Fejer-type kernel Fn d = F d : R d R Jodeit (1969) Let m : Z d C and let m : R d C be m F d, then T m : L p (R d ) L p (R d Tm ) C d : L p (T) L p (T). Let m : [ π, π] d C and let m be its periodic extension, then T m : L p (R d ) L p (R d Tm ) C d : L p (Z) L p (Z).

18 Extension of multipliers Let F : R R, F = 1 [ 1, 1 ] 1 [ 1, 1 ] be a Fejer-type kernel Fn d = F d : R d R Jodeit (1969) Let m : Z d C and let m : R d C be m F d, then T m : L p (R d ) L p (R d Tm ) C d : L p (T) L p (T). Let m : [ π, π] d C and let m be its periodic extension, then T m : L p (R d ) L p (R d Tm ) C d : L p (Z) L p (Z). He also treated restrictions. This gives the hard way in the compactication theorem.

19 G LCA, H G be closed. Figà-Talamanca and Gaudry (1970) Assume H is discrete, let m : H C and let m = m F F for a Fejer-type kernel, then T m : L p (Ĝ) Lp(Ĝ) Tm. : L p (Ĥ) Lp(Ĥ) More satisfactory : Better constant but F F instead of F.

20 G LCA, H G be closed. Figà-Talamanca and Gaudry (1970) Assume H is discrete, let m : H C and let m = m F F for a Fejer-type kernel, then T m : L p (Ĝ) Lp(Ĝ) Tm. : L p (Ĥ) Lp(Ĥ) More satisfactory : Better constant but F F instead of F. Cowling (1975) Assume H is closed, let m : H C and let m = m F F for a Fejer-type kernel, then. T m : L p (Ĝ) L p (Ĝ) T m : L p (Ĥ) Lp(Ĥ) Use of disintegration and structure theories for LCA groups. He also looked at periodization.

21 Noncommutative G? Lots of works on convolutors on L p (G) and L p (G/H) (Doodley, Gaudry, Derighetti,...)

22 Noncommutative G? Lots of works on convolutors on L p (G) and L p (G/H) (Doodley, Gaudry, Derighetti,...) Lots of works on (vector valued) transference (Coiman-Weiss,...)

23 Noncommutative G? Lots of works on convolutors on L p (G) and L p (G/H) (Doodley, Gaudry, Derighetti,...) Lots of works on (vector valued) transference (Coiman-Weiss,...) Using noncommutative L p space we have L p (Ĝ) = L p (L(G)) for G LC. Restriction, periodization, compactication make sense in this setting.

24 Noncommutative G? Lots of works on convolutors on L p (G) and L p (G/H) (Doodley, Gaudry, Derighetti,...) Lots of works on (vector valued) transference (Coiman-Weiss,...) Using noncommutative L p space we have L p (Ĝ) = L p (L(G)) for G LC. Restriction, periodization, compactication make sense in this setting. Pb : what are the right multipliers? Bounded or completely bounded? Arhancet (2011) For any innite LCA group G and 1 < p 2 <, there is an L p -Fourier multiplier which is not completely bounded. Extension of a result by Pisier for T using transference.

25 Operator spaces results For m : G C, T m : L p (L(G)) L p (L(G)) Fourier multiplier S m : S p (L 2 (G)) S p (L 2 (G)) the equivariant Schur multiplier φ(s, t) = m(s 1 t).

26 Operator spaces results For m : G C, T m : L p (L(G)) L p (L(G)) Fourier multiplier S m : S p (L 2 (G)) S p (L 2 (G)) the equivariant Schur multiplier φ(s, t) = m(s 1 t). Neuwirth-R (G discrete), Caspers-de la Salle (G LC) Assume G is amenable and m is bounded T m : L p (L(G)) L p (L(G)) cb = S m : S p (L 2 (G)) S p (L 2 (G)) cb It suces to look at restriction, periodization, compactication for Schur multipliers.

27 Operator spaces results For m : G C, T m : L p (L(G)) L p (L(G)) Fourier multiplier S m : S p (L 2 (G)) S p (L 2 (G)) the equivariant Schur multiplier φ(s, t) = m(s 1 t). Neuwirth-R (G discrete), Caspers-de la Salle (G LC) Assume G is amenable and m is bounded T m : L p (L(G)) L p (L(G)) cb = S m : S p (L 2 (G)) S p (L 2 (G)) cb It suces to look at restriction, periodization, compactication for Schur multipliers.

28 Using basic results by Haagerup, Laorgue-de la Salle An easy compactication theorem Assume G is amenable, 1 p, let m : G C continuous then T m : L p (L(G disc )) L p (L(G disc )) = T m : L p (L(G)) L p (L(G)). cb cb

29 Using basic results by Haagerup, Laorgue-de la Salle An easy compactication theorem Assume G is amenable, 1 p, let m : G C continuous then T m : L p (L(G disc )) L p (L(G disc )) = T m : L p (L(G)) L p (L(G)). cb cb An easy restriction theorem Assume H G with compatible modular functions and H amenable, 1 p, let m : G C continuous, m = m H then T m : L p (L(H)) L p (L(H)) T m : L p (L(G)) L p (L(G)). cb cb When p = 1,, one can remove H amenable (Bo»ejko-Fendler). Similarly there are is an easy periodization theorem.

30 An easy Jodeit's theorem Assume G is amenable and H G be a lattice with fd X, 1 p, let m : H C, put m = 1 X m 1 X then T m : L p (L(G)) L p (L(G)) = T m : L p (L(H)) L p (L(H)). cb cb 1 X 1 X = F cst 1 : better than Jodeit's result for Z R but cb.

31 An easy Jodeit's theorem Assume G is amenable and H G be a lattice with fd X, 1 p, let m : H C, put m = 1 X m 1 X then T m : L p (L(G)) L p (L(G)) = T m : L p (L(H)) L p (L(H)). cb cb 1 X 1 X = F cst 1 : better than Jodeit's result for Z R but cb. Question : What is the right constant in Jodeit's thm (not cb)?

32 Drawbacks : Amenability of G : hard to get rid of it

33 Drawbacks : Amenability of G : hard to get rid of it cb assumption, one does not recover the classical results general result for bounded maps cb version

34 Drawbacks : Amenability of G : hard to get rid of it cb assumption, one does not recover the classical results general result for bounded maps cb version Basic idea's : to adapt de Leeuw's approach

35 Drawbacks : Amenability of G : hard to get rid of it cb assumption, one does not recover the classical results general result for bounded maps cb version Basic idea's : to adapt de Leeuw's approach to use other transferences if possible

36 Drawbacks : Amenability of G : hard to get rid of it cb assumption, one does not recover the classical results general result for bounded maps cb version Basic idea's : to adapt de Leeuw's approach to use other transferences if possible to relate L p (L(H)) and L p (L(G))

37 The basic restriction thm Assume G is LC and H G amenable discrete with ( G ) H = 1, let m : G C continuous, m = m H T m : L p (L(H)) L p (L(H)) T m : L p (L(G)) L p (L(G)).

38 The basic restriction thm Assume G is LC and H G amenable discrete with ( G ) H = 1, let m : G C continuous, m = m H T m : L p (L(H)) L p (L(H)) T m : L p (L(G)) L p (L(G)). Idea of the proof : To embed L p (L(H)) approximately in L p (L(G)) in a way that intertwines multipliers φ L p (L(H)) i Lp (L(G)) T m L p (L(H)) φ i Tm Lp (L(G))

39 Take V a small symmetric neighborhood of 1 G y = 1 λ(1 V ) L 2 (L(G)) µ(v )

40 Take V a small symmetric neighborhood of 1 G y = 1 λ(1 V ) L 2 (L(G)) µ(v ) φ p y : L p (L(H)) L p (L(G)) ; λ(h) λ(h)u y 2/p

41 Take V a small symmetric neighborhood of 1 G y = 1 λ(1 V ) L 2 (L(G)) µ(v ) φ p y : L p (L(H)) L p (L(G)) ; λ(h) λ(h)u y 2/p φ p y is a contraction by interpolation if V is small enough.

42 Take V a small symmetric neighborhood of 1 G y = 1 λ(1 V ) L 2 (L(G)) µ(v ) φ p y : L p (L(H)) L p (L(G)) ; λ(h) λ(h)u y 2/p φ p y is a contraction by interpolation if V is small enough. One would think of lim V {e} φ p y (f ) p = f p.

43 Take V a small symmetric neighborhood of 1 G y = 1 λ(1 V ) L 2 (L(G)) µ(v ) φ p y : L p (L(H)) L p (L(G)) ; λ(h) λ(h)u y 2/p φ p y is a contraction by interpolation if V is small enough. One would think of lim V {e} φ p y (f ) p = f p. Obvious for p = 2

44 Take V a small symmetric neighborhood of 1 G y = 1 λ(1 V ) L 2 (L(G)) µ(v ) φ p y : L p (L(H)) L p (L(G)) ; λ(h) λ(h)u y 2/p φ p y is a contraction by interpolation if V is small enough. One would think of lim V {e} φ p y (f ) p = f p. Obvious for p = 2 Using L 2 -duality obvious when G is commutative

45 Take V a small symmetric neighborhood of 1 G y = 1 λ(1 V ) L 2 (L(G)) µ(v ) φ p y : L p (L(H)) L p (L(G)) ; λ(h) λ(h)u y 2/p φ p y is a contraction by interpolation if V is small enough. One would think of lim V {e} φ p y (f ) p = f p. Obvious for p = 2 Using L 2 -duality obvious when G is commutative To get it we need that V is almost invariant by conjugation by H h H, µ(hvh 1 V )/µ(v ) 0 If this is true G is [SAIN] H

46 Take V a small symmetric neighborhood of 1 G y = 1 λ(1 V ) L 2 (L(G)) µ(v ) φ p y : L p (L(H)) L p (L(G)) ; λ(h) λ(h)u y 2/p φ p y is a contraction by interpolation if V is small enough. One would think of lim V {e} φ p y (f ) p = f p. Obvious for p = 2 Using L 2 -duality obvious when G is commutative To get it we need that V is almost invariant by conjugation by H h H, µ(hvh 1 V )/µ(v ) 0 If this is true G is [SAIN] H H amenable G is [SAIN] H

47 The commutation relation with T m is a more delicate technical issue.

48 The commutation relation with T m is a more delicate technical issue. Obvious for p = 2 : good space location of y in L 2.

49 The commutation relation with T m is a more delicate technical issue. Obvious for p = 2 : good space location of y in L 2. It suces to do it for ucp T m using continuity of m.

50 The commutation relation with T m is a more delicate technical issue. Obvious for p = 2 : good space location of y in L 2. It suces to do it for ucp T m using continuity of m. What is the support of y t?

51 The commutation relation with T m is a more delicate technical issue. Obvious for p = 2 : good space location of y in L 2. It suces to do it for ucp T m using continuity of m. What is the support of y t? De Leeuw multiplication with γ ɛ instead of y : nice convolution semi-group : γ t ɛ = γ tɛ

52 The commutation relation with T m is a more delicate technical issue. Obvious for p = 2 : good space location of y in L 2. It suces to do it for ucp T m using continuity of m. What is the support of y t? De Leeuw multiplication with γ ɛ instead of y : nice convolution semi-group : γ t ɛ = γ tɛ good approximations of identity

53 The commutation relation with T m is a more delicate technical issue. Obvious for p = 2 : good space location of y in L 2. It suces to do it for ucp T m using continuity of m. What is the support of y t? De Leeuw multiplication with γ ɛ instead of y : nice convolution semi-group : γ t ɛ = γ tɛ good approximations of identity One needs a local control on approximations of identity for dierent values of p.

54 Almost multiplicative maps Multiplicative domains Let A be a C -algebra, T : A A be ucp and x = x A, then T (x 2 ) = T (x) 2 T (f (x)) = f (T (x)), f C(σ(x))

55 Almost multiplicative maps Multiplicative domains Let A be a C -algebra, T : A A be ucp and x = x A, then T (x 2 ) = T (x) 2 T (f (x)) = f (T (x)), f C(σ(x)) Using an ultraproduct argument if x 1 and f C([ 1, 1]) T (x 2 ) T (x) 2 ɛ T (f (x)) f (T (x)) δ

56 Almost multiplicative maps Multiplicative domains Let A be a C -algebra, T : A A be ucp and x = x A, then T (x 2 ) = T (x) 2 T (f (x)) = f (T (x)), f C(σ(x)) Using an ultraproduct argument if x 1 and f C([ 1, 1]) T (x 2 ) T (x) 2 ɛ T (f (x)) f (T (x)) δ Assume T : (M, τ) (M, τ) is ucp trace preserving Then T : L p L p What can we say if x L p?

57 Almost multiplicativity on L p Let x L + p and T : M M ucp τ-preserving then T (x) T ( x) 2 2p 1 2 T (x 2 ) T (x) 2 p.

58 Almost multiplicativity on L p Let x L + p and T : M M ucp τ-preserving then T (x) T ( x) 2 2p 1 2 T (x 2 ) T (x) 2 p. Local approximations of identity Let y L 2 with y = u y and T : M M ucp τ-preserving then T (u y θ ) u y θ 2 θ This gives the commutation relation! C T (y) y θ 4 2 y 3θ 4 2.

59 Almost multiplicativity on L p Let x L + p and T : M M ucp τ-preserving then T (x) T ( x) 2 2p 1 2 T (x 2 ) T (x) 2 p. Local approximations of identity Let y L 2 with y = u y and T : M M ucp τ-preserving then T (u y θ ) u y θ 2 θ This gives the commutation relation! No easy ultraproduct argument (type III) C T (y) y θ 4 2 y 3θ 4 2.

60 More elaborated versions Recall De Leeuw's idea R = k 0 2 k Z.

61 More elaborated versions Recall De Leeuw's idea R = k 0 2 k Z. We say that G is ADS if there is are lattices Γ i G with fd X i shrinking to {e}.

62 More elaborated versions Recall De Leeuw's idea R = k 0 2 k Z. We say that G is ADS if there is are lattices Γ i G with fd X i shrinking to {e}. Examples : LCA, Heisenberg groups, Nilpotent matricial groups.

63 More elaborated versions Recall De Leeuw's idea R = k 0 2 k Z. We say that G is ADS if there is are lattices Γ i G with fd X i shrinking to {e}. Examples : LCA, Heisenberg groups, Nilpotent matricial groups. The restriction thm Assume G is LC and H G with ( G ) H = 1 and H ADS, G [SAIN] H, let m : G C continuous, m = m H for 1 p : T m : L p (L(H)) L p (L(H)) T m : L p (L(G)) L p (L(G)).

64 The compactication theorem Let 1 p, let m : G C continuous, If G is ADS T m : L p (L(G)) L p (L(G)) T m : L p (L(G disc )) L p (L(G disc )), If G disc is amenable T m : L p (L(G disc )) L p (L(G disc )) T m : L p (L(G)) L p (L(G)). There is = for LCA, Heisenberg, Nilpotent triangular matricial groups. One can also get some periodization results.

65 An application Motivation : work by M. Junge, T. Mei, J. Parcet on transference of smooth multipliers

66 An application Motivation : work by M. Junge, T. Mei, J. Parcet on transference of smooth multipliers To decide if some Idempotent multipliers are bounded on R

67 An application Motivation : work by M. Junge, T. Mei, J. Parcet on transference of smooth multipliers To decide if some Idempotent multipliers are bounded on R γ slope(γ)=β Ω