Noncommutative de Leeuw theorems

Noncommutative de Leeuw theorems

Éric Ricard

Laboratoire de Mathématiques Nicolas Oresme Université de Caen Normandie

June, 2016

Joint work with M. Caspers, J. Parcet and M. Perrin

Motivation

Work by M. Junge, T. Mei, J. Parcet on transference of smooth multipliers To decide if some Idempotent multipliers are bounded on R

Ω

slope/∈Q

γ 6

-

De Leeuw's Theorems

General setting :H ⊂R^d =Rc^d subgroup,H =Zandd =1 to simplify.

Question : Relating Lp-multipliers with symbols m:R^d →C: Tcm :Lp(Rc^d)→Lp(Rc^d)

Td_mf(ξ) = m(ξ)fb(ξ), Tmf =

Z

Rⁿ

m(ξ)fb(ξ)λ(ξ)dξ, to H⊂R^d.

Recall that R^d Hb.

De Leeuw's Theorems

The restriction theorem (1965)

H ⊂R^d closed subgroup, m:R^d →Ccontinuous, let m˜ =m_|H then

T_m˜ :L_p(H)b →L_p(H)b 6

T_m:L_p(Rc^d)→L_p(Rc^d) .

Basic case : Transference of multipliers on Lp(R) toLp(T) m continuous can be weakened :

Riesz Transforms on Lp(R) ⇒Riesz Transforms on Lp(T)

De Leeuw's Theorems

The periodization theorem

H ⊂R^d closed subgroup, m:R^d/H→C, letm_p :R^d →Cwhere m_p =m◦q with q :R^d R^d/H then

Tmp :Lp(Rc^d)→Lp(Rc^d) 6

Tm:Lp(R\^d/H)→Lp(R\^d/H) .

Basic case : Transference of convolutors on `_p(Z) toL_p(R)

De Leeuw's Theorems

Let R^d_disc beR^d with the discrete topology as a LCA group.

R[^d_disc =R^d_bohr

The compactication theorem Let m:R^d →C continuous then

T_m:L_p(R^dbohr)→L_p(R^dbohr) =

T_m:L_p(Rc^d)→L_p(Rc^d) .

Compactication is somehow the strongest result.

It implies the restriction thm as it is clear for discreteG. All these theorems are about transference.

Ideas behind the proof of compactication

Ris commutative

Ris well approximated by discrete subgroups : R= ∪

k>02^−kZ.

Ris amenable : existence of Folner sets (intervals) Convolution with Gaussians γ(x) = ^√1

2πe^−x22,γ = ¹γ

x

Good approximations on both frequency and space sides.

Nice semi-group of convolution

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

Extension of multipliers Let F :R→R,F =1_[−¹

2,¹₂]∗1_[−¹

2,¹₂] be a Fejer-type kernel F_n^d =F^⊗d:R^d →R

Jodeit (1969)

Let m:Z^d →C and letm˜ :R^d →Cbem∗F^d, then

T_m˜ :L_p(R^d)→L_p(R^d) 6C_d

T_m :L_p(T^d)→L_p(T^d) . Let m:R^d →C and letm˜ be its periodic extension from[−π, π], then

Tm :`p(Z<sup>d</sup>)→`p(Z^d) 6C_d

Tm˜ :Lp(R^d)→Lp(R^d) . He also treated restrictions.

This gives the hard way in the compactication theorem.

G LCA, H⊂G be closed.

Figà-Talamanca and Gaudry (1970)

AssumeH is discrete, letm:H→C and letm˜ =m∗F ∗F for a Fejer-type kernel, then

T_m˜ :Lp( ˆG)→Lp( ˆG) 6

Tm :Lp(H)b →Lp(H)b . More satisfactory : Better constant butF ∗F instead of F. Cowling (1975)

AssumeH is closed, let m:H→Cand letm˜ =m∗F ∗F for a Fejer-type kernel, then

T_m˜ :L_p( ˆG)→L_p( ˆG) 6

T_m :L_p(H)b →L_p(H)b . Use of disintegration and structure theories for LCA groups.

He also looked at periodization.

Noncommutative G?

→ Lots of works on convolutors on L_p(G) andL_p(G/H) (Doodley, Gaudry, Derighetti,...)

→ Lots of works on (vector valued) transference (Coiman-Weiss,...)

→ Another direction : Noncommutative analysis How to dene Lp( ˆG)when G is non abelian ? For p=∞

L∞( ˆG)⊂B(L₂( ˆG))'B(L₂(G))

where L∞( ˆG) : convolution operator onB(L₂(G)) G l.c. group : left representation λ:G →B(`₂G)

L(G) =λ(G)⁰⁰=Span{λ(g) ;g ∈G}^w Noncommutative Lp-space

Non commutative analysis

M ⊂B(`₂) von Neumann algebra with a normal faithful seminite traceτ (M, τ) ↔ (L∞(Ω), µ)

τ(xy) =τ(yx) ↔ fg =gf normal ↔ monotone cv thm faithful ↔ full support

seminite ↔ σ-nite

L₀(M, τ) ↔ L₀(Ω, µ)

via functional calculus on L₀(M, τ),|y|= (y^∗y)¹² Lp(M, τ) ={x∈L₀(M, τ)| kxk^p_p =τ(|x|^p)<∞}

Not all vN admit a trace.

If not, equivalent denitions by Connes and Haagerup

Noncommutative G?

We denie L_p( ˆG) =L_p(L(G))for G LC.

Denition of the trace τ via FT (G unimodular) x =R

fgλ(g)dg,f ∈Cc(G) :τ(x) =fe. Notion of multiplier with symbolφ:G →C:

M_φ( Z

fλ(g)dg) = Z

φ_gf_gλ(g)dg

Restriction, periodization, compactication make sense in this setting.

Noncommutative G?

Pb : What is the best notion of boundedness ?

→ Boundedness of Mφ on Lp

→ Complete Boundedness : boundedness of Mφ×1 on G ×G⁰ for anyG⁰. If G andG⁰ are abelian :Mφ×1 bounded⇔M_φ bounded

Arhancet / Dutta-Mohanty-Tewari

For any innite LCA group G and 1<p 6=2<∞, there is an L_p-Fourier multiplier which is not completely bounded.

Extension of a result by Pisier for compact groups using transference.

Cb norm results

For m:G →C,Tm :Lp(L(G))→Lp(L(G))Fourier multiplier S_m :S_p(L₂(G))→S_p(L₂(G))the equivariant Schur multiplier φ(s,t) =m(s⁻¹t).

Neuwirth-R (G discrete), Caspers-de la Salle (G LC) AssumeG is amenable andm is bounded

T_m:L_p(L(G))→L_p(L(G))k_cb=

S_m:S_p(L₂(G))→S_p(L₂(G))k_cb

It suces to look at restriction, periodization, compactication for Schur multipliers.

Using basic results by Haagerup, Laorgue-de la Salle An easy compactication theorem

AssumeG is amenable, 16p6∞, letm:G →C continuous then

Tm:Lp(L(G_disc))→Lp(L(G_disc)) cb =

Tm:Lp(L(G))→Lp(L(G)) cb.

An easy restriction theorem

AssumeH ⊂G with compatible modular functions andH amenable, 16p 6∞, letm:G →Ccontinuous, m˜ =m|H then

Tm˜ :Lp(L(H))→Lp(L(H)) cb 6

Tm :Lp(L(G))→Lp(L(G)) cb. When p =1,∞, one can removeH amenable (Bo»ejko-Fendler).

Similarly there are is an easy periodization theorem.

An easy Jodeit's theorem

AssumeG is amenable andH ⊂G be a lattice with fdX, 16p 6∞, let m:H →C, putm˜ =1X ∗m∗1X then

Tm˜ :Lp(L(G))→Lp(L(G)) cb=

Tm :Lp(L(H))→Lp(L(H)) cb.

1X ∗1X =F → cst 1 : better than Jodeit's result for Z⊂Rbut cb.

Question : What is the right constant in Jodeit's thm (not cb) ?

Drawbacks :

Amenability of G : hard to get rid of it

cb assumption, one does not recover the classical results general result for bounded maps⇒ cbversion Basic idea's :

to adapt de Leeuw's approach to use other transferences if possible to relateL_p(L(H))andL_p(L(G))

The basic restriction thm

AssumeG is LC andH⊂G amenable discrete with(∆_G)_|H =1, let m:G →Ccontinuous, m˜=m_|H

T_m˜ :L_p(L(H))→L_p(L(H)) 6

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

L_p(L(H)) −→^φi L_p(L(G))

T_m˜ ↓ ↓ _Tm

Lp(L(H)) −→^φi Lp(L(G))

Take V a small symmetric neighborhood of 1∈G y = √¹

µ(V)λ(1V)∈L₂(L(G))

φ^p_y :L_p(L(H))→L_p(L(G)) ; λ(h)7→λ(h)u|y|^2/p

φ^py is a contraction by interpolation if V is small enough.

One would think of limV→{e}kφ^p_y(f)k_p=kfk_p. Obvious forp =2

UsingL₂-duality → obvious whenG is commutative

To get it we need that V is almost invariant by conjugation byH

∀h∈H, µ(hVh⁻¹∆V)/µ(V)→0 If this is true G is [SAIN]_H

H amenable ⇒G is[SAIN]_H

The commutation relation with T_m is a more delicate technical issue.

→ Obvious for p=2 : good space location of y inL₂.

→ It suces to do it for ucpTm using continuity ofm.

What is the support of |y|^t?

De Leeuw→ multiplication withγ instead of y : nice convolution semi-group : γ^t =γ_t

good approximations of identity

One needs a local control on approximations of identity for dierent values of p.

Almost multiplicative maps

Multiplicative domains

Let Abe a C^∗-algebra, T :A→Abe ucp and x =x^∗ ∈A, then T(x²) =T(x)² ⇒ T(f(x)) =f(T(x)), f ∈C(σ(x))

Using an ultraproduct argument if kxk61 and f ∈C([−1,1]) kT(x²)−T(x)²k6 ⇒ kT(f(x))−f(T(x))k6δ

If A=C([0,1]), this is a strong quantitative Korovkin theorem.

AssumeT : (M, τ)→(M, τ) is ucp trace preserving Then T :Lp→Lp

What can we say if x∈Lp?

Almost multiplicativity on L_p

Let x∈L⁺_p andT :M →M ucpτ-preserving then kT(x)−T(√

x)²k_2p6 1

2kT(x²)−T(x)²k_p. Local approximations of identity

Let y ∈L₂ with y =u|y|andT :M →M ucpτ-preserving then kT(u|y|^θ)−u|y|^θk2

θ 6CkT(y)−yk

θ

24kyk³

θ

24 . This gives the commutation relation !

No easy ultraproduct argument (type III)

More elaborated versions Recall De Leeuw's idea R= ∪

k>02^−kZ.

We say that G is ADS if there is are latticesΓi ⊂G with fdXi shrinking to {e}.

Examples : LCA, Heisenberg groups, Nilpotent matricial groups.

The restriction thm

AssumeG is LC andH⊂G with(∆_G)|H =1 and H∈ADS, G ∈[SAIN]_H, letm:G →C continuous,m˜ =m_|H for 16p6∞:

T_m˜ :L_p(L(H))→L_p(L(H)) 6

T_m :L_p(L(G))→L_p(L(G)) .

The compactication theorem

Let 16p6∞, letm:G →C continuous, If G is ADS

T_m :L_p(L(G))→L_p(L(G)) 6

T_m :L_p(L(G_disc))→L_p(L(G_disc)) , If Gdisc is amenable

T_m :L_p(L(G_disc))→L_p(L(G_disc)) 6

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.