Self-absorption of Hankel systems on monoids --a seemingly universal property

Given any cancellative monoid M, we study the Hankel system determined by its multiplication table. We prove that the Hankel system admits self-absorption property provided that the monoid M has the local algebraic structure: \[ (ax = by, cx=dy, az=bw \,\, in M) (cz=dw \,\, in M). \] Our result holds for all group-embeddable monoids and goes beyond. In particular, it works for all cancellative Abelian monoids and most common non-Abelian cancellative monoids such as SLd(N): = \[aᵢⱼ]1≤ i,j≤ d∈ SLd(Z)| aᵢⱼ ∈ N\. The Hankel system determined by the multiplication table of a monoid is further generalized to that determined by level sets of any abstract two-variable map. We introduce an algebraic notion of lunar maps and establish a stronger hereditary self-absorption property for the corresponding generalized Hankel systems. As a consequence, we prove the self-absorption property for arbitrary spatial compression of the regular representation system \λG(g)∈ G of any discrete group G, as well as the Hankel system \Γ_^Φ\ determined by the level sets of any rational map of the form Φ(x,y)=a xᵐ + b yⁿ with a,b,m,n∈ Z^*: \[ Γ_Φ(x, y)= 1(a x^m + b y^n= ), x, y∈ N^*, \, ∈ Φ (N^*× N^*). \] The self-absorption property is applied to the study of completely bounded Fourier multipliers between Hardy spaces. Further applications are: i) exact complete bounded norm of the Carleman embedding in any dimension; ii) mixed Fourier-Schur multiplier inequalities with critical exponent $4/3$; iii) failure of hyper-complete-contractivity for the Poisson semigroup.