LIMITWISE MONOTONIC FUNCTIONS, SETS, AND DEGREES ON COMPUTABLE DOMAINS
Sidon sets in discrete groupsmath.iisc.ac.in/~naru/dmha/Wednesday/Pisier.pdf · Sidon sets in the...
Transcript of 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.
Gilles Pisier Sidon sets in discrete groups
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
Gilles Pisier Sidon sets in discrete groups
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‖∞
Gilles Pisier Sidon sets in discrete groups
Fundamental Example
G = TN
∀z = (zn) ∈ TN ϕn(z) = zn
‖∑
anϕn‖∞ =∑|an| (C = 1)
Note: (ϕn) are independent random variables
Gilles Pisier Sidon sets in discrete groups
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 ?
Gilles Pisier Sidon sets in discrete groups
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 ?
Gilles Pisier Sidon sets in discrete groups
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 )
Gilles Pisier Sidon sets in discrete groups
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 )
Gilles Pisier Sidon sets in discrete groups
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 ?
Gilles Pisier Sidon sets in discrete groups
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 ?
Gilles Pisier Sidon sets in discrete groups
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)
Gilles Pisier Sidon sets in discrete groups
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 ?
Gilles Pisier Sidon sets in discrete groups
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
Gilles Pisier Sidon sets in discrete groups
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
Gilles Pisier Sidon sets in discrete groups
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.
Gilles Pisier Sidon sets in discrete groups
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
Gilles Pisier Sidon sets in discrete groups
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)]
Gilles Pisier Sidon sets in discrete groups
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.
Gilles Pisier Sidon sets in discrete groups
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
Gilles Pisier Sidon sets in discrete groups
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
Gilles Pisier Sidon sets in discrete groups
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|
Gilles Pisier Sidon sets in discrete groups
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.
Gilles Pisier Sidon sets in discrete groups
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
Gilles Pisier Sidon sets in discrete groups
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)).
Gilles Pisier Sidon sets in discrete groups
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 ?
Gilles Pisier Sidon sets in discrete groups
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 ?
Gilles Pisier Sidon sets in discrete groups
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 ?
Gilles Pisier Sidon sets in discrete groups
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...
Gilles Pisier Sidon sets in discrete groups
All the relevant preprints are available on arxiv
Thank you !
Gilles Pisier Sidon sets in discrete groups