Riesz Groups Research Articles

Asymptotic behavior of 𝐿^{𝑝} estimates for a class of multipliers with homogeneous unimodular symbols

We study Fourier multiplier operators associated with symbols ξ ↦ exp ⁡ ( i λ ϕ ( ξ / | ξ | ) ) \xi \mapsto \exp (\mathbb {i}\lambda \phi (\xi /|\xi |)) , where λ \lambda is a real number and ϕ \phi is a real-valued C ∞ \mathrm {C}^\infty function on the standard unit sphere S n − 1 ⊂ R n \mathbb {S}^{n-1}\subset \mathbb {R}^n . For 1 > p > ∞ 1>p>\infty we investigate asymptotic behavior of norms of these operators on L p ( R n ) \mathrm {L}^p(\mathbb {R}^n) as | λ | → ∞ |\lambda |\to \infty . We show that these norms are always O ( ( p ∗ − 1 ) | λ | n | 1 / p − 1 / 2 | ) O((p^\ast -1) |\lambda |^{n|1/p-1/2|}) , where p ∗ p^\ast is the larger number between p p and its conjugate exponent. More substantially, we show that this bound is sharp in all even-dimensional Euclidean spaces R n \mathbb {R}^n . In particular, this gives a negative answer to a question posed by Maz’ya. Concrete operators that fall into the studied class are the multipliers forming the two-dimensional Riesz group, given by the symbols r exp ⁡ ( i φ ) ↦ exp ⁡ ( i λ cos ⁡ φ ) r\exp (\mathbb {i}\varphi ) \mapsto \exp (\mathbb {i}\lambda \cos \varphi ) . We show that their L p \mathrm {L}^p norms are comparable to ( p ∗ − 1 ) | λ | 2 | 1 / p − 1 / 2 | (p^\ast -1) |\lambda |^{2|1/p-1/2|} for large | λ | |\lambda | , solving affirmatively a problem suggested in the work of Dragičević, Petermichl, and Volberg.

Riesz and pre-Riesz monoids

Call a directed partially ordered cancellative divisibility monoid M a Riesz monoid if for all $$x,y_{1},y_{2}\ge 0$$ in M, $$x\le y_{1}+y_{2}\Rightarrow x=x_{1}+x_{2}$$ where $$0\le x_{i}\le y_{i}$$ . We explore the necessary and sufficient conditions under which a Riesz monoid M with $$M^{+}=\{x\ge 0\mid x\in M\}=M$$ generates a Riesz group and indicate some applications. We call a directed p.o. monoid M $$\Pi $$ -pre-Riesz if $$ M^{+}=M$$ and for all $$x_{1},x_{2}, \dots ,x_{n}\in M$$ , $${{\,\mathrm{glb}\,}}(x_{1},x_{2},\dots ,x_{n})=0$$ or there is $$r\in \Pi $$ such that $$0<r\le x_{1},x_{2},\dots ,x_{n},$$ for some subset $$\Pi $$ of M. We explore examples of $$\Pi $$ -pre-Riesz monoids of $$*$$ -ideals of different types. We show for instance that if M is the monoid of nonzero (integral) ideals of a Noetherian domain D and $$\Pi $$ the set of invertible ideals, M is $$\Pi $$ -pre-Riesz if and only D is a Dedekind domain. We also study factorization in pre-Riesz monoids of a certain type and link it with factorization theory of ideals in an integral domain.

L-algebras and topology

L-algebras are based on an equation which is fundamental in the construction of various torsion-free groups, including spherical Artin groups, Riesz groups, certain mapping class groups, para-unitary groups, and structure groups of set-theoretic solutions to the Yang–Baxter equation. A topological study of L-algebras is initiated. A prime spectrum is associated to certain (possibly all) L-algebras, including three classes of L-algebras where the ideals are determined in a more explicite fashion. Known results on orthomodular lattices, Heyting algebras, or quantales are extended and revisited from an L-algebraic perspective.

Classification of spatial Lp AF algebras

We define spatial [Formula: see text] AF algebras for [Formula: see text], and prove the following analog of the Elliott AF algebra classification theorem. If [Formula: see text] and [Formula: see text] are spatial [Formula: see text] AF algebras, then the following are equivalent: [Formula: see text] and [Formula: see text] have isomorphic scaled preordered [Formula: see text]-groups. [Formula: see text] as rings. [Formula: see text] (not necessarily isometrically) as Banach algebras. [Formula: see text] is isometrically isomorphic to [Formula: see text] as Banach algebras. [Formula: see text] is completely isometrically isomorphic to [Formula: see text] as matricial [Formula: see text] operator algebras. As background, we develop the theory of matricial [Formula: see text] operator algebras, and show that there is a unique way to make a spatial [Formula: see text] AF algebra into a matricial [Formula: see text] operator algebra. We also show that any countable scaled Riesz group can be realized as the scaled preordered [Formula: see text]-group of a spatial [Formula: see text] AF algebra.

Fractability for partially ordered groups and Boolean algebras

In the present paper there are introduced and investigated the notions of fractal and semifractal lattice ordered groups or Riesz groups, respectively. The definitions are related to those which have been applied in the lattice theory. Besides, there is shown the existence of a proper class of Boolean algebras which are semifractal lattices but fail to be fractal lattices.

On congruences and ideals of partially ordered quasigroups

Some results concerning congruence relations on partially ordered quasigroups (especially, Riesz quasigroups) and ideals of partially ordered loops are presented. These results generalize the assertions which were proved by Fuchs in [5] for partially ordered groups and Riesz groups.

Factoriality in Riesz groups

Factoriality in Riesz groups

Simple Riesz groups of rank one having wild intervals

We prove that every partially ordered simple group of rank one which is not Riesz embeds into a simple Riesz group of rank one if and only if it is not isomorphic to the additive group of the rationals. Using this result, we construct examples of simple Riesz groups of rank one G, containing unbounded intervals ( D n ) n ⩾ 1 and D, that satisfy: (a) for each n ⩾ 1 , t D n ≠ G + for every ( t < q n ), but q n D n = G + (where ( q n ) is a sequence of relatively prime integers); (b) for every n ⩾ 1 , n D ≠ G + . We sketch some potential applications of these results in the context of K-theory.

Embedding rank one simple groups into rank one simple Riesz groups

We give a method for embedding a large family of partially ordered simple groups of rank one into simple Riesz groups of rank one. In particular, we answer in the affirmative a question of Wehrung, by constructing a torsion-free, simple Riesz group G of rank one containing an interval D\varsubsetneqq G^+ such that 2D=G^+. We sketch some potential applications of this result in the context of monoids of intervals and K-Theory of rings.

On a Generalization of the Notion of Orthogonality and on the Riesz Groups

A method for generalizing the notion of orthogonality to arbitrary partially ordered groups is considered. In the paper, the properties of almost orthogonal elements in Riesz groups are investigated. A series of results on convex directed subgroups associated with a pair of almost orthogonal elements of a Riesz group is obtained. For these subgroups, some theorems on homomorphisms are proved.

On the representation of simple riesz groups

In this paper we answer Open Problem 2 of Goodearl's book on partially ordered abelian groups in the case of partially ordered simple groups.As a consequence, we obtain a version of the Theorem of structure of dimension groups in the case of simple Riesz groups.Also, we give a method for constructing torsion-free strictly perforated simple Riesz groups of rank one, and we see that every dense additive subgroup of Q can be obtained using this method.

The Structure of Positive Elements for C*-Algebras with Real Rank Zero

In this paper we give a representation theorem for the Cuntz monoid S(A) of a σ-unital C*-algebra A with real rank zero and stable rank one, which allows to prove several Riesz decomposition properties on the monoid. As a consequence, it is proved that the comparability conditions (FCQ), stable (FCQ) and (FCQ+) are equivalent for simple C*-algebras with real rank zero. It is also shown that the Grothendieck group [Formula: see text] of S(A) is a Riesz group, and lattice-ordered under some additional assumptions on A.

On dilations and contractions in Riesz groups

On dilations and contractions in Riesz groups

The D + XDS[ X] construction from GCD-domains

Let D be an integral domain, S a multiplicative set in D and X an indeterminate over D. We show that if D is a GCD-domain, then the behaviour of D ( S) = { a 0 + Σ a i X i | a 0 ϵ D, a i ϵ D S } = D + XD S [ X] depends upon the relationship between S and the prime ideals P of D such that D P is a valuation domain. We use this study to construct locally GCD-domains which are not GCD-domains. These domains have, each, at least one prime t-ideal P such that PD P is not a t-ideal. We also give an example of an ideal A with A⊊A t⊊A v⊊D. The D ( S) construction is also used to construct, from lattice ordered groups, Riesz groups which are not lattice ordered.

Isometries in Riesz groups

Isometries in Riesz groups

A regular ring whose k0is not a riesz group∗

A regular ring whose k₀is not a riesz group^∗

Riesz groups with a finite number of disjoint elements

Riesz groups with a finite number of disjoint elements

Subdirect representation of tight Riesz groups by hybrid products.

Subdirect representation of tight Riesz groups by hybrid products.

Representations of ordered groups with compatible tight Riesz orders

A tight Riesz group G is a partially ordered group G that satisfies a strengthened form of the Riesz interpolation property. The term “tight” was introduced by Miller in (1970) and the tight interpolation property has been considered by Fuchs (1965), Miller (1973), (to appear), (preprint), Loy and Miller (1972) and Wirth (1973). If G is free of elements called pseudozeros then G is a non-discrete Hausdorff topological group with respect to the open interval topology U. Moreover the closure P of the cone P of the given order is the cone of an associated order on G. This allows an interesting interplay between the associated order, the tight Riesz order and the topology U. Loy and Miller found of particular interest the case in which the associated partial order is a lattice order. This situation was considered in reverse by Reilly (1973) and Wirth (1973), who investigated the circumstances under which a lattice ordered group, and indeed a partially ordered group, permits the existence of a tight Riesz order for which the initial order is the associated order. These tight Riesz orders were then called compatible tight Riesz orders. In Section one we relate these ideas to the topologies denned on partially ordered groups by means of topological identities, as described by Banaschewski (1957), and show that the topologies obtained from topological identities are precisely the open interval topologies from compatible tight Riesz orders.

The Completion of an Abelian l-Group

A directed partially ordered abelian group (G, ≦ ) is a tight Riesz group if for a1, a2, b1, b2 ∈ G with ai &lt; bj, i, j = 1,2, there is an x ∈ G with ai &lt; x &lt; bj, i, j = 1, 2. The open interval topology on G is the topology having as a base the set of all open intervals (a, b) = {x ∈ G|a &lt; x &lt; b}. For any x ∈ G, a neighborhood base at x is the set of all open intervals (x — a, x + a) = x + ( — a, a) for a &gt; 0.