Abstract. We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue-Orlicz spaces of a discrete group Γ and relative Toeplitz-Schur multipliers on Schatten-vonNeumann-Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum Λ ⊆ Γ , the norm of the Hilbert transform and the Riesz projection on Schatten-von-Neumann classes with exponent a power of 2, and the norm of Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less than 1.

Let Λ be a set of three integers and let CΛ be the space of 2π-periodic functions with spectrum in Λ endowed with the maximum modulus norm. We isolate the maximum modulus points x of trigonometric trinomials T ∈ CΛ and prove that x is unique unless |T | has an axis of symmetry. This permits to compute the exposed and the extreme points of the unit ball of CΛ, to describe how the maximum modulus of T varies with respect to the arguments of its Fourier coefficients and to compute the norm of unimodular relative Fourier multipliers on CΛ. We obtain in particular the Sidon constant of Λ.

Unbounded entailment relations, introduced by Paul Lorenzen (1951), are a slight variant of a notion which plays a fundamental rôle in logic (see Scott 1974) and in algebra (see Lombardi and Quitté 2015). We call systems of ideals their single-conclusion counterpart. If they preserve the order of a commutative ordered monoid G and are equivariant w.r.t. its law, we call them equivariant systems of ideals for G: they describe all morphisms from G to meet-semilattice-ordered monoids generated by (the image of) G. Taking an article by Lorenzen (1953) as a starting point, we also describe all morphisms from a commutative ordered group G to lattice-ordered groups generated by G through unbounded entailment relations that preserve its order, are equivariant, and satisfy a regularity property invented by Lorenzen (1950); we call them regular entailment relations. In particular, the free lattice-ordered group generated by G is described through the finest regular entailment relation for G, and we provide an explicit description for it; it is order-reflecting if and only if the morphism is injective, so that the Lorenzen-Clifford-Dieudonné theorem fits into our framework. Lorenzen's research in algebra starts as an inquiry into the system of Dedekind ideals for the divisibility group of an integral domain R, and specifically into Wolfgang Krull's "Fundamentalsatz" that R may be represented as an intersection of valuation rings if and only if R is integrally closed: his constructive substitute for this representation is the regularisation of the system of Dedekind ideals, i.e. the lattice-ordered group generated by it when one proceeds as if its elements are comparable. Keywords: Ordered monoid; system of ideals; equivariant system of ideals; morphism from an ordered monoid to a meet-semilattice-ordered monoid; ordered group; unbounded entailment relation; regular entailment relation; regular system of ideals; morphism from an ordered group to a lattice-ordered group; Lorenzen-Clifford-Dieudonné theorem; Fundamentalsatz for integral domains; Grothendieck ℓ-group; cancellativity.MSC 2010: Primary 06F20; Secondary 06F05, 13A15, 13B22.

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