sidon sets in discrete groupsmath.iisc.ac.in/~naru/dmha/wednesday/pisier.pdf · sidon sets in the...

Sidon sets in discrete groups

Gilles PisierTexas A&M University

Conference in honour of Professor ThangaveluIISc. Dec. 20, 2017

Gilles Pisier Sidon sets in discrete groups

Sidon sets in the last century (Golden age)

Λ ⊂ ZZ is Sidon if∑n∈Λ

ane int ∈ C (T)⇒∑n∈Λ

|an| <∞

Sidon sets (and more generally “thin sets” e.g. Helson sets) werea very active subject in the 1960’s and 1970’s: Kahane, Malliavin,Varopoulos, Yves Meyer, Bonami +others (in France), Edwards &Gaudry (Australia), Figa-Talamanca (Italy), Rudin, Hewitt & Ross,Rider (USA), Hartman & Ryll-Nardzewski, Bozejko (Poland),Katznelson (Israel), Herz, Drury (Canada)...The first period culminated with Sam Drury’s solution of

“the Union problem”:Drury (1970): The union of two Sidon sets is again a Sidon set.


References

1970: E. Hewitt and K. Ross, Abstract harmonic analysis, VolumeII, Structure and Analysis for Compact Groups, Analysis on LocallyCompact Abelian Groups, Springer, Heidelberg, 1970.1975: J. Lopez and K.A. Ross, Sidon sets. Lecture Notes inPure and Applied Mathematics, Vol. 13. Marcel Dekker, Inc., NewYork, 1975.1981: M.B. Marcus and G. Pisier, Random Fourier series withApplications to Harmonic Analysis, Annals of Math. Studiesn◦101, Princeton Univ. Press, 1981.1985: J. P. Kahane, Some random series of functions. Secondedition, Cambridge University Press, 1985.2013: C. Graham and K. Hare, Interpolation and Sidon sets forcompact groups. Springer, New York, 2013. xviii+249 pp.More recent work 2015-2017: J. Bourgain-M.LewkoAnnales Inst. Fourier 2017G.P. Math. Res. Letters 2017 + papers to appear,all on arxiv


Sidon

Λ ⊂ ZZ is Sidon if∑n∈Λ

ane int ∈ C (T)⇒∑n∈Λ

|an| <∞

Equivalently: ∃C such that ∀A ⊂ Λ with |A| <∞∑n∈A|an| ≤ C‖

∑n∈A

ane int‖∞

More generally, let G be a compact Abelian group, Λ = {ϕn} ⊂ G(characters on G ), Λ is Sidon if ∃C such that ∀A with |A| <∞∑

n∈A|an| ≤ C‖

∑n∈A

anϕn‖∞


Fundamental Example

G = TN

∀z = (zn) ∈ TN ϕn(z) = zn

‖∑

anϕn‖∞ =∑|an| (C = 1)

Note: (ϕn) are independent random variables


More Examples

Hadamard lacunary sequences n1 < n2 < · · · < nk , · · · such that

infk

nk+1

nk> 1

Explicit examplenk = 2k

Basic Example: Quasi-independent setsΛ is quasi-independent if all the sums

{∑n∈A

n | A ⊂ Λ, |A| <∞} are distinct numbers

quasi-independent ⇒ SidonMain Open ProblemIs every Sidon set a finite union of quasi-independent sets ?


The recent rebirth

Bourgain and Lewko (Ann. Inst. Fourier 2017) wondered whethera group environment is needed for the known results about SidonsetsQuestionWhat remains valid if Λ ⊂ G is replaced by a uniformly boundedorthonormal system ?


Let Λ = {ϕn} ⊂ L∞(T ,m) orthonormal in L2(T ,m) ((T ,m) anyprobability space)

(i) We say that (ϕn) is Sidon with constant C if for any N andany complex sequence (an) we have∑N

1|an| ≤ C‖

∑N

1anϕn‖∞.

(ii) Let k ≥ 1. We say that (ϕn) is ⊗k -Sidon with constant C ifthe system {ϕn(t1) · · ·ϕn(tk)} (or equivalently {ϕ⊗kn }) isSidon with constant C in L∞(T k ,m⊗k).

Crucial remark: For characters on a compact group T

Sidon ⇔ ⊗k − Sidon

because

‖∑N

1anϕn‖∞ = ‖

∑N

1anϕn(t1) · · ·ϕk(tk)‖L∞(T k )


The Return of Union Problem

Theorem (Union problem for unif.bded o.n. systems)

Let (ϕn) be an orthonormal system bounded in L∞. Assume that(ϕn) is the union of two (or finitely many) Sidon systems. Then(ϕn) is ⊗4-Sidon.

But is it Sidon ?


No !Let (εn) be i.i.d. ±1-valued symmetric random variables(e.g. the Rademacher functions)

Proposition

There are two orthonormal martingale difference sequences(ϕ+

n ) and (ϕ−n ) in L2 with span[ϕ+n ] ⊥ span[ϕ−n ] such that

(ϕ+n ) = (ϕ+

n ) = (εn) in distribution

but their union is not a Sidon system.More precisely the union of {ϕ+

k | k ≤ n} and {ϕ−k | k ≤ n} has aSidon constant Cn ≈

√n.

Same holds with (εn) replaced by our fundamental example (zn)


About Randomly Sidon

We say that (ϕn) is randomly Sidon with constant C if for any Nand any complex sequence (an) we have∑N

1|an| ≤ C Average±1‖

∑N

1±anϕn‖∞,

Theorem

Let (ϕn) be an orthonormal system system bounded in L∞. Thefollowing are equivalent:

(i) The system (ϕn) is randomly Sidon.

(ii) The system (ϕn) is ⊗4-Sidon.

(iii) The system (ϕn) is ⊗k -Sidon for all k ≥ 4.

(iv) The system (ϕn) is ⊗k -Sidon for some k ≥ 4.

This generalizes Rider’s 1975 result that randomly Sidon impliesSidon for charactersOpen question: What about k = 2 or k = 3 ?


Non-commutative case

Two different possibilities have been consideredEITHER(I) Replace the compact Abelian group (e.g. T) by anon-Abelian compact group G such as SO(n),SU(n),U(n), ....Then the set Λ ⊂ G is a subset of the dual object i.e. the set ofunitary irreducible representationsOR(II) Replace the discrete Abelian group (e.g. ZZ) by anon-Abelian discrete group Γ such as a free group IFn

Then Λ ⊂ ΓIn both cases I have obtained the analogues of the preceding i.e.results for general orthonormal functions, that imply the case ofcharacters as special case using the notion of ⊗k -Sidon


Sidon sets in duals of compact non-commutativegroups

G compact non-commutative groupG the set of distinct irreps, dπ = dim(Hπ)Λ ⊂ G is called Sidon if ∃C such that ∀aπ ∈ Mdπ (π ∈ Λ) we have∑

π∈Λdπtr|aπ| ≤ C‖

∑π∈Λ

dπtr(πaπ)‖∞.

Λ ⊂ G is called randomly Sidon if ∃C such that ∀aπ ∈ Mdπ

(π ∈ Λ) we have∑π∈Λ

dπtr|aπ| ≤ C IE‖∑

π∈Λdπtr(εππaπ)‖∞

where (επ) are an independent family such that each επ isuniformly distributed over O(dπ).Important Remark (easy proof) Different randomizations (e.g.Gaussian random matrices) lead to equivalent definitions


Fundamental example

G =∏n≥1

U(dn)

Λ = {πn | n ≥ 1}

πn : G → U(dn) n-th coordinate

C = 1 :∑

n≥1dntr|an| = ‖

∑n≥1

dntr(πnan)‖∞.

−−−−−−−−−−−−−−−−−−−−−−−−−−−−

Observe that for the functions ϕn(i , j) defined on (G ,mG ) by

ϕn(i , j)(g) = πn(g)ij

{d1/2n ϕn(i , j) | n ≥ 1, 1 ≤ i , j ≤ dn} is an orthonormal system.


Rider (1975, unpublished) extended all results previouslymentioned to arbitrary compact groupsin particular: randomly Sidon implies Sidon (solving thenon-commutative union problem)I posted a paper on this on arxiv including (presumably) his proof


General matricial systems

Assume given a sequence of finite dimensions dn.For each n let (ϕn) be a random matrix of size dn × dn on (T ,m).We call this a “matricial system”:

ϕn = [ϕn(i , j)]

or rather for t ∈ Tϕn(t) = [ϕn(i , j)(t)]


The uniform boundedness condition becomes

∃C ′ ∀n ‖ϕn‖L∞(Mdn ) ≤ C ′.

The orthonormality condition becomes :

{d1/2n ϕn(i , j) | n ≥ 1, 1 ≤ i , j ≤ dn}

is an orthonormal system.


The definition of ⊗k-Sidon now means that the family of matrix

products (ϕn(t1) · · ·ϕn(tk)) is Sidon on (T ,m)⊗k

Theorem (The union problem)

The union of two “orthogonal” Sidon sets is ⊗4-Sidon

t 7→ ψ1(t) ∈ Md t 7→ ψ2(t) ∈ Md

(ψ1⊗ψ2)(t1, t2) = ψ1(t1)ψ2(t2)

Analogous result for Randomly Sidon


Sidon sets in discrete non-commutative groups

Γ (arbitrary) discrete group, C ∗(Γ) the full (or maximal)C ∗-algebra of Γ i.e. the C ∗-algebra generated by the universalrepresentation UΓ : G → B(H) of Γ Consider a subset Λ ⊂ ΓWe set

ϕt = UΓ(t)

Definition

Λ is called is “operator Sidon” if there is a constant C such thatfor any finitely supported a : Λ→ B(H) (H arbitrary, say H = `2)we have

supzt∈B(H)‖zt‖≤1

{‖∑

t∈Λa(t)⊗ zt‖B(H⊗2H) ≤ C‖

∑t∈Λ

a(t)⊗ ϕt‖B(H⊗2H).

Remark: This is much stronger than Sidon but when dim(H) = 1this reduces to the previous definition of Sidon sets, because

sup zn∈B(H)‖zn‖≤1

{|∑N

1 zn ⊗ an| =∑N

1 |an|


Proposition

Consider Λ ⊂ Γ. The Following Are Equivalent:

(i) Λ is operator Sidon

(ii) span[ϕt | t ∈ Λ] ' `1(Λ) completely isomorphically

(iii) ∀f : Λ→ B(H) in `∞(B(H)) ∃f : Γ→ B(H) of the form

∀t ∈ Γ f (t) = V ∗π(t)W

for some unitary representation π : Γ→ B(Hπ) andV ,W ∈ B(H,Hπ).

Proof is easy:(i) ⇔ (ii) is essentially obvious from definitionsproof of (i) ⇒ (iii) is by (Arveson) Hahn-Banach:To any f associate uf : span[ϕt | t ∈ Λ]→ C with ‖uf ‖cb ≤ C

−−−−−−−−−−−−Variants of interpolation pty (iii) were considered for generaldiscrete groups in the 1980’s by Bozejko, Picardello and others.


Fundamental Example

Γ = IF∞ with free generators (gn)

Λ = {gn}

or more generally any free set is operator Sidon

Remark

If ∃Λ ⊂ Γ infinite operator Sidon set then Γ is non-amenable, butwe do not know whether IF2 ⊂ Γ


Recall Λ is operator Sidon IFF

(iii) ∀f : Λ→ B(H) in `∞(B(H)) ∃F : Γ→ B(H) of the form

∀t ∈ Γ F (t) = V ∗π(t)W

for some unitary representation π : Γ→ B(Hπ) andV ,W ∈ B(H,Hπ).

Natural operator valued analogue of“Fourier-Stieltjes algebra”:For F : Γ→ B(H)

‖F‖B(Γ;B(H)) = inf{‖V ‖‖W ‖ | F ( ) = V ∗π( )W }

Recall when Γ is Abelian, Γ is compact then in the case B(H) = C

B(Γ) = M(Γ) and ‖F‖B(Γ) = ‖F‖M(Γ)

(iii) ∀f ∈ `∞(B(H)) ∃F ∈ B(Γ;B(H)) such that FΛ = f and

‖F‖B(Γ;B(H)) ≤ C‖f ‖`∞(B(H)).


The following was proved very recently:

Theorem

Operator Sidon sets are stable by union.

Corollary

Finite union of translates of free sets are operator Sidon

Open problem: Is every operator Sidon set the finite union oftranslates of free sets ?


Sidon sets in C ∗-algebras

The right framework for the preceding is C ∗-algebrasLet (ϕn) be a bounded sequence in a C ∗-algebra

A ⊂ B(H)

Let K be another infinite dimensional Hilbert space (say K = `2)We say that (ϕn) is completely Sidon if there is C such that ∀Nand all an ∈ B(K ) we have

supzn∈B(K)‖zn‖≤1

{‖∑N

1zn ⊗ an‖ ≤ C‖

∑N

1an ⊗ ϕn‖.

We have also extended to this operator valued setting the result onunions being ⊗4-Sidon...


All the relevant preprints are available on arxiv

Thank you !


sidon sets in discrete groupsmath.iisc.ac.in/~naru/dmha/wednesday/pisier.pdf · sidon sets in the...

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