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

Abstract

Given any cancellative monoid $\mathcal{M}$, we study the Hankel system determined by its multiplication table. We prove that the Hankel system admits self-absorption property provided that the monoid $\mathcal{M}$ has the local algebraic structure: \[ \big(ax = by, cx=dy, az=bw \,\, \text{in $\mathcal{M}$}\big)\Longrightarrow \big(cz=dw \,\, \text{in $\mathcal{M}$}\big). \] 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 $${\mathrm{SL}}_d(\mathbb{N}):=\{[a_{ij}{]}_{1\le i,j\le d}\in {\mathrm{SL}}_d(\mathbb{Z})|a_{ij}\in \mathbb{N}\}.$$ 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 $\{\hat{I}{»}_G(g){\}}_{g\in G}$ of any discrete group $G$, as well as the Hankel system $\{\hat{I}{“}_{\ell }^{\hat{I}}¦\}$ determined by the level sets of any rational map of the form $\hat{I}¦(x,y)=a{x}^m+b{y}^n$ with $a,b,m,n\in {\mathbb{Z}}^{\ast }$: \[ Γ_\ell^Φ(x, y)= \mathbf{1}(a x^m + b y^n= \ell), \quad x, y\in \mathbb{N}^*, \, \ell\in Φ(\mathbb{N}^*\times \mathbb{N}^*). \] 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.

45 pages