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

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: $$(ax=by,cx=dy,az=bw\text{in }\mathcal{M})⟹(cz=dw\text{in }\mathcal{M})\mathrm{.}$$ 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 $${\mathrm{S}\mathrm{L}}_d(\mathbb{N}):=\{[a_{ij}{]}_{1\le i,j\le d}\in {\mathrm{S}\mathrm{L}}_d(\mathbb{Z})\mid a_{ij}\in \mathbb{N}\}\mathrm{.}$$ 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 $\{{\lambda }_G(g){\}}_{g\in G}$ of any discrete group $G$, as well as the Hankel system $\{{\mathrm{\Gamma }}_{\mathrm{\ell }}^{\mathrm{\Phi }}\}$ determined by the level sets of any rational map of the form $\mathrm{\Phi }(x,y)=a{x}^m+b{y}^n$ with $a,b,m,n\in {\mathbb{Z}}^{\ast }$: $${\mathrm{\Gamma }}_{\mathrm{\ell }}^{\mathrm{\Phi }}(x,y)=\mathbf{1}(a{x}^m+b{y}^n=\mathrm{\ell }),x,y\in {\mathbb{N}}^{\ast },\mathrm{\ell }\in \mathrm{\Phi }({\mathbb{N}}^{\ast }\times {\mathbb{N}}^{\ast })\mathrm{.}$$ 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 $4\mathrm{/}3$; iii) failure of hyper-complete-contractivity for the Poisson semigroup.