arXiv subjects

Xingni Jiang

Publications and source records attributed to Xingni Jiang.

7 recordsLinked to original sources

arXiv2026

Free Banach lattices over pre-ordered Banach spaces

We study free Banach lattices over pre-ordered Banach spaces in the category of Banach lattices of a given convexity type. These generalise the free Banach lattices under convexity conditions over Banach spaces in the literature. Their existence is shown from the existence of free vector lattices over pre-ordered vector spaces, which are also investigated. We determine when the positive contraction from the pre-ordered Banach space into the free Banach lattice is injective or bipositive, and when it has closed range. It is a bipositive embedding with closed range if and only if the positive wedge of the space is a closed normal cone. Even for a Banach lattice it can be non-isometric. By analysing the norm of the free $p$-convex Banach lattice with convexity constant 1 over a pre-ordered Banach space, it becomes clear that it can be realised as a function lattice on the positive part of the dual unit ball. This generalises the known realisation for a free Banach lattice of that type over a Banach space. As a preparation for this analysis of the norm, characterisations of $p$-convex Banach lattices in terms of vector lattice homomorphisms into $\mathrm{L}_p(μ)$-spaces for probability measures $μ$ are given.

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arXiv2026

Riesz representation theorems for vector lattices and Banach lattices of regular operators

For a non-empty locally compact Hausdorff space $X$ and a Dedekind complete normal vector lattice $E$, we show that the vector lattice of norm to order bounded operators from ${\text C}_{\text c}(X)$ or ${\text C}_0(X)$ into $E$ is isomorphic to the vector lattice of $E$-valued regular Borel measures on $X$. When $E$ is an order continuous Banach lattice, the isomorphism is an isometric isomorphism between Banach lattices. When $X$ is compact, every regular operator from $\mathrm{C}(X)$ into $E$ is norm to order bounded. For some spaces $E$, such as KB-spaces or the regular operators on a KB-space, every regular operator from ${\mathrm C}_0(X)$ into $E$ is norm to order bounded. Additional results are obtained for the whole space of regular operators from ${\text C}_{\text c}(X)$ into an order continuous Banach lattice. As a preparation, vector lattices and Banach lattices, resp. cones, of measures with values in a Dedekind complete vector lattice $E$, resp. in the extended positive cone of $E$, are investigated, as well as vector and Banach lattices of norm to order bounded operators. When $E$ is the real numbers, our results specialise to the well-known Riesz representation theorems for the order and norm duals of ${\text C}_{\text c}(X)$ and ${\text C}_0(X)$.

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arXiv2025

Central operators on complex Banach lattices

We show that the centre of a Dedekind complete complex Banach lattice is a commutative $\mathrm{C}^\ast$-algebra in the order unit norm. This implies that the order unit norm and the operator norm coincide. As an application of the latter, a Fuglede--Putnam--Rosenblum-type theorem is established. Under an extra condition on the underlying real Banach lattice, which is satisfied when it is order continuous, a spectral theorem is given for the centre of the complex Banach lattice as a whole and for an individual central operator. The ensuing functional calculus for an individual operator is applied to show that central operators have many spectral properties similar to those of normal operators on complex Hilbert spaces. For example, when the spectrum is countable every element is the sum of an order convergent series of eigenvectors.

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arXiv2025

L-valued integration

We develop integration theory for integrating functions taking values into a Dedekind complete unital $f$-algebra $\mathbb{L}$ with respect to $\mathbb{L}$-valued measures. We then discuss and prove completeness results of $\mathbb{L}$-valued $L^p$-spaces.

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arXiv2024

Spectral theorems for positive algebra homomorphisms

Let $X$ be a locally compact Hausdorff space, let $A$ be a partially ordered algebra, and let $π\colon \mathrm{C}_{\mathrm c}(X)\to A$ be a positive algebra homomorphism. Under conditions on $A$ that are satisfied in a good number of cases of practical interest, it is shown that $π$ is represented by a unique regular spectral measure $μ$ on the Borel $σ$-algebra of $X$, taking its values in the positive idempotents in $A$. The measure $μ$, which is $σ$-additive in an ordered sense, represents $π$ via the order integral (a generalisation of the Lebesgue integral) that goes back to J.D.M. Wright and which was investigated earlier by the authors. The positive algebra homomorphism $π$ can be extended from $\mathrm{C}_{\mathrm c}(X)$ to a positive linear map from the accompanying $L^1$-space of $μ$ into $A$. It is shown that, quite often, this $L^1$-space is closed under multiplication, so that it is a vector lattice algebra, and that the extended map from $L^1$ into $A$ is not only an algebra homomorphism but, even when $A$ is not a vector lattice, also a vector lattice homomorphism in a sense that is explained in the paper. When $ A$ has the countable sup property, the image of $L^1$ (or of its positive cone) is described in terms of consecutive ups and downs of the image of ${\mathrm C}_{\mathrm c}(X)$ (or of its positive cone). The general results are applied in three different contexts, showing how various spectral theorems have a common order-theoretical root: representations on Banach lattices, on Hilbert spaces, and (the algebra need not consist of operators) spectral theory for JBW-algebras.

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arXiv2021

Order Integrals

We define an integral of real-valued functions with respect to a measure that takes its values in the extended positive cone of a partially ordered vector space $E$. The monotone convergence theorem, Fatou's lemma, and the dominated convergence theorem are established; the analogues of the classical ${\mathcal L}^1$- and ${\mathrm L}^1$-spaces are investigated. The results extend earlier work by Wright and specialise to those for the Lebesgue integral when $E$ equals the real numbers. The hypothesis on $E$ that is needed for the definition of the integral and for the monotone convergence theorem to hold ($σ$-monotone completeness) is a rather mild one. It is satisfied, for example, by the space of regular operators between a directed partially ordered vector space and a $σ$-monotone complete partially ordered vector space, and by every JBW-algebra. Fatou's lemma and the dominated convergence theorem hold for every $σ$-Dedekind complete space. When $E$ consists of the regular operators on a Banach lattice with an order continuous norm, or when it consists of the self-adjoint elements of a strongly closed complex linear subspace of the bounded operators on a complex Hilbert space, then the finite measures as in the current paper are precisely the strongly $σ$-additive positive operator-valued measures. When $E$ is a partially ordered Banach space with a closed positive cone, then every positive vector measure is a measure in our sense, but not conversely. Even when a measure falls into both categories, the domain of the integral as defined in this paper can properly contain that of any reasonably defined integral with respect to the vector measure using Banach space methods.

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arXiv2016

Simultaneous power factorization in modules over Banach algebras

Let $A$ be a Banach algebra with a bounded left approximate identity $\{e_λ\}_{λ\inΛ}$, let $π$ be a continuous representation of $A$ on a Banach space $X$, and let $S$ be a non-empty subset of $X$ such that $\lim_λπ(e_λ)s=s$ uniformly on $S$. If $S$ is bounded, or if $\{e_λ\}_{λ\inΛ}$ is commutative, then we show that there exist $a\in A$ and maps $x_n: S\to X$ for $n\geq 1$ such that $s=π(a^n)x_n(s)$ for all $n\geq 1$ and $s\in S$. The properties of $a\in A$ and the maps $x_n$, as produced by the constructive proof, are studied in some detail. The results generalize previous simultaneous factorization theorems as well as Allan and Sinclair's power factorization theorem. In an ordered context, we also consider the existence of a positive factorization for a subset of the positive cone of an ordered Banach space that is a positive module over an ordered Banach algebra with a positive bounded left approximate identity. Such factorizations are not always possible. In certain cases, including those for positive modules over ordered Banach algebras of bounded functions, such positive factorizations exist, but the general picture is still unclear. Furthermore, simultaneous pointwise power factorizations for sets of bounded maps with values in a Banach module (such as sets of bounded convergent nets) are obtained. A worked example for the left regular representation of $\mathrm{C}_0({\mathbb R})$ and unbounded $S$ is included.

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