Quaestiones Mathematicae

Editorial note: To the memory of W.A.J. Luxemburg and his mathematical legacy

Publishing Manager — Tue, 16 Apr 2024 00:00:00 +0000

No Abstract

Generating functions in Riesz spaces

Youssef Azouzi, Youssef Nasri — Tue, 16 Apr 2024 00:00:00 +0000

We introduce and study the concept of generating function for natural elements in a Dedekind complete Riesz space equipped with a conditional expectation operator. This allows us to study discrete processes in a free-measure setting. In particular we improve a result obtained by Kuo, Vardy and Watson concerning Poisson approximation.

Truncated vector lattices: A Maeda-Ogasawara type representation

Karim Boulabiar, Rawaa Hajji — Tue, 16 Apr 2024 00:00:00 +0000

Let L be a truncated Archimedean vector lattice whose truncation is denoted by ∗. In a recent paper, we proved that there exists a locally compact Hausdorff space X such that L is a lattice isomorphic with a truncated vector lattice of functions in C ∞ (X) whose truncation is provided by meet with some characteristic function on X. This representation, no matter how interesting it is, has a major drawback, namely, C ∞ (X) need not be a vector lattice, unless X is extremally disconnected. The main purpose of this paper is to remedy this shortcoming by proving that, indeed, an extremally disconnected locally compact space X can be found such that L can be seen as an order dense vector sublattice of C ∞ (X) whose truncation is provided, again, by meet with some characteristic function on X.

The order Bidual of C(x) for a real compact space

Marcel de Jeu, Jan Harm van der Walt — Tue, 16 Apr 2024 00:00:00 +0000

It is well known that the bidual of C(X) for a compact space X, supplied with the Arens product, is isometrically isomorphic as a Banach algebra to C(X˜) for some compact space X˜. The space X˜ is unique up to homeomorphism. We establish a similar result for realcompact spaces: The order bidual of C(X) for a realcompact space X, when supplied with the Arens product, is isomorphic as an f-algebra to C(X˜) for some realcompact space X˜. The space X˜ is unique up to homeomorphism.

The modulus of a vector measure

Ben de Pagter, Werner J. Ricker — Tue, 16 Apr 2024 00:00:00 +0000

It is known that if L is a Dedekind complete Riesz space and (Ω, Σ) is a measurable space, then the partially ordered linear space of all L- valued, finitely additive and order bounded vector measures m on Σ is also a Dedekind complete Riesz space (for the natural operations). In particular, the modulus |m|_o of m exists in this space of measures and |m|o is given by a well known formula. Some 20 years ago L. Drewnowski and W. Wnuk asked the question (for L not Dedekind complete) if there is an m for which |m|o exists but, |m|_o is not given by the usual formula? We show that such a measure m does indeed exist.

On the diagonal of Riesz operators on Banach Lattices

R. Drnovsek, M. Kandic — Tue, 16 Apr 2024 00:00:00 +0000

This paper extends the well-known Ringrose theory for compact operators to polynomially Riesz operators on Banach spaces. The particular case of an ideal-triangularizable Riesz operator on an order continuous Banach lattice yields that the spectrum of such operator lies on its diagonal, which motivates the systematic study of an abstract diagonal of a regular operator on an order complete vector lattice E. We prove that the class D of regular operators for which the diagonal coincides with the atomic diagonal is always a band in Lr(E), which contains the band of abstract integral operators. If E is also a Banach lattice, then D contains positive Riesz and positive AM-compact operators.

Scaling and instantaneous blow up

Gisele Ruiz Goldstein, Jerome A. Goldstein, Ismail Kombe, Abdelaziz Rhandi — Tue, 16 Apr 2024 00:00:00 +0000

The main result is a simple proof of the Baras-Goldstein (1984) instantaneous blow up result for the heat equation with the inverse square potential. The proof relies heavily, indeed mainly, on scaling. Remarks are also given concerning the case when the underlying space R^N is replaced by the Heisenberg group H^N .

Order boundedness and order continuity properties of positive operator semigroups

Jochen Gluck, Michael Kaplin — Tue, 16 Apr 2024 00:00:00 +0000

Relatively uniformly continuous (ruc) semigroups were recently introduced and studied by Kandic, Kramar-Fijavˇz, and the second-named author, in order to make the theory of one-parameter operator semigroups available in the setting of vector lattices, where no norm is present in general. In this article, we return to the more standard Banach lattice setting – where both ruc semigroups and C0-semigroups are well-defined concepts – and compare both notions. We show that the ruc semigroups are precisely those positive C0- semigroups whose orbits are order bounded for small times. We then relate this result to three different topics: (i) equality of the spectral and the growth bound for positive C0-semigroups; (ii) a uniform order boundedness principle which holds for all operator families between Banach lattices; and (iii) a description of unbounded order convergence in terms of almost everywhere convergence for nets which have an uncountable index set containing a co-final sequence.

Hereditarily projectable archimedean lattice-ordered groups with unit

Anthony W. Hager, Brian Wynne — Tue, 16 Apr 2024 00:00:00 +0000

No Abstract

Remarks on weak compactness criteria in variable exponent Lebesgue spaces

Francisco L. Hernandez , Cesar Ruiz, Mauro Sanchiz — Tue, 16 Apr 2024 00:00:00 +0000

We give two weak compactness Andˆo type criteria in variable exponent Lebesgue spaces L p(·) (Ω) on infinite measures. This extends some results of [13] given in the case of finite measures. Spaces L p(·) (Ω) on infinite measures are weakly Banach-Saks when p + < ∞. Suitable weak compactness criteria in Nakano sequence spaces ℓpn are also deduced.

On a special Kapteyn series

A.J.E.M. Janssen — Tue, 16 Apr 2024 00:00:00 +0000

We investigate the mathematical properties of the function T(ε) =∞n=1 n−1Jn(nε), ε ∈ [−1, 1], which is a special Kapteyn series of the first kind. Unlike various other special Kapteyn series, the function T(ε) does not seem to possess a closed-form expression. We derive an integral representation for T(ε) from which various properties of T(ε) can be established. In particular, monotonicity and convexity properties of T(ε) and ε ⁻¹ T(ε) can be shown. Also, the behaviour of T(ε) as ε ↑ 1 can be determined from the integral representation. Furthermore, while the Kapteyn series representation of T(ε) is very slowly convergent when ε is close to ±1, a regularized form of the integral representation of T(ε) allows to compute T(ε) accurately using Simpson’s rule with relatively few sample points of the involved integrand.

A Hahn-Jordan decomposition and Riesz-Frechet representation theorem in Riesz spaces

Anke Kalauch, Wenchi Kuo, Bruce A. Watson — Tue, 16 Apr 2024 00:00:00 +0000

We give a Hahn-Jordan decomposition in Riesz spaces which generalizes that of [B.A. Watson, An Andˆo-Douglas type theorem in Riesz spaces with a conditional expectation, Positivity, 13 (2009), 543–558] and a Riesz-Frechet representation theorem for the T-strong dual, where T is a Riesz space conditional expectation operator. The result of Watson was formulated specifically to assist in the proof of the existence of Riesz space conditional expectation operators with given range space, i.e., a result of Andˆo-Douglas type. This was needed in the study of Markov processes and martingale theory in Riesz spaces. In the current work, our interest is a Riesz-Frechet representation theorem, for which another variant of the Hahn-Jordan decomposition is required.

Derivations in disjointly complete commutative regular algebras

Aleksey Ber, Vladimir Chilin, Fedor Sukochev — Tue, 16 Apr 2024 00:00:00 +0000

We show that any nonexpansive derivation on a subalgebra of a disjointly complete commutative regular algebra A extends up to a derivation on A. For an algebra C∞(X, K) of functions X → K, continuous on a dense open subset of Stone compact X, we establish that the lack of nontrivial derivation is equivalent to σ-distributivity of the Boolean algebra of clopen subsets of X. The field K is an arbitrary normed field of charachteristic zero containing a complete non-discrete subfield. Our work is motivated by two seemingly unrelated problems due to Ayupov [2] and Wickstead [32].

Towards a geometrical equivalence of norms

Eder Kikianty — Tue, 16 Apr 2024 00:00:00 +0000

Angular equivalence of norms, introduced by Kikianty and Sinnamon (2017), is a notion of norm equivalence that is more attuned to the geometry of the norms. For certain geometrical properties and two angularly equivalent norms, it is the case that if one of the norms has a property, then so does the other. In this paper, we show further results in this direction, namely angularly equivalent norms share the property of non-squareness; and that the exposed points of the unit balls are in the same direction, under the condition that these points are assumed to be smooth with respect to both norms. A discussion on (the angular equivalence of) the dual norms of angularly equivalent norms is also given, giving a partial answer to an open problem as stated in the paper by Kikianty and Sinnamon (2017).

Stone-Weierstrass approximation revisited

Anatoly G. Kusraev, Semen S. Kutateladze — Tue, 16 Apr 2024 00:00:00 +0000

The aim of the present article is to extend the Stone-Weierstrass theorem to functions ranging in a lattice normed space and admitting order rather than topological approximation. We proceed with the machinery of Boolean-valued transfer from lattice normed space to normed space.

Weak and strong type inequalities for Orlicz spaces

Louis Labuschagne — Tue, 16 Apr 2024 00:00:00 +0000

We revisit the generalisation of Calderon’s Transfer Principle as espoused in [7]. This principle is used to generate weak type maximal inequalities for ergodic operators in the setting of σ-compact locally compact Hausdorff groups acting measure-preservingly on σ-finite measure spaces. In particular we develop a much more robust protocol for transferring weak and strong type inequalities from Orlicz spaces in the group setting to Orlicz spaces in the measure space setting. This is an important addition to the protocol developed in [7], which to date has only yielded weak type inequalities. The current approach also places fewer restrictions on the underlying Young functions describing the Orlicz spaces involved.

Generalized domination of ergodic elements in ordered Banach algebras

S. Mouton, A.D. Rabearivony — Tue, 16 Apr 2024 00:00:00 +0000

No Abstract

A rigid Riesz space

A.W. Wickstead — Tue, 16 Apr 2024 00:00:00 +0000

We give an example of an Archimedean Riesz space on which every automorphism is a strictly positive multiple of the identity.