An Axiomatic Theory Of Normed Modules Via Riesz Spaces

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY-NC-ND 4.0 https://creativecommons.org/licenses/by-nc-nd/4.0/ An Axiomatic Theory Of Normed Modules Via Riesz Spaces © The Author(s) 2024. Published by Oxford University Press. Published version Lučić, Danka; Pasqualetto, Enrico Lučić, D., & Pasqualetto, E. (2024). An Axiomatic Theory Of Normed Modules Via Riesz Spaces. Quarterly Journal of Mathematics, Early online. https://doi.org/10.1093/qmath/haae053 2024 The Quarterly Journal of Mathematics, 2024, 00, 1–51 doi:https://doi.org/10.1093/qmath/haae053 Article AN AXIOMATIC THEORY OF NORMED MODULES VIA RIESZ SPACES by DANKA LUCI ˇC ´† and ENRICO PASQUALETTO‡ Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), Jyväskylä FI-40014, Finland †Corresponding author. E-mail: [email protected] ‡E-mail: [email protected] A B S T R A C T We introduce and study an axiomatic theory of V-normed U-modules, where V is a Riesz space and U is an f-algebra; the spaces U and V also have some additional structure and are required to satisfy a compatibility condition. Roughly speaking, a V-normed U-module is a module over U that is endowed with a pointwise norm operator taking values in V. The aim of our approach is to develop a unified framework, which is tailored to the differential calculus on metric measure spaces, where U and V can take many different spaces of functions. Received 7 July 2023; Revised 16 August 2024 © The Author(s) 2024. Published by Oxford University Press. This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs licence (https://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial reproduction and distribution of the work, in any medium, provided the original work is not altered or transformed in any way, and that the work is properly cited. For commercial re-use, please contact [email protected] for reprints and translation rights for reprints. All other permissions can be obtained through our RightsLink service via the Permissions link on the article page on our site–for further information please contact [email protected]. 1. INTRODUCTION 1.1. General overview and motivations In this paper, we introduce and study a class of structures named V-normed U-modules, where V is a Riesz space and U is an f-algebra (that is, a Riesz space together with a multiplication operation), which fulfil suitable compatibility requirements. Roughly speaking, a V-normed U-module is a module over U (thus in particular, a vector space) equipped with a ‘pointwise norm’ operator that takes values into the positive cone of V. Several structures of these kinds—where, typically, U and V are function spaces—have been investigated in the literature: we refer to them as ‘functional’ normed modules. Different theories of functional normed modules were developed in the last 35 years, with a variety of applications, for example, in analysis, geometry and mathematical finance. Before delving into a more precise description of our notion of V-normed U-module (in Subsection 1.2), let us provide a brief overview of various classes of spaces that are covered by our axiomatization. •Normed spaces, which are ℝ-normed ℝ-modules. Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 2 • D. LUˇ CI´ C AND E. PASQUALETTO •Lebesgue–Bochner spaces, that is, spaces of p-integrable maps from a measure space to a normed space. These spaces are V-normed U-modules, where V is the space of p-integrable functions and U is the space of bounded measurable functions. •More generally, direct integrals of Banach spaces [31] and different spaces of measurable sections of a measurable Banach bundle [34]. •Random normed modules, which were introduced by Guo [25, 26] after the work of Schweizer and Sklar [37] on probabilistic metric spaces. The theory of random normed modules has been thoroughly developed in a long series of works (mostly by Guo and his coauthors); see the survey paper [27]. Particular attention was devoted to the study of random conjugate spaces (see, for example, [29]), which have applications in mathematical finance and in the modelling of conditional risk measures [15]. •Randomly normed L0-modules, which were developed by Haydon, Levy and Raynaud in [31] as a tool to study ultraproducts of Lebesgue–Bochner spaces. In this case, the Riesz spaces under consideration are Köthe function spaces, which are order-dense order-ideals in the space L0(𝜇) of measurable functions on a given measure space. This theory and Guo’s one—which were developed independently and concurrently—are fully consistent. •Lp-normed L∞-modules and their variants, which were introduced by Gigli [20], with the goal of developing an effective theory of measurable 1-forms and vector fields in the non-smooth setting of metric measure spaces. This approach was based on the work of Weaver [40] and on his definition of L∞-module. Similar structures have been widely considered in the framework of Dirichlet forms (see, for example, [14, 33]) and in the investigation of 1-forms induced by Dirichlet spaces [5]. Gigli’s theory is consistent with the above-mentioned notions of random normed modules (cf. with [20, Section 1.4] and [28]). •Normed A-modules in the sense of [8, 9], where A is a suitable f-algebra. This approach, which is similar in spirit to the one that we adopt in this paper, has been applied to the study of mathematical models in finance. Our interest in the language of normed modules is motivated by its applications in the differential calculus on metric measure spaces. Below, we briefly describe some important concepts and results from [20]. The goal of this description is 2-fold: to give a heuristic presentation of our definition of V- normed U-module and to expound the advantages of an axiomatic approach. However, we underline that our theory may be relevant even beyond the analysis of metric measure spaces. On metric measure spaces (X,d,𝔪), the study of Sobolev spaces W1,p(X) for p∈(1,∞) has been a fruitful field of research in the last decades (see, for example, [4, 10, 30, 38]). In order to develop a differential calculus modelled over W1,p(X), several notions of ‘measurable (co)vector fields’ were studied, for example, by [10] in the setting of doubling spaces supporting a Poincaré inequality. One of the objectives of [20] was to provide a meaningful notion of ‘space of measurable 1-forms’ for arbitrary metric measure spaces. This is encoded in the concept of cotangent module, which we are going to remind. It is proved in [20, Section 2.2.1] that it is possible to construct a vector space Lp(T*X) and a linear operator d: W1,p(X) →Lp(T*X) having the following features: (i) The elements of Lp(T*X) can be multiplied by L∞(𝔪)-functions; to be precise, Lp(T*X) is a module over the commutative ring L∞(𝔪). (ii) There exists a map |⋅|:Lp(T*X) →Lp(𝔪)+ that vanishes only at 0, that satisfies |𝜔+𝜂|≤|𝜔|+|𝜂|, for every 𝜔,𝜂∈Lp(T*X), and that is compatible with the L∞(𝔪)-module structure, in the sense that |f⋅𝜔|=|f||𝜔| for every f∈L∞(𝔪) and 𝜔∈Lp(T*X). The map |⋅| is said to be a pointwise norm operator. Moreover, the norm on Lp(T*X) induced by the pointwise norm via integration, that is, ‖𝜔‖Lp(T*X) := ‖|𝜔|‖Lp(𝔪), for every 𝜔∈Lp(T*X), Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 AN AXIOMATIC THEORY OF NORMED MODULES VIA RIESZ SPACES • 3 is required to be complete. (iii) The operator d: W1,p(X) →Lp(T*X), which is called the differential, satisfies ‖f‖W1,p(X) =(‖f‖p Lp(𝔪)+‖|df|‖p Lp(𝔪))1/p, for every f∈W1,p(X), and has the property that the L∞(𝔪)-module generated by its image is dense in Lp(T*X). Following [20, Definition 1.2.10], any couple (ℳ,|⋅|) verifying i) and ii) is called an Lp(𝔪)-Banach L∞(𝔪)-module (in fact, in [20] the term ‘Lp(𝔪)-normed L∞(𝔪)-module’ is used, but in this paper we need to distinguish between complete and non-complete modules). Nevertheless, a number of variants of this notion have been considered in [20] and in the subsequent literature: •It might be convenient (and sometimes necessary) to drop the Lp-integrability assumption. Technically speaking, this is made precise by the notion of L0(𝔪)-Banach L0(𝔪)-module; see [20, Section 1.3]. For example, the notion of L0-Banach L0-module becomes essential in the construction of tensor products of L2(𝔪)-Hilbert L∞(𝔪)-modules, cf. with [20, Section 1.5]. •The case p=∞ has been studied as well. Indeed, L∞(𝔪)-Banach L∞(𝔪)-modules are fundamental in order to apply the lifting theory by von Neumann in the Banach module setting [13], which in turn allows us to provide ‘fibrewise descriptions’, that is, to show that any Banach module is the space of sections of some generalized Banach bundle [21]. At present, using fibres is the only way to provide an explicit characterization of duals and of pullbacks of Banach modules, which are useful objects for the applications in metric measure geometry. •Under suitable curvature bounds (for example, in the setting of RCD(K,∞) spaces), one is often interested in extending the differential calculus to codimension-one measures (for example, to perimeter measures). The functional-analytic framework that allows us to achieve this goal is based on the concept of L0(Cap)-Banach L0(Cap)-module, which was introduced in [11]. Here, Cap denotes the Sobolev capacity, which is an outer measure on X that is not Borel regular. The aim of this work is to provide a unified theory of Banach modules, which covers—at least—all the notions of Banach modules discussed above. Indeed, albeit similar on some aspects, the several variants of Banach module often required different ad hoc definitions and proof strategies. Our goal is to introduce an ‘axiomatic framework’, where instead of function spaces we consider more general classes of Riesz spaces and f-algebras, as well as to obtain rather general existence results, which can be applied in all the specific cases we described above, whenever needed. 1.2. Main definitions Let us now discuss the various objects we are going to introduce, also motivating the reasons behind our definitions. First, a key feature of all the ‘functional’ Banach modules from Section 1.1 is the possibility to multiply by characteristic functions. This is fundamental, for example, when constructing the cotangent module. Observe that in L∞(𝔪) the characteristic functions of Borel sets are given exactly by the idempotent elements, that is, by those f∈L∞(𝔪) satisfying f2=f. Moreover, two different function spaces appear in the definition of Banach module: the ring of functions that can be multiplied by the elements of the Banach module (for example, L∞(𝔪) ), and the vector space of functions where the pointwise norm takes values (for example, Lp(𝔪) ). These two function spaces must be related. For example, the compatibility requirement between the pointwise norm and the module structure uses the fact that fg ∈Lp(𝔪) whenever f∈L∞(𝔪) and g∈Lp(𝔪). Taking all these features into account, we propose in Definition 2.23 the concept of metric f-structure (𝒰,U,V). Let us describe informally what a metric f-structure (𝒰,U,V) is: Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 4 • D. LUˇ CI´ C AND E. PASQUALETTO •𝒰 is an ambient localizable f -algebra (see Definition 2.15), which means that it is an f-algebra (that is, a Riesz space together with a compatible multiplication operation, see Definition 2.5) having plenty of idempotent elements (see (eq: idem)). This corresponds, for example, to the fact that simple functions are order-dense in L0(𝔪). •(U,dU) is a metric f-algebra (see Definition 2.22) that is an f-subalgebra of 𝒰. This means that U is an f-algebra endowed with a complete distance dU that verifies suitable compatibility conditions. For example, the space L∞(𝔪) is an f-subalgebra of L0(𝔪), and together with (the distance induced by) its norm, L∞(𝔪) is a metric f-algebra. •(V,dV) is a metric Riesz space (see Definition 2.20) that is also a Riesz subspace of 𝒰 satisfying UV =V. For example, Lp(𝔪) is a metric Riesz space and L∞(𝔪)⋅Lp(𝔪) = Lp(𝔪). Our axiomatization of a metric f-structure is tailored to the kinds of Banach modules we are interested in. However, already in the framework of differential calculus on metric measure spaces, some important objects studied in the literature (for example, Lipschitz derivations [39] or local vector measures [6]) are not covered by our theory, roughly speaking because the f-algebra of bounded continuous functions is not localizable (as characteristic functions are typically not continuous). As we discussed above, an example of metric f-structure is (L0(𝔪),L∞(𝔪),Lp(𝔪)). Taking into consideration the notion of Lp(𝔪)-Banach L∞(𝔪)-module from Section 1.1, one can think of the elements of U as those that can be multiplied by the elements of the Banach module, and the role of V is ‘the space where the pointwise norm takes values’, while 𝒰 is an ambient space where both U and V can be embedded (which is convenient to formulate the requirement that UV =V ). Having this discussion in mind, we propose in Definition 3.1 the concept of V-BanachU-module ℳ. The definition of V-Banach U-module roughly states the following: •ℳ is a module over the commutative ring U endowed with a pointwise norm |⋅|:ℳ→V+, which verifies the pointwise triangle inequality and is compatible with the module operations. •ℳ has the gluing property, which means every admissible sequence of disjoint elements (vn)n∈ℕ of ℳ can be ‘glued together’, thus obtaining a new element ∑n∈ℕvn∈ℳ. The order structure of (𝒰,U,V) comes into play here, that is, when declaring which sequences are admissible, see Definition 3.1 (ii). We also point out that, in general, ∑n∈ℕvn is just a formal series, which does not necessarily coincide with any kind of limit of finite sums. •The distance dℳ(v,w) := dV(|v−w|,0) on ℳ is complete. In the class of Lp(𝔪)-Banach L∞(𝔪)-modules with p∈[1,∞) we described in Section 1.1, we did not mention the gluing property, the reason being that in that specific framework it follows automatically from the other axioms. On the other hand, this is not always the case. For example, the gluing property has to be required when dealing with L∞(𝔪)-Banach L∞(𝔪)-modules (see [20, Example 1.2.5] or [13, Remark 2.22]). Moreover—different from what happens with Lp(𝔪)-Banach L∞(𝔪)-modules, where |∑k n=1 vn−∑n∈ℕvn|→0 in Lp(𝔪) – on L∞(𝔪)-Banach L∞(𝔪)-modules it is clear that the expression ∑n∈ℕvn might be only formal: in the space L∞(ℝ) itself (which is an L∞(ℝ)-Banach L∞(ℝ)-module), the elements fn:= 𝟙[n,n+1) for n∈ℤ can be ‘glued together’, obtaining the constant function 1 ; however, 1 is not the limit in the L∞(ℝ)-norm of the partial sums ∑k n=−kfk=𝟙[−k,k+1) as k→∞. In this example, it is still true that the partial sums converge in some sense to the glued object (for example, in the weak * topology), but this needs not be the case for arbitrary L∞(𝔪)-Banach L∞(𝔪)-modules, which do not always have a predual. We also mention that taking duals is very useful in differential calculus on metric measure spaces. For instance, the so-called tangent module Lq(TX), which can be regarded as the space of ‘q-integrable vector fields’ on a metric measure space (X,d,𝔪), is defined as the Banach module dual of the cotangent module Lp(T*X) ; see [20, Definition 2.3.1]. An important observation is that, according to Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 AN AXIOMATIC THEORY OF NORMED MODULES VIA RIESZ SPACES • 5 [20, Proposition 1.2.14 i)], the dual of an Lp(𝔪)-Banach L∞(𝔪)-module ℳ is an Lq(𝔪)-Banach L∞(𝔪)-module ℳ*, where p and q are conjugate exponents. This means that in our axiomatization when constructing the dual of a Banach module we have to change also the underlying metric f-structure. To address this issue, we propose in Definition 2.25 the concept of dual system of metricf-structures (𝒰,U,V,W,Z). We omit the details here. However, the definition of dual system is given so that the module dual of a V-Banach U-module ℳ is a W-Banach U-module, see Definition 3.15. More generally, the space Hom(ℳ,𝒩) of all homomorphisms (Definition 3.11) from a V-Banach U-module ℳ to a Z-Banach U-module 𝒩 inherits a natural structure of W-Banach U-module (Theorem 3.12). An example of dual system of metric f-structures is (L0(𝔪),L∞(𝔪),Lp(𝔪),Lq(𝔪),L1(𝔪)). When proving finer results about homomorphisms and dual modules, one often has to require a further regularity on the underlying f-algebras and Riesz spaces, namely, that they are Dedekind complete and they have the countable sup property (or CSP, for short); see Definition 2.4. The above assumptions amount to saying that every set that is bounded from above (resp. from below) has a supremum (resp. an infimum) and that such supremum (resp. infimum) can be expressed as a countable supremum (resp. a countable infimum) of elements of the given set. These properties are enjoyed, for example, by Lp(𝔪) whenever p∈{0}∪[1,∞] and 𝔪 is a 𝜎-finite measure (Proposition 4.3), but they fail in L0(Cap) (Example 4.4). Dedekind completeness and CSP are also needed, for instance, to construct local inverses (Proposition 3.6) or to define the support of a metric f-structure (Definition 3.7). 1.3. Main results Another objective of this work is to provide a rather complete toolbox of results and techniques concerning Banach modules over a metric f-structure, which we plan to apply in the future, as a ‘black box’, to many particular cases of interest. Our two main achievements are the following: •Theorem 3.19: Given a metric f-structure (𝒰,U,V), a vector space 𝒱 and an even sublinear map 𝜓:𝒱→V+, there exists a unique couple (ℳ⟨𝜓⟩,T⟨𝜓⟩), where ℳ⟨𝜓⟩ is a V-Banach U-module, while T⟨𝜓⟩:𝒱→ℳ⟨𝜓⟩ is a linear operator with ‘generating image’ (in a suitable sense) such that |T⟨𝜓⟩v|=𝜓(v) for every v∈𝒱. The uniqueness is formulated in categorical terms, that is, via a universal property (see also Corollary 3.22). This quite general existence result incorporates most of the existence results for Banach modules considered so far in the related literature. For example, the cotangent module Lp(T*X) and the differential d are given by (Lp(T*X),d) ≅(ℳ⟨𝜓p⟩,T⟨𝜓p⟩), where the map 𝜓p:W1,p(X) →Lp(𝔪)+ is defined as 𝜓p(f) := |Df |. See Section 4.2.5 for this example, as well as for other relevant constructions of Banach modules induced by an even sublinear map. •Theorem 3.16 is an existence criterion for homomorphisms of Banach modules. Indeed, given that the theory of V-Banach U-modules fits well in a categorical framework (see Definition 3.14), it is natural to couple Theorem 3.19 with an existence result for homomorphisms. For simplicity of presentation, let us state here only a corollary of Theorem 3.12: given a dual system (𝒰,U,V,W,Z), a V-Banach U-module ℳ, a Z-Banach U-module 𝒩, a ‘generating’ vector subspace 𝒱 of ℳ and a linear operator T:𝒱→𝒩 satisfying |Tv|≤b|v| for some b∈W+, there is a unique extension ˉ T∈Hom(ℳ,𝒩) of T, which still satisfies |ˉ Tv|≤b|v|. Finally, we conclude the introduction by briefly mentioning other results we obtain in the paper: •Using Theorems 3.19 and 3.12, we prove that each homomorphism of metric f-structures induces a pushforward functor (or, to be more precise, a ‘direct image functor’) in the categories of Banach modules; see Section 3.3.2. Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 6 • D. LUˇ CI´ C AND E. PASQUALETTO •We prove a version of the Hahn–Banach extension theorem for V-Banach U-modules; see Theorem 3.30. It is used, for example, for studying module duals and embedding operators into the bidual; see Sections 3.4 and 3.4.1. •We study Hilbert modules, for example, Banach modules whose pointwise norm verifies a pointwise parallelogram identity; see Definition 3.5. Among the several results we obtain, let us mention a Hilbert projection theorem and a Riesz representation theorem; see Section 3.5. •We prove that V-Banach U-modules admit a dimensional decomposition (assuming Dedekind completeness and CSP of the metric f-structure); see Section 3.6. In Sections 2 and 3 the whole treatment is at the level of ‘abstract’ Riesz spaces and f-algebras, without ever mentioning any kind of function spaces. The applications of our axiomatic theory to the various classes of Banach modules over spaces of functions are discussed in Section 4. 2. LOCALIZABLE F-ALGEBRAS AND METRIC F-STRUCTURES In Section 2.1 we recall many useful definitions and results concerning Riesz spaces and f-algebras, which are quite standard and well-established; our presentation is essentially taken from [18, 17] (see also [1, 2]). In Section 2.2 we study the set of idempotent elements, while in Sections 2.3 and 2.4 we introduce the language of localizable f-algebras and of (dual systems of) metric f-structures, respectively. 2.1. Reminder on Riesz spaces and f-algebras Let (P,≤) be a partially ordered set, and S≠∅ a subset of P. We recall the following definitions: (i) We say that S is upwards directed if for every p,p′∈S there exists q∈S such that p≤q and p′≤q. We say that S is downwards directed if for every p,p′∈S there exists q∈S such that q≤p and q≤p′. (ii) A sequence (pn)n∈ℕ⊂P is said to be non-decreasing provided pn≤pn+1 for every n∈ℕ, while it is said to be non-increasing provided pn≥pn+1 for every n∈ℕ. (iii) An element p∈P is said to be an upper bound for S provided that s≤p holds for every s∈S. We say that p is the supremum of S, and we write p=supS, provided that p≤p′ holds for any other upper bound p′∈P for S. If supS exists, then it is uniquely determined. (iv) An element q∈P is said to be a lower bound for S provided that q≤s holds for every s∈S. We say that q is the infimum of S, and we write q=infS, provided that q′≤q holds for any other lower bound q′∈P for S. If infS exists, then it is uniquely determined. (v) We say that S is order-bounded provided that it has both an upper bound and a lower bound. (vi) We say that P is Dedekind 𝜎-complete provided that every countable non-empty subset of P with an upper bound has a supremum and every countable non-empty subset of P with a lower bound has an infimum. (vii) P is Dedekind complete if every non-empty subset of P with an upper bound has a supremum or equivalently every non-empty subset of P with a lower bound has an infimum. A map 𝜙:P→Q between partially ordered sets P and Q is said to be order-preserving provided 𝜙(p)≤𝜙(q), for every p,q∈Pwith p≤q. An order-preserving map 𝜙:P→Q is said to be order-continuous provided that it holds that ∃sup{𝜙(p)∣p∈R}=𝜙(ˉ p), whenever R⊂P is upwards directed and ∃ˉ p:= supR∈P, ∃inf{𝜙(q)∣q∈S}=𝜙(ˉ q), whenever S⊂P is downwards directed and ∃ˉ q:= infS∈P. Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 AN AXIOMATIC THEORY OF NORMED MODULES VIA RIESZ SPACES • 7 We say that an order-preserving map 𝜙:P→Q is 𝜎-order-continuous provided it holds that ∃sup n∈ℕ𝜙(pn) = 𝜙(sup n∈ℕ pn), whenever (pn)n∈ℕ⊂P is non-decreasing and sup n∈ℕ pn exists, ∃inf n∈ℕ𝜙(qn) = 𝜙(inf n∈ℕqn), whenever (qn)n∈ℕ⊂P is non-increasing and inf n∈ℕqn exists, where supn∈ℕpn stands for sup{pn}n∈ℕ. Note that order-continuity implies 𝜎-order-continuity. A lattice is a partially ordered set (P,≤) such that p∨q:= sup{p,q} and p∧q:= inf{p,q} exist for all p,q∈P. A set S⊂P is called a sublattice of P if it is closed under ∨ and ∧, that is, p∨q,p∧q∈S, for every p,q∈S. A map 𝜙:P→Q between lattices P and Q is said to be a lattice homomorphism provided 𝜙(p∨q) = 𝜙(p)∨𝜙(q), 𝜙(p∧q) = 𝜙(p)∧𝜙(q), for every p,q∈P. For an arbitrary family {Pi}i∈I of partially ordered sets Pi= (Pi,≤i), the product P:= ∏i∈IPi can be endowed with the following partial order: for any (pi)i∈I,(qi)i∈I∈∏i∈IPi, we declare that (pi)i∈I≤ (qi)i∈I if and only if pi≤iqi for every i∈I. Observe that (P,≤) is a lattice if and only if (Pi,≤i) is a lattice for every i∈I. 2.1.1. The theory of Riesz spaces A partially ordered linear space (U,≤) is a vector space U= (U,+,⋅) over the field ℝ of real numbers, together with a partial order ≤ on U such that the following properties are verified: u+w≤v+w, for every u,v,w∈Uwith u≤v, 𝜆u≥0, for every 𝜆∈ℝ+and u∈Uwith u≥0. A Riesz space is a partially ordered linear space U= (U,+,⋅,≤) that is a lattice. We define u+:= u∨0, u−:= (−u)∨0, |u|:= (−u)∨u, for every u∈U. We have that |u|≥0 holds for every u∈U, with equality if and only if u= 0. For a proof of the next result, we refer, for example, to [17, 352D] or [1, Theorem 1.3]. Proposition 2.1 (Basic properties of Riesz spaces) Let U be a Riesz space. Then it holds that 𝜆(u∨v) = 𝜆u∨𝜆v,for every 𝜆∈ℝwith 𝜆> 0 and u,v∈U, (1a) |𝜆u|=𝜆|u|,for every 𝜆∈ℝ+and u ∈U, (1b) −u∨v= (−u)∧(−v), for every u,v∈U, (1c) u+v∨w= (u+v)∨(u+w), for every u,v,w∈U, (1d) u+v∧w= (u+v)∧(u+w), for every u,v,w∈U, (1e) u∨v+u∧v=u+v,for every u,v∈U, (1f) u=u+−u−,for every u ∈U, (1g) |u|=u+∨u−=u++u−,for every u ∈U, (1h) u+∧u−= 0, for every u ∈U, (1i) Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 8 • D. LUˇ CI´ C AND E. PASQUALETTO (u+v)+≤u++v+,for every u,v∈U, (1j) |u+v|≤|u|+|v|,for every u,v∈U, (1k) u∧(v+w)≤u∧v+u∧w,for every u,v,w∈U+. (1l) A Riesz subspace of U is a linear subspace which is also a sublattice. A homomorphism of Riesz spaces 𝜙:U→V is a linear operator such that 𝜙(u)∧𝜙(v) = 0, for every u,v∈Usuch that u∧v= 0. By virtue of [17, 352G], each homomorphism of Riesz spaces 𝜙:U→V has the following property: |𝜙(u)|=𝜙(|u|), for every u∈U. (2) We denote by U+ the positive cone of a Riesz space U, namely, U+:= {u∈U∣u≥0}. We recall from [1, Definition 1.22] that a Riesz subspace V of a given Riesz space U is said to be superorder-dense in U if for any u∈U+ there exists a non-decreasing sequence (un)n∈ℕ⊂V+ such that u=supn∈ℕun. Moreover, a Riesz subspace V of a Riesz space U is said to be solid provided that v∈V holds whenever v∈U, and there exists u∈V such that |v|≤|u|. We also recall from [18, Proposition 15B] the following result: Proposition 2.2 Any Dedekind 𝜎-complete Riesz space U is Archimedean, that is, for any u,v∈U nu ≤v,for every n ∈ℕ⟹u≤0. Definition 2.3 (Disjoint set) Let U be a Riesz space. Let S be a non-empty subset of U. Then we say that S is disjoint provided that it holds that |u|∧|v|= 0, for every u,v∈S such that u≠v. When S is a finite disjoint set {u1,…,un}⊂U, we say that the elements u1,…,un are pairwise disjoint. Observe that if 𝜙:U→V is a homomorphism of Riesz spaces, then it holds that {𝜙(u)∣u∈S}⊂Vis disjoint, for every ∅≠S⊂U disjoint. (3) Indeed, if u,v∈S and 𝜙(u)≠𝜙(v), then u≠v and |𝜙(u)|∧|𝜙(v)|=𝜙(|u|)∧𝜙(|v|) = 0 by (2). We also recall (see [2, p. 3] or [1, Definition 1.43]) the following notion: Definition 2.4 (CSP) Let U be a Riesz space. Then we say that U has the CSP (or that U is a CSP space ) if it holds that ∀∅≠V⊂U such that ∃supV,∃(vn)n∈ℕ⊂V:sup n∈ℕ vn=supV, ∀∅≠ ˜ V⊂U such that ∃infV,∃(˜ vn)n∈ℕ⊂˜ V:inf n∈ℕ˜ vn=inf ˜ V. Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 AN AXIOMATIC THEORY OF NORMED MODULES VIA RIESZ SPACES • 15 In Definitions 2.22 and 2.23, given two metric spaces (X,dX) and (Y,dY), we will consider the distance dX×dY on the Cartesian product X ×Y, which is given by (dX×dY)((x,y),(˜ x,˜ y)):= dX(x,˜ x) + dY(y,˜ y), for every (x,y),(˜ x,˜ y)∈ X ×Y. Next, we introduce the family of metric f-algebras: Definition 2.22 (Metric f-algebra) By a metric f-algebra we mean a couple (U,dU), where (i) U is a localizable f-algebra, and (U,dU) is a metric Riesz space. (ii) The multiplication ×:U×U→U is continuous from (U×U,dU×dU) to (U,dU). (iii) The family 𝒮(U) of all simple elements of U is dense in (U,dU). (iv) dU(𝜀1U,0) →0 as 𝜀↘0. A homomorphism of metric f-algebras is a Lipschitz homomorphism of localizable f-algebras. For some examples of metric f-algebras in the case of function spaces we are interested in, see Subsection 4.2.1. Having the notions of metric Riesz space and of metric f-algebra at our disposal, we can finally introduce metric f-structures: Definition 2.23 (Metric f-structure) A metric f-structure is a triple (𝒰,U,V), where (i) 𝒰 is a localizable f-algebra. (ii) U= (U,dU) is a metric f-algebra such that U is a solid f-subalgebra of 𝒰. (iii) V= (V,dV) is metric Riesz space such that V is a solid Riesz subspace of 𝒰. (iv) It holds UV =V, and the multiplication is continuous from (U×V,dU×dV) to (V,dV). (v) Given any (un)n∈ℕ∈𝒫(1U) and 𝜀> 0, there exists 𝛿> 0 such that for any (vn)n∈ℕ⊂V+ with ∑n∈ℕdV(unvn,0) ≤𝛿 it holds that (unvn)n∈ℕis order-bounded in V,dV(sup n∈ℕ unvn,0)≤𝜀. We say that a metric f-structure (𝒰,U,V) is Dedekind complete (resp. CSP), provided that the spaces 𝒰, U and V are Dedekind complete (resp. CSP). A homomorphism of metric f-structures between two metric f-structures (𝒰1,U1,V1) and (𝒰2,U2,V2) is a homomorphism 𝜑:𝒰1→𝒰2 of f-algebras such that 𝜑|U1:U1→U2 is a homomorphism of metric f-algebras and 𝜑|V1:V1→V2 a is homomorphism of metric Riesz spaces. In the case of function spaces, some examples of metric f-structures are listed in Subsection 4.2.2. Example 2.24 If U= (U,dU) is a metric f-algebra, then (U,U,U) is a metric f-structure. As we discussed in Section 1, in order to study dual modules (and, more generally, spaces of homomorphisms) we also have to define the dual systems of metric f-structures: Definition 2.25 (Dual system of metric f-structures) A quintuplet (𝒰,U,V,W,Z) is said to be a dual system of metric f-structures provided that the following conditions are verified: (i) (𝒰,U,V), (𝒰,U,W) and (𝒰,U,Z) are metric f-structures. (ii) It holds Z=VW and the multiplication is continuous from (V×W,dV×dW) to (Z,dZ). Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 16 • D. LUˇ CI´ C AND E. PASQUALETTO We say that (𝒰,U,V,W,Z) is a complete dual system of metric f-structures if in addition: (iii) 𝒰, U, V, W, Z are Dedekind complete and the multiplication is order-continuous from U+×V+ to V+, from U+×W+ to W+, from U+×Z+ to Z+ and from V+×W+ to Z+. Also, we say that a complete dual system (𝒰,U,V,W,Z) is CSP if in addition (𝒰,U,V), (𝒰,U,W) and (𝒰,U,Z) are CSP. By a homomorphism of dual systems between two given dual systems of metric f-structures (𝒰1,U1,V1,W1,Z1) and (𝒰2,U2,V2,W2,Z2) we mean a map 𝜑:𝒰1→𝒰2 that is a homomorphism of metric f-structures from (𝒰1,U1,V1) to (𝒰2,U2,V2), from (𝒰1,U1,W1) to (𝒰2,U2,W2) and from (𝒰1,U1,Z1) to (𝒰2,U2,Z2). Some relevant examples of dual systems in the case of function spaces are presented in Section 4.2.3. Example 2.26 Let (𝒰,U,V) be a metric f-structure. Then (𝒰,U,V,U,V) is a dual system of metric f-structures. If U is Dedekind complete and the multiplication map is order-continuous from U+×V+ to V+, then (𝒰,U,V,U,V) is a complete dual system of metric f-structures. Remark 2.27 If (𝒰,U,V,W,Z) is a dual system of metric f-structures, then (𝒰,U,W,V,Z) is a dual system of metric f-structures as well. Moreover, if (𝒰,U,V,W,Z) is a complete (resp. CSP complete) dual system, then (𝒰,U,W,V,Z) is a complete (resp. CSP complete) dual system. 3. NORMED MODULES OVER A METRIC F-STRUCTURE In Sections 3.1 and 3.2 we introduce the category of Banach modules over a metric f-structure; in the former we study the objects, while in the latter we study the morphisms. In Section 3.3 we prove some existence results concerning Banach modules and their homomorphisms, as well as some of their consequences. In Sections 3.4, 3.5 and 3.6 we study the Hahn–Banach theorem, the class of Hilbert modules and the dimensional decomposition of a Banach module, respectively. 3.1. Definitions and basic properties First of all, let us give the definition of normed/Banach module over a metric f-structure: Definition 3.1 (Normed module) Let (𝒰,U,V) be a metric f-structure and ℳ a module over U. Then we say that ℳ is a V-normed U-module provided that it is endowed with a map |⋅|:ℳ→V+ – called a V-pointwise norm operator on ℳ—such that the following properties are verified: (i) Given any u∈U and v,w∈ℳ, it holds that |v|= 0 ⟺v= 0, (12a) |v+w|≤|v|+|w|, (12b) |u⋅v|=|u||v|. (12c) (ii) Gluing property. Let (un)n∈ℕ∈𝒫(1U) and (vn)n∈ℕ⊂ℳ be chosen so that the family (|un⋅vn|)n∈ℕ is order-bounded in V. Then there exists an element v∈ℳ Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 AN AXIOMATIC THEORY OF NORMED MODULES VIA RIESZ SPACES • 17 such that un⋅v=un⋅vn, for every n∈ℕ. (13) We call Adm(ℳ) the set of all families (un,vn)n∈ℕ as above, while ∑n∈ℕun⋅vn stands for the element v∈ℳ satisfying (eq: glueing)—whose uniqueness follows from Lemma 3.2. Moreover, we endow the space ℳ with the distance dℳ, which is defined as dℳ(v,w) := dV(|v−w|,0), for every v,w∈ℳ. (14) Whenever (ℳ,dℳ) is a complete metric space, we say that ℳ is a V-Banach U-module. Some examples of functional normed/Banach modules that are covered by the above definition are presented in Section 4.2.4. Lemma 3.2 (Locality property) Let (𝒰,U,V) be a metric f-structure, and let ℳ be a V-normed U-module. Let (un)n∈ℕ∈𝒫(1U) and v ∈ℳ satisfy un⋅v= 0 for every n ∈ℕ. Then v =0. Proof. Given that the multiplication map is 𝜎-order-continuous on 𝒰+×𝒰+, we deduce that |v|=|v|sup n∈ℕ un=sup n∈ℕ un|v|=sup n∈ℕ|un⋅v|= 0, whence it follows that v= 0, as we claimed in the statement. Given any non-empty subset S of a V-normed U-module ℳ, we denote by 𝒢(S)⊂ℳ the family of those elements that can be obtained by gluing together elements of S. Namely, we set 𝒢(S) = 𝒢ℳ(S) := {∑ n∈ℕ un⋅vn∣(un)n∈ℕ∈𝒫(1U), (vn)n∈ℕ⊂S, (un,vn)n∈ℕ∈Adm(ℳ)}. Observe that if S is a vector subspace of ℳ, then 𝒢(S) is a vector subspace of ℳ as well. Proposition 3.3 Let (𝒰,U,V) be a metric f-structure. Then V is a V-Banach U-module, with the scalar multiplication ⋅:U×V→V being given by the multiplication × in 𝒰. Moreover, it holds ∑ n∈ℕ unvn=sup n∈ℕ unv+ n−sup n∈ℕ unv− n,for every (un,vn)n∈ℕ∈Adm(V). (15) Proof. The fact that V is a U-module verifying item (i) of Definition 3.1 readily follows from the very definition of a metric f-algebra. Moreover, the distance on V defined as in (14) coincides with the original distance dV itself, which is complete by assumption. It only remains to check the validity of the gluing property. To this aim, fix any (un,vn)n∈ℕ∈Adm(V). In particular, both sequences (unv+ n)n∈ℕ and (unv− n)n∈ℕ are order-bounded, and thus the Dedekind 𝜎-completeness of V yields existence of w+:= supn∈ℕunv+ n∈V+ and w−:= supn∈ℕunv− n∈V+. We claim that un(w+−w−) = unvn, for every n∈ℕ. (16) To prove it, notice that unw+=unv+ n for every n∈ℕ : the inequality ≥ is trivial, while to get the converse one it suffices to observe that (1− un)w++unv+ n is an upper bound for (umv+ m)m∈ℕ. Similarly, one can show that unw−=unv− n, whence it follows that un(w+−w−) = unv+ n−unv− n, yielding (16). This proves the validity of the gluing property, as well as formula (15). Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 18 • D. LUˇ CI´ C AND E. PASQUALETTO Proposition 3.4 (Continuity of normed module operations) Let (𝒰,U,V) be a metric f-structure and ℳ a V-normed U-module. Then the following properties are verified: (i) The map |⋅|:ℳ→V+ is 1-Lipschitz from (ℳ,dℳ) to (V,dV). (ii) The map +: ℳ×ℳ→ℳ is 1-Lipschitz from (ℳ×ℳ,dℳ×dℳ) to (ℳ,dℳ). (iii) The map ⋅:U×ℳ→ℳ is continuous from (U×ℳ,dU×dℳ) to (ℳ,dℳ). Proof. (i) Given that |v|≤|v−w|+|w| and |w|≤|w−v|+|v| hold for every v,w∈ℳ, we deduce that ∣|v|−|w|∣≤|v−w|, for every v,w∈ℳ. In particular, for any v,w∈ℳ one has dV(|v|,|w|) = dV(∣|v|−|w|∣,0) ≤dV(|v−w|,0) = dℳ(v,w), which shows that |⋅|:ℳ→V+ is a 1-Lipschitz mapping from (ℳ,dℳ) to (V,dV), as required. (ii) For any v,v′,w,w′∈ℳ we have |(v+w) − (v′+w′)|≤|v−v′|+|w−w′|, thus accordingly dℳ(v+w,v′+w′) = dV(∣(v+w) − (v′+w′)∣,0)≤dV(|v−v′|+|w−w′|,0) ≤dV(|v−v′|,0) + dV(|w−w′|,0) = dℳ(v,v′) + dℳ(w,w′) = (dℳ×dℳ)((v,v′),(w,w′)), for every v,v′,w,w′∈ℳ. This proves that + is 1-Lipschitz from (ℳ×ℳ,dℳ×dℳ) to (ℳ,dℳ). (iii) Fix (un)n∈ℕ⊂U and u∈U with limn→∞dU(un,u) = 0. Fix (vn)n∈ℕ⊂ℳ and v∈ℳ with limn→∞dℳ(vn,v) = 0. The continuity of |⋅|:ℳ→V+ from (i) ensures that |un|→|u| in (U,dU) and |vn−v|→0 in (V,dV). Hence, by letting n→∞ in |un⋅vn−u⋅v|≤|un||vn−v|+|un−u||v| we obtain limn→∞dℳ(un⋅vn,u⋅v) = 0, which shows that ⋅:U×ℳ→ℳ is continuous. Let (𝒰,U,V) be a metric f-structure, and let ℳ be a V-normed U-module. Then a given U- submodule 𝒩 of ℳ is said to be a V-normed U-submodule of ℳ, provided that it satisfies 𝒢ℳ(𝒩) = 𝒩. In the case where ℳ is a V-Banach U-module and 𝒩 is dℳ-closed in ℳ, we say that 𝒩 is a V-Banach U-submodule of ℳ. These are some useful examples of V-Banach U-submodule: •The ‘localized’ module u⋅ℳ:= {u⋅v:v∈ℳ} for every u∈Idem(U). •The ‘one-dimensional’ module U⋅v:= {u⋅v:u∈U} for every v∈ℳ. •The sum 𝒩1+𝒩2:= {v+w:v∈𝒩1,w∈𝒩2} where 𝒩1, 𝒩2 are V-Banach U-submodules of ℳ. We say that ℳ is the direct sum of 𝒩1 and 𝒩2, and we write ℳ=𝒩1⊕𝒩2, if ℳ=𝒩1+𝒩2 and 𝒩1∩𝒩2={0}. In this case, the map 𝒩1×𝒩2∋(v,w)↦v+w∈ℳ is bijective. If (𝒰,U,V) is a metric f-structure such that V is Dedekind complete, ℳ is a V-Banach U-module and 𝒩 is a V-Banach U-submodule of ℳ, then the quotient module ℳ/𝒩 is a V-Banach U-module if Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 AN AXIOMATIC THEORY OF NORMED MODULES VIA RIESZ SPACES • 19 endowed with the following V-pointwise norm operator: |v+𝒩|:= inf{|v+w|∣w∈𝒩}∈V+, for every v+𝒩∈ℳ/𝒩. Definition 3.5 (Hilbert module) Let (𝒰,U,V,V,Z) be a dual system of metric f-structures. Then by a V-Hilbert U-module we mean a V-Banach U-module ℋ such that |v+w|2+|v−w|2= 2|v|2+ 2|w|2, for every v,w∈ℋ. (17) We refer to (17) as the pointwise parallelogram law of ℋ. The pointwise scalar product on ℋ is defined as follows: ℋ×ℋ∋(v,w)↦v⋅w:= 1 2(|v+w|2−|v|2−|w|2)∈Z, for every v,w∈ℋ. One can readily check that the pointwise scalar product is U-bilinear, which means that ℋ∋v↦v⋅z∈Z, is U-linear, ℋ∋w↦z⋅w∈Z, is U-linear, for any given z∈ℋ. We will study Hilbert modules more in detail in Section 3.5. 3.1.1. Local invertibility Let (𝒰,U,V) be a Dedekind complete CSP metric f-structure, and ℳ a V-normed U-module. Then each v∈ℳ is associated with the element 𝜒{v=0}∈Idem(U), which we define as 𝜒{v=0}:= sup{u∈Idem(U)∣u⋅v= 0}∈Idem(U). The idempotency of 𝜒{v=0} follows from Lemma 2.17 and the fact that U is Dedekind complete CSP. The gluing property of ℳ ensures that 𝜒{v=0}⋅v= 0. We also define 𝜒{v≠0}:= 1 − 𝜒{v=0}∈Idem(U), so that v=𝜒{v≠0}⋅v. Similarly, we define 𝜒{v=w}:= 𝜒{v−w=0} and so on. Proposition 3.6 (Local inverses) Let U be a Dedekind complete CSP metric f-algebra. Let u∈U+ be given. Then there exist a partition (un)n∈ℕ of 𝜒{u>0} and a sequence (wn)n∈ℕ⊂U+ such that un(uwn− 1) = 0, for every n ∈ℕ. Proof. First, recall that (U,U,U) is a metric f-structure (Example 2.24) and that U is a U-Banach U-module (Proposition 3.3). Since 𝒮(U) is super-order-dense in U (as guaranteed by the very definition of a localizable f-algebra), we can find a non-decreasing sequence (sn)n∈ℕ⊂𝒮+(U) such that u=supn∈ℕsn. Note that, setting bn:= 𝜒{sn>0} for every n∈ℕ, we have 𝜒{u>0}=supn∈ℕbn. Indeed, on the one hand 𝜒{sn>0}≤𝜒{u>0} for every n∈ℕ and thus supnbn≤𝜒{u>0}. On the other hand, if we denote s:= 𝜒{u>0}−supnbn∈Idem(U), then s= 0 in view of the fact that su =ssup n∈ℕ sn=sup n∈ℕ ssn= 0. Moreover, for any n∈ℕ there exists a real number 𝜆n> 0 with 𝜆nbn≤u. More precisely, if sn is written as ∑kn i=1 𝜆i nui n for some kn∈ℕ, (𝜆i n)kn i=1 ⊂ℝ∩(0,+∞) and (ui n)kn i=0 ∈𝒫f(1U), then Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 20 • D. LUˇ CI´ C AND E. PASQUALETTO we have that bn=u1 n∨⋯∨ukn n and that 𝜆1 n∧⋯∧𝜆kn n can be chosen as 𝜆n. Now we define u1:= b1 and un+1 := bn+1(1 − bn) for every n∈ℕ. Observe that (un)n∈ℕ is a partition of 𝜒{u>0}. Next we consider the simple elements tj:= ∑kj i=1 1 𝜆i j ui j for every j∈ℕ. Given any n∈ℕ, we have that the sequence (untj)∞ j=n is non-increasing and satisfies 0 ≤untj≤1 𝜆n un for every j≥n. Then the infimum wn:= infj≥nuntj∈U+ exists and, since sjtj=bj≥un for every j≥n, it holds that unuwn=𝜆−1 nunu−unu(𝜆−1 nun−wn) = 𝜆−1 nunu−un(sup j≥n sj)(𝜆−1 nun−inf j≥nuntj) =𝜆−1 nunu−(sup j≥n sj)sup j≥n (𝜆−1 nun−untj) = 𝜆−1 nunu−sup j≥n (𝜆−1 nunsj−unsjtj) =𝜆−1 nunu−sup j≥n (𝜆−1 nunsj−un) = 𝜆−1 nunu−un(𝜆−1 nu− 1) = un. Consequently, the statement is finally achieved. 3.1.2. Support of a metric f-structure Given a metric f-structure (𝒰,U,V) that is Dedekind complete and CSP, we can define its support, which is the ‘largest idempotent element where at least an element of V does not vanish’. Namely: Definition 3.7 (Support) Let (𝒰,U,V) be a Dedekind complete CSP metric f-structure. Then we define S(V) as S(V) := sup{𝜒{v≠0}∣v∈V+}∈Idem(U). We say that S(V) is the support of V or of the metric f-structure (𝒰,U,V). Let us check that the previous definition is well-posed. Since 𝜒{v≠0}≤1 for every v∈V+, the Dedekind completeness of V ensures that S(V) exists. Moreover, the countable representability assumption ensures the existence of a sequence (vn)n∈ℕ⊂V+ such that S(V) = supn𝜒{vn≠0}. Taking into account Lemma 2.17, it also follows that S(V)∈Idem(U). Remark 3.8 If (𝒰,U,V,W,Z) is a CSP complete dual system of metric f-structures, then it holds that S(V)∧S(W)≤S(Z). In order to prove it, fix two sequences (vi)i∈ℕ⊂V+ and (wj)j∈ℕ⊂W+ with S(V) = supi𝜒{vi≠0} and S(W) = supj𝜒{wj≠0}. Notice that 𝜒{vi≠0}∧𝜒{wj≠0}≤𝜒{viwj≠0}≤S(Z) for every i,j∈ℕ. Taking the supremum over i,j∈ℕ, we conclude that S(V)∧S(W)≤S(Z), as we claimed. Next we prove two technical results concerning the support of a metric f-structure. Lemma 3.9 Let (𝒰,U,V) be a Dedekind complete CSP metric f-structure. Then there exists an element h ∈V+∩U+ such that h ≤1 and 𝜒{h>0}=S(V). Proof. Pick a sequence (vn)n∈ℕ⊂V+ such that vn≤1 for every n∈ℕ and S(V) = supn𝜒{vn≠0}. Define u1:= 𝜒{v1≠0}∈Idem(U) and, recursively, un+1 := 𝜒{vn+1≠0}(1 − u1)…(1 − un)∈ Idem(U) for every n∈ℕ. Notice that (un)n∈ℕ is a Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 AN AXIOMATIC THEORY OF NORMED MODULES VIA RIESZ SPACES • 21 partition of S(V). Recalling item (v) of Definition 2.23, we can find 𝛿> 0 such that (unwn)n∈ℕ is order-bounded in V whenever (wn)n∈ℕ⊂V+ is chosen so that ∑n∈ℕdV(unwn,0) ≤𝛿. Now pick (𝜆n)n∈ℕ⊂(0,1) with dV(𝜆nunvn,0) ≤𝛿 2n for all n∈ℕ. Therefore, the element h:= ∑n∈ℕ𝜆nunvn∈V+ exists. Notice that h≤1 and 𝜒{h≠0}= S(V). In particular, 𝜒{h=0}|v|= (1 − S(V))𝜒{v=0}|v|= 0 for every v∈V, whence the statement follows. Lemma 3.10 Let (𝒰,U,V) be a Dedekind complete CSP metric f-structure. Let W ⊂𝒰 be a Dedekind complete CSP metric Riesz space such that (𝒰,U,W) is a metric f-structure and S(V)≤S(W). Let v ∈V+ be given. Then there exists a partition (un)n∈ℕ⊂W∩Idem(U) of 𝜒{v≠0} such that unv∈W∩U holds for every n ∈ℕ. Proof. Thanks to Lemma 3.9, we can find an element h∈W+∩U+ such that 𝜒{h≠0}= S(W). We then define ˜ si:= 𝜒{0<v≤ih} and ˜ tj:= 𝜒{v≠0}𝜒{h≥j−11U} for every i,j∈ℕ. We claim that sup i∈ℕ ˜ si=𝜒{v≠0}=sup j∈ℕ ˜ tj. (18) We prove the first equality, since the proof of the second one is similar. Clearly, supi˜ si≤𝜒{v≠0}. For the converse inequality, we argue by contradiction: suppose that s:= 𝜒{v≠0}−supi˜ si≠0. Then ish ≤v for every i∈ℕ, whence it follows (since 𝒰 is Archimedean) that sh = 0, which leads to a contradiction. Therefore, the claim (18) is proved. Now let us define s1:= ˜ s1, t1:= ˜ t1 and, recursively, si+1 := ˜ si+1 −s1…si˜ si+1 and tj+1 := ˜ tj+1 −t1…tj˜ tj+1 for every i,j∈ℕ. Notice that (18) implies that (si)i∈ℕ and (tj)j∈ℕ are partitions of 𝜒{v≠0}. Moreover, siv≤ih ∈W+∩U+ and tj≤jh ∈W+ for every i,j∈ℕ, thus accordingly siv∈W+∩U+ and tj∈W+. Relabelling the family {sitj:i,j∈ℕ} as {un}n∈ℕ, we finally obtain a partition (un)n∈ℕ⊂W∩Idem(U) of the element 𝜒{v≠0} satisfying unv∈W∩U for every n∈ℕ, as desired. 3.2. Homomorphisms of normed modules To begin with, we introduce the notion of a homomorphism between normed modules. Definition 3.11 (Homomorphism of normed modules) Let (𝒰,U,V,W,Z) be a dual system of metric f-structures, ℳ a V-normed U-module and 𝒩 a Z-normed U-module. Then we define Hom(ℳ,𝒩) := {T:ℳ→𝒩U-linear ∣∃w∈W+:|Tv|≤w|v|,for every v∈ℳ}. Next, we endow Hom(ℳ,𝒩) with a U-module structure. Given any T,S∈Hom(ℳ,𝒩) and u∈ U, we define the elements T+S∈Hom(ℳ,𝒩) and u⋅T∈Hom(ℳ,𝒩) as (T+S)v:= Tv +Sv, for every v∈ℳ, (u⋅T)v:= u⋅Tv, for every v∈ℳ, respectively. One can readily check that the triple (Hom(ℳ,𝒩),+,⋅) is a module over U. In the case where W is Dedekind complete, for any given T∈Hom(ℳ,𝒩) it holds that ∃|T|:= inf{w∈W+∣|Tv|≤w|v|,for every v∈ℳ}∈W+. The space of homomorphisms between two normed modules inherits a normed module structure: Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 22 • D. LUˇ CI´ C AND E. PASQUALETTO Theorem 3.12 Let (𝒰,U,V,W,Z) be a complete dual system of metric f-structures. Let ℳ be a V-normed U-module, and 𝒩 a Z-normed U-module. Let T ∈Hom(ℳ,𝒩) be given. Then |Tv|≤|T||v|,for every v ∈ℳ. Moreover, the space (Hom(ℳ,𝒩),|⋅|) is a W-normed U-module and ∣∑ n∈ℕ un⋅Tn∣=∑ n∈ℕ un|Tn|,for every (un,Tn)n∈ℕ∈Adm(Hom(ℳ,𝒩)). (19) If in addition 𝒩 is a Z-Banach U-module, then Hom(ℳ,𝒩) is a W-Banach U-module. Proof. Verification of the W-pointwise norm axioms. Given any T∈Hom(ℳ,𝒩), we define ℱT:= {w∈W+∣|Tv|≤w|v|,for every v∈ℳ}≠∅. (20) Since ℱT is a sublattice of W+—thus in particular it is downwards directed—we deduce that |Tv|≤ inf w∈ℱT w|v|=|v|inf w∈ℱT w=|T||v|, for every v∈ℳ, as a consequence of the order continuity of the multiplication from V+×W+ to Z+. It readily follows that |⋅|: Hom(ℳ,𝒩)→W+ satisfies (12a) and (12b). It is also easy to check that the identity |𝜆T|=|𝜆||T| holds for every 𝜆∈ℝ and T∈Hom(ℳ,𝒩). We now pass to the verification of (12c). For any u∈U and T∈Hom(ℳ,𝒩), one has |(u⋅T)v|=|u||Tv|≤|u||T||v| for every v∈ℳ, whence it follows that |u⋅T|≤|u||T|. On the other hand, we claim that also |u||T|≤|u⋅T|, for every u∈Uand T∈Hom(ℳ,𝒩). (21) In the case where u∈Idem(U), the inequality stated in (21) follows from the observation that u|T|=u|(1 − u)⋅T+u⋅T|≤u|(1 − u)⋅T|+u|u⋅T|≤u(1 − u)|T|+|u⋅T|=|u⋅T|. Moreover, if u=∑k i=1 𝜆iui∈𝒮+(U) is given, then for any j= 1,…,k it holds that uj k ∑ i=1 𝜆i|ui⋅T|≤ k ∑ i=1 𝜆iuiuj|T|=𝜆ju2 j|T|=|uju⋅T|≤uj|u⋅T|, which implies that u|T|=∑k i=1 𝜆iui|T|=∑k i=1 𝜆i|ui⋅T|≤|u⋅T|, proving (21) for all u∈𝒮+(U). Given any u∈U+, we can pick (un)n∈ℕ⊂𝒮+(U) such that limn→∞dU(un,u) = 0 and thus |u||T|=lim n→∞|un||T|≤ lim n→∞|un⋅T|=|u⋅T|, which proves (21) for all u∈U+. Finally, given an arbitrary element u∈U we have that |u+⋅T|∧|u−⋅T|≤(u+|T|)∧(u−|T|)=(u+∧u−)|T|(5a) = 0, so that |u⋅T|=|u+⋅T|+|u−⋅T| by (5d) and thus |u||T|=u+|T|+u−|T|≤|u+⋅T|+|u−⋅T|=|u⋅T|. This proves (21) for general u∈U. Therefore, the proof of the validity of (12c) is complete. Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 AN AXIOMATIC THEORY OF NORMED MODULES VIA RIESZ SPACES • 23 Verification of the gluing property. To prove that (Hom(ℳ,𝒩),|⋅|) is a W-normed U-module, it only remains to show the validity of the gluing property. Fix any (un)n∈ℕ∈𝒫(1U) and (Tn)n∈ℕ⊂Hom(ℳ,𝒩) with (|un⋅Tn|)n∈ℕ order-bounded in W. Since |un⋅Tn|=|un||Tn|, this means that (un,|Tn|)n∈ℕ∈Adm(W) and thus w:= ∑n∈ℕun|Tn|∈W+ exists, cf. with Proposition 3.3. Given any v∈ℳ, we have that |un⋅Tnv|≤|un⋅Tn||v| for every n∈ℕ, which ensures that (un,Tnv)n∈ℕ∈Adm(𝒩), so it makes sense to consider Tv := ∑n∈ℕun⋅Tnv∈𝒩. The U-linearity of the resulting map T:ℳ→𝒩 can be easily checked. Given that for any n∈ℕ and v∈ℳ one has that un|Tv|=|un⋅Tv|=|un⋅Tnv|≤un|Tn||v|=unw|v|, we deduce that |Tv|=sup n∈ℕ un|Tv|≤sup n∈ℕ unw|v|=w|v|, for every v∈ℳ. This yields T∈Hom(ℳ,𝒩) and |T|≤w. Note that (un⋅T)v=un⋅Tv =un⋅Tnv= (un⋅Tn)v for every v∈ℳ, so that T=∑n∈ℕun⋅Tn. Finally, for any n∈ℕ we have un⋅Tn=un⋅T and thus un|Tn|=un|T|, which gives w=∑n∈ℕun|Tn|=supn∈ℕun|T|=|T|. This proves (19). Completeness. Suppose (𝒩,d𝒩) is complete. Let (Tn)n∈ℕ⊂Hom(ℳ,𝒩) be a Cauchy sequence. Until taking a not relabeled subsequence, we may assume that dW(|Tn+1 −Tn|,0) ≤2−n for every n∈ℕ. Define Sn:= |T1|+∑k<n|Tk+1 −Tk|∈W+ for every n∈ℕ. If n,m∈ℕ are such that n<m, then we have that Sm−Sn=∑m−1 k=n|Tk+1 −Tk|, and thus accordingly dW(|Sm−Sn|,0) ≤m−1 ∑ k=n dW(|Tk+1 −Tk|,0) ≤m−1 ∑ k=n 1 2k≤1 2n−1 . This shows that (Sn)n∈ℕ is a Cauchy sequence in W+, so that it dW-converges to some S∈W+. Notice that (Sn)n∈ℕ is a non-decreasing sequence by construction, and thus in particular it holds |Tn|=∣(Tn−Tn−1) + (Tn−1 −Tn−2) + ⋯+ (T2−T1) + T1∣≤Sn≤S,∀n∈ℕ. (22) Since the multiplication is continuous from V×W to Z and dW(|Tm−Tn|,0) →0 as n,m→∞, for any fixed element v∈ℳ we have that d𝒩(Tmv,Tnv)≤dZ(|Tm−Tn||v|,0) →0 as n,m→∞, which shows that (Tnv)n∈ℕ is a Cauchy sequence in 𝒩. We then define Tv := limn→∞Tnv∈𝒩. The resulting mapping T:ℳ→𝒩 is U-linear, as it is a pointwise limit of U-linear maps. Also, |Tv|=lim n→∞|Tnv|(22) ≤S|v|, for every v∈ℳ, whence it follows that T∈Hom(ℳ,𝒩) and |T|≤S. Now set Pn m:= ∑m−1 k=n|Tk+1 −Tk|∈W+ for every n,m∈ℕ with n<m. Arguing as we did before, we see that (Pn m)m∈ℕ is dW-Cauchy, and thus limm→∞Pn m=Pn for some Pn∈W+. Note that Pn m≤Pn and dW(Pn,0) ≤2−n+1. Hence, |(T−Tn)v|=lim m→∞|(Tm−Tn)v|≤ lim m→∞Pn m|v|=Pn|v|, for every v∈ℳ, which implies |T−Tn|≤Pn→0 as n→∞. The completeness of Hom(ℳ,𝒩) follows. In the case where the dual system under consideration is CSP complete, the W-pointwise norm |T| of any given T∈Hom(ℳ,𝒩) can be also characterized as follows: Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 24 • D. LUˇ CI´ C AND E. PASQUALETTO Lemma 3.13 Let (𝒰,U,V,W,Z) be a CSP complete dual system of metric f-structures, ℳ a V-normed U-module and 𝒩 a Z-normed U-module. Then it holds that |T|=sup{|Tv|∣v∈ℳ,|v|≤1},for every T ∈Hom(ℳ,𝒩). (23) Proof. On the one hand, |Tv|≤|T||v|≤|T| for all v∈ℳ with |v|≤1, and thus the right-hand side in (23) exists and defines an element b∈W+ with b≤|T|. On the other hand, we claim that |Tv|≤b|v|, for every v∈ℳ. (24) In order to prove it, fix any v∈ℳ. By using Lemma 3.10 and Proposition 3.6, we can find a partition (un)n∈ℕ of 𝜒{v≠0} and a sequence (wn)n∈ℕ⊂U+ such that un|v|∈U and unwn|v|=un for every n∈ℕ. Letting vn:= (unwn)⋅v∈ℳ, we have |vn|=un≤1 and thus |Tvn|≤b. Then |Tv|=∑ n∈ℕ (unwn|v|)|Tv|=∑ n∈ℕ|v||Tvn|≤∑ n∈ℕ unb|v|=b|v|. This proves (24) and accordingly that b=|T|, thus concluding the proof of the statement. Given a dual system of metric f-structures (𝒰,U,V,W,Z), a V-Banach U-module ℳ and a Z- Banach U-module 𝒩, we define the kernel of a homomorphism T∈Hom(ℳ,𝒩) as ker(T) := T−1({0}) = {v∈ℳ|Tv = 0}. It can be readily checked that ker(T) is a V-Banach U-submodule of ℳ. It is natural to introduce the categories of normed modules and of Banach modules: Definition 3.14 (Category of normed modules) Let (𝒰,U,V) be a metric f-structure. Then we define the category NormMod(𝒰,U,V) of normed modules over (𝒰,U,V) as follows: (i) The objects of NormMod(𝒰,U,V) are given by the V-normed U-modules. (ii) For any two objects ℳ and 𝒩 of NormMod(𝒰,U,V), the morphisms between ℳ and 𝒩 are given by those homomorphisms T∈Hom(ℳ,𝒩) satisfying |Tv|≤|v| for every v∈ℳ. Moreover, we denote by BanMod(𝒰,U,V) the full subcategory of NormMod(𝒰,U,V) whose objects are the V-Banach U-modules. It would be interesting—but outside the scope of this work—to investigate which results of [36] are valid for Banach modules over a more general class of metric f-structures. Let us also define the dual of a Banach module: Definition 3.15 (Dual Banach module) Let (𝒰,U,V,W,Z) be a complete dual system of metric f-structures, and ℳ a V-normed U-module. Then the dual of ℳ is the W-Banach U-module ℳ*:= Hom(ℳ,Z). The duality pairing between ℳ and ℳ* is given by ℳ*×ℳ∋(𝜔,v)↦⟨𝜔,v⟩:= 𝜔(v)∈Z. Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 AN AXIOMATIC THEORY OF NORMED MODULES VIA RIESZ SPACES • 31 (33), so that we have |(𝜙*T)w|≤𝜙(|T|)|w| for every w∈𝜙*ℳ, whence it follows that |𝜙*T|≤𝜙(|T|). Combining Theorem 3.23 and Corollary 3.25, any homomorphism 𝜙: (𝒰1,U1,V1)→(𝒰2,U2,V2) induces a functor 𝜙*:BanMod(𝒰1,U1,V1)→BanMod(𝒰2,U2,V2). We point out that, even though we chose the term ‘pushforward’ by analogy with [20, Section 1.6], from the categorical perspective the correct term would be the direct image functor. Remark 3.26 Let (𝒰1,U1,V1,V1,Z1), (𝒰2,U2,V2,V2,Z2) be dual systems of metric f-structures. Let 𝜙:𝒰1→𝒰2 be a homomorphism of dual systems. Let ℋ be a V1-Hilbert U1-module. Then 𝜙*ℋ, is a V2-Hilbert U2-module, (34a) 𝜙*v⋅𝜙*w=𝜙(v⋅w), for every v,w∈ℋ. (34b) To prove (34a), notice that Theorem 3.23 (i) implies that the elements of 𝒢(𝜙*(ℋ))—thus all the elements of 𝜙*ℋ, thanks to Theorem 3.23 (ii)—satisfy the pointwise parallelogram law. Also, 𝜙*v⋅𝜙*w=1 2(|𝜙*(v+w)|2−|𝜙*v|2−|𝜙*w|2)=𝜙(1 2(|v+w|2−|v|2−|w|2))=𝜙(v⋅w) hold for every v,w∈ℋ, which shows that (34b) is verified. Theorem 3.27 Let (𝒰1,U1,V1,W1,Z1) and (𝒰2,U2,V2,W2,Z2) be CSP complete dual systems of metric f-structures. Let 𝜙:𝒰1→𝒰2 be a homomorphism of dual systems. Let ℳ be a V1-Banach U1-module. Then there exists a unique homomorphism I𝜙∈Hom(𝜙*ℳ*,(𝜙*ℳ)*) such that ⟨I𝜙(𝜙*𝜔),𝜙*v⟩=𝜙(⟨𝜔,v⟩), for every 𝜔∈ℳ*and v ∈ℳ. (35) Moreover, it holds that |I𝜙(𝜂)|=|𝜂|,for every 𝜂∈𝜙*ℳ*. (36) Proof. Given any 𝜔∈ℳ*, we define the operator ˜ i𝜙(𝜔): ℳ→Z2 as ˜ i𝜙(𝜔)v:= 𝜙(⟨𝜔,v⟩), for every v∈ℳ. Note that ˜ i𝜙(𝜔) is linear and satisfies |˜ i𝜙(𝜔)v|≤𝜙(|𝜔|)𝜙(|v|) for all v∈ℳ. Hence, we know from Proposition 3.24 that there is a unique element i𝜙(𝜔)∈(𝜙*ℳ)* such that |i𝜙(𝜔)|≤𝜙(|𝜔|) and ⟨i𝜙(𝜔),𝜙*v⟩=˜ i𝜙(𝜔)v=𝜙(⟨𝜔,v⟩), for every v∈ℳ. Since the resulting operator i𝜙:ℳ*→(𝜙*ℳ)* is linear, by applying Proposition 3.24 again we deduce that there exists a unique I𝜙∈Hom(𝜙*ℳ*,(𝜙*ℳ)*) with |I𝜙|≤1 such that (35) holds. Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 32 • D. LUˇ CI´ C AND E. PASQUALETTO It remains to check (36). Thanks to Proposition 3.4 (i) and Theorem 3.23 (ii), it suffices to prove (36) for 𝜂∈𝒢(𝜙*(ℳ*)), say 𝜂=∑n∈ℕun⋅𝜙*𝜔n with (un)n∈𝒫(1U2) and (𝜔n)n⊂ℳ*. Given any n∈ℕ, we deduce from Lemma 3.13 that |𝜔n|=sup{⟨𝜔n,v⟩:v∈ℳ,|v|≤1}, so we can pick a sequence (vi n)i∈ℕ⊂ℳ such that |vi n|≤1 for all i∈ℕ and |𝜔n|=supi⟨𝜔n,vi n⟩. Then 𝜙(|𝜔n|) = |𝜙*𝜔n|=sup i∈ℕ𝜙(⟨𝜔n,vi n⟩)(23) =sup i∈ℕ⟨I𝜙(𝜙*𝜔n),𝜙*vi n⟩. (37) Multiplying by un and summing over n, we deduce (using Lemma 3.13 again and |𝜙*vi n|≤1 ) that |𝜂|=∑ n∈ℕ un|𝜙*𝜔n|(37) =∑ n∈ℕ unsup i∈ℕ⟨I𝜙(𝜙*𝜔n),𝜙*vi n⟩=∑ n∈ℕ sup i∈ℕ⟨un⋅I𝜙(𝜙*𝜔n),𝜙*vi n⟩ =∑ n∈ℕ sup i∈ℕ⟨I𝜙(un⋅𝜂),𝜙*vi n⟩≤∑ n∈ℕ|I𝜙(un⋅𝜂)|=∑ n∈ℕ un|I𝜙(𝜂)|=|I𝜙(𝜂)|. Since |I𝜙|≤1 yields the converse inequality |I𝜙(𝜂)|≤|𝜂|, the statement is finally achieved. 3.3.3. Completion of a normed module It follows from Theorem 3.23 that each V-normed U-module can be ‘completed’ to a V-Banach U- module, and much like the metric, completion of a normed space has a Banach space structure: Theorem 3.28 (Completion of a normed module) Let (𝒰,U,V) be a metric f-structure, and let ℳ be a V-normed U-module. Then there exists a unique couple (ˉ ℳ,𝜄) such that ˉ ℳ is a V-Banach U-module and 𝜄:ℳ→ˉ ℳ is a U-linear map with a dense range satisfying |𝜄v|=|v| for every v ∈ℳ. Uniqueness is up to unique isomorphism: given another couple (˜ ℳ,˜ 𝜄) with the same properties, there is a unique isomorphism Φ:ˉ ℳ→˜ ℳ of V-Banach U-modules such that is a commutative diagram. We say that the couple (ˉ ℳ,𝜄), or just ˉ ℳ, is the completion of ℳ. Moreover, if ℳ,𝒩 are V-normed U-modules and T ∈Hom(ℳ,𝒩) is given, then there exists a unique homomorphism ˉ T∈Hom(ˉ ℳ,ˉ 𝒩) such that ˉ T|ℳ=T, where we regard ℳ and 𝒩 as subsets of ˉ ℳ and ˉ 𝒩, respectively. If in addition U is Dedekind complete, then it holds |ˉ T|=|T|. Proof. The identity mapping id𝒰:𝒰→𝒰 is a homomorphism of metric f-structures from (𝒰,U,V) to itself, and thus we can define ˉ ℳ:= (id𝒰)*ℳ and 𝜄:= (id𝒰)*:ℳ→ˉ ℳ. By Theorem 3.23, to prove the first part of the statement amounts to showing that 𝜄 is U-linear and that 𝜄(ℳ) is dense in ˉ ℳ. The former property follows from Corollary 3.18. About the latter, recall that 𝒢(𝜄(ℳ)) is dense in ˉ ℳ, and thus it only remains to show that 𝜄(ℳ) = 𝒢(𝜄(ℳ)). The inclusion 𝜄(ℳ)⊂𝒢(𝜄(ℳ)) is trivial. Conversely, fix w=∑n∈ℕun⋅𝜄vn∈𝒢(𝜄(ℳ)). Since (un,𝜄vn)n∈ℕ∈Adm( ˉ ℳ), we have that (un,vn)n∈ℕ∈Adm(ℳ), and thus it makes sense to consider v:= ∑n∈ℕun⋅vn∈ℳ. We claim that 𝜄v=w, whence it follows that w∈𝜄(ℳ) and thus 𝒢(𝜄(ℳ)) ⊂𝜄(ℳ). Given that un⋅𝜄v=𝜄(un⋅v) = 𝜄(un⋅vn) = un⋅𝜄vn=un⋅w, for every n∈ℕ, we deduce from Lemma 3.2 that 𝜄v=w. Therefore, the first part of the statement is proved. Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 AN AXIOMATIC THEORY OF NORMED MODULES VIA RIESZ SPACES • 33 About the second part of the statement, observe that ˉ T:= (id𝒰)*T∈Hom( ˉ ℳ,ˉ 𝒩) is the unique homomorphism extending T. Note also that if ˉ u∈U+ satisfies |Tv|≤ˉ u|v| for all v∈ℳ, then |ˉ Tv|≤ˉ u|v| for every v∈ˉ ℳ by approximation. Finally, borrowing the notation from the proof of Theorem 3.12 (see (20)), we get that ℱT=ℱˉ T, so that (assuming that U is Dedekind complete) we conclude that |ˉ T|=|T|. 3.4. Hahn–Banach extension theorem The aim of this section is to obtain a normed module version of the Hahn–Banach extension theorem as well as to investigate some of its basic consequences. Let (𝒰,U,V) be a metric f-structure, and ℳ a module over U. Then we say that a given map p:ℳ→V+ is U-sublinear if it is subadditive (that is, p(v+w)≤p(v) + p(w) for every v,w∈ℳ ) and positively U-homogeneous, which means that p(u⋅v) = up(v) for all u∈U+ and v∈ℳ. Lemma 3.29 (One-dimensional dominated extension) Let (𝒰,U,V) be a Dedekind complete CSP metric f-structure, ℳ a V-normed U-module and 𝒩⊊ℳ a V-normed U-submodule of ℳ. Fix any z ∈ℳ such that u ⋅z∉𝒩 for every u ∈Idem(U)\{0}, with u ≤𝜒{z≠0}. Let f:𝒩→V be a U-linear map, and p:ℳ→V+ a U-sublinear map with f ≤p on 𝒩. Define 𝒩+z:= 𝒩+U⋅z. Then there exists a U-linear map ˉ f:𝒩+z→V with ˉ f|𝒩=f such that ˉ f≤p on 𝒩+z. Proof. Given any v,w∈𝒩, we can estimate f(v) − f(w) = f(v−w)≤p(v−w) = p(v+z− (w+z))≤p(v+z) + p(−w−z), whence it follows that −p(−w−z) − f(w)≤p(v+z) − f(v) for every v,w∈𝒩. Substituting w= 0, we obtain that p(z)≤p(v+z) − f(v) for every v∈𝒩, and thus the Dedekind completeness of V ensures that b:= inf{p(v+z) − f(v) : v∈𝒩}∈V exists. Then we have that −p(−w−z) − f(w)≤b≤p(v+z) − f(v), for every v,w∈𝒩. (38) Substituting v=w= 0 in (38) and multiplying by 𝜒{z=0}, we deduce that 𝜒{z=0}⋅b= 0, so that 𝜒{z=0}≤𝜒{b=0}. (39) Next we claim that for any v,˜ v∈𝒩 and u,˜ u∈U it holds that v+u⋅z=˜ v+˜ u⋅z⟹v=˜ vand u⋅b=˜ u⋅b. (40) To prove it, suppose that (u−˜ u)⋅z=˜ v−v. Pick a partition (un)n∈ℕ of 𝜒{u≠˜ u} and (wn)n∈ℕ⊂U such that un(u−˜ u)wn=un for every n∈ℕ. Multiplying (u−˜ u)⋅z=˜ v−v by unwn, we obtain un⋅z= (un(u−˜ u)wn)⋅z= (unwn)⋅(˜ v−v)∈𝒩, for every n∈ℕ. Hence, the gluing property of 𝒩 ensures that 𝜒{u≠˜ u}⋅z∈𝒩, and thus accordingly 𝜒{u≠˜ u}≤𝜒{z=0}. This implies that ˜ v−v=u⋅z−˜ u⋅z= 0 and (recalling (39)) that u⋅b=˜ u⋅b, proving (40). Therefore, the map ˉ f:𝒩+z→V defined as follows is well-posed: ˉ f(v+u⋅z) := f(v) + u⋅b, for every v∈𝒩and u∈U. It is immediate to check that ˉ f is a U-linear extension of f. It only remains to show that ˉ f≤p on 𝒩+z. To this aim, fix v∈𝒩 and u∈U\{0}. Pick a partition (un)n∈ℕ of {u> 0}, a Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 34 • D. LUˇ CI´ C AND E. PASQUALETTO partition (˜ un)n∈ℕ of {u< 0} and (wn)n∈ℕ,(˜ wn)n∈ℕ⊂U+ such that unu+wn=un and ˜ unu−˜ wn=˜ un for all n∈ℕ. It follows from (38) that p(wn⋅v+z) − f(wn⋅v)≥b and −p(˜ wn⋅v−z) + f(˜ wn⋅v)≤b. Multiplying by unu+ and ˜ unu−, respectively, we obtain un⋅(f(v) + u⋅b)≤un⋅p(v+u⋅z) and ˜ un⋅(f(v) + u⋅b) = ˜ un⋅(f(v) − u−⋅b)≤˜ un⋅p(v−u−⋅z) = ˜ un⋅p(v+u⋅z), respectively. Using the gluing property, we conclude that ˉ f(v+u⋅z) = f(v) + u⋅b≤p(v+u⋅z). Theorem 3.30 (Hahn–Banach theorem for normed modules) Let (𝒰,U,V) be a Dedekind complete CSP metric f-structure. Let ℳ be a V-normed U-module, and let 𝒩⊊ℳ be a V-normed U-submodule of ℳ. Let f :𝒩→V be a U-linear map, and p:ℳ→V+ a U-sublinear map with f ≤p on 𝒩. Then there exists a U-linear map ˉ f:ℳ→V such that ˉ f|𝒩=f such that ˉ f≤p on ℳ. Proof. Let us denote by ℱ the family of all couples (𝒬,g), where 𝒬 is a V-normed U-submodule of ℳ containing 𝒩 and g:𝒬→V is a U-linear extension of f satisfying q≤p on 𝒬. Clearly ℱ is non-empty, as it contains (𝒩,f). We endow ℱ with the partial order ⪯ defined as follows: given (𝒬,g),( ˜ 𝒬,˜ g)∈ℱ, we declare that (𝒬,g)⪯(˜ 𝒬,˜ g) provided 𝒬⊂˜ 𝒬 and ˜ g|𝒬=g. It is easy to check that any totally ordered subset 𝒞 of (ℱ,⪯) has an upper bound, namely, (𝒬,g0), where 𝒬0:= ⋃(𝒬,g)∈ℱ𝒬 and g0:𝒬0→V is given by g0(v) := g(v) for every (𝒬,g)∈ℱ with v∈𝒬. Hence, an application of Zorn’s lemma yields the existence of a maximal element (𝒩0,f0) of (ℱ,⪯). In order to conclude, we aim to show that 𝒩0=ℳ. We argue by contradiction: suppose that ℳ\𝒩0≠∅. Fix any ˜ z∈ℳ\𝒩0. The fact that U is Dedekind complete and CSP ensures that ∃q:= sup{u∈Idem(U)∣u⋅˜ z∈𝒩0}∈Idem(U). Moreover, the gluing property implies that q⋅˜ z∈𝒩0. Then we define z:= (1 − q)⋅˜ z∈ℳ. Observe that u⋅z∉𝒩0 holds for every u∈Idem(U)\{0}, with u≤1 − q=𝜒{z≠0}. Therefore, Lemma 3.29 yields the existence of a map ˉ f:𝒩0+U⋅z→V such that (𝒩0+U⋅z,ˉ f)∈ℱ and (𝒩0,f0)⪯(𝒩0+U⋅z,ˉ f), which leads to a contradiction with the maximality of (𝒩0,f0). Corollary 3.31 Let (𝒰,U,V,W,Z) be a CSP complete dual system of metric f-structures. Let ℳ be a V-Banach U-module. Let v∈ℳ satisfy |v|∈Z∩U and 𝜒{v≠0}∈W. Then there exists an element 𝜔∈ℳ* such that ⟨𝜔,v⟩=|v| and |𝜔|=𝜒{v≠0}. Proof. Notice that U⋅v is a V-Banach U-submodule of ℳ. We define the map T:U⋅v→Z as T(u⋅v) := u|v|, for every u∈U. Clearly, T is well-posed and U-linear. Moreover, we have |T(u⋅v)|=𝜒{v≠0}|u⋅v| for every u∈U, and thus T∈Hom(U⋅v,Z) and |T|≤𝜒{v≠0}. Now let us define p:ℳ→Z+ as p(w) := 𝜒{v≠0}|w| for every w∈ℳ. It can be readily checked that p is U-sublinear. Since T≤p on U⋅v, an application of Theorem 3.30 yields a U-linear map 𝜔:ℳ→Z satisfying 𝜔|U⋅v=T and 𝜔≤p on ℳ. The latter gives |𝜔(w)|≤𝜒{v≠0}|w| for every w∈ℳ, which shows that 𝜔∈ℳ* and |𝜔|≤𝜒{v≠0}. Notice also that ⟨𝜔,v⟩=|v| by construction. Therefore, in order to conclude it suffices to check that |𝜔|≥𝜒{v≠0}. To this aim, pick a partition (un)n∈ℕ of {v≠0} and a sequence (wn)n∈ℕ⊂U+ such that unwn|v|=un holds for Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 AN AXIOMATIC THEORY OF NORMED MODULES VIA RIESZ SPACES • 35 every n∈ℕ. Now fix any g∈W+ satisfying |⟨𝜔,w⟩|≤g|w| for every w∈ℳ. Then for any n∈ℕ we can estimate ung=unwn|v|g=g|(unwn)⋅v|≥|⟨𝜔,(unwn)⋅v⟩|=unwn|v|=un, which implies that g≥𝜒{v≠0} thanks to the arbitrariness of n∈ℕ. The statement is achieved. 3.4.1. Reflexive Banach modules Let (𝒰,U,V,W,Z) be a complete dual system of metric f-structures, and let ℳ be a V-Banach U- module. Then we denote the bidual of ℳ by ℳ** := (ℳ*)*. Here, we are considering the dual of ℳ* with respect to the dual system (𝒰,U,W,V,Z) (recall Remark 2.27), so that ℳ** is a V-Banach U-module. Definition 3.32 (Embedding into the bidual) Let (𝒰,U,V,W,Z) be a complete dual system of metric f-structures. Let ℳ be a V-Banach U-module. Then we define Jℳ:ℳ→ℳ** as ⟨Jℳ(v),𝜔⟩:= ⟨𝜔,v⟩, for every v∈ℳand 𝜔∈ℳ*. Notice that the map ℳ×ℳ*∋(v,𝜔)↦⟨Jℳ(v),𝜔⟩∈Z is U-bilinear. Moreover, one has |⟨Jℳ(v),𝜔⟩|≤|𝜔||v|, for every v∈ℳand 𝜔∈ℳ*. It follows that Jℳ(v)∈ℳ** for every v∈ℳ, Jℳ∈Hom(ℳ,ℳ**) and |Jℳ|≤1. Under suitable assumptions, the homomorphism Jℳ actually preserves the pointwise norm: Proposition 3.33 Let (𝒰,U,V,W,Z) be a CSP complete dual system of metric f-structures. Suppose that S(V)≤S(W). Let ℳ be a V-Banach U-module. Then it holds that ∣Jℳ(v)∣=|v|,for every v ∈ℳ. Proof. Let v∈ℳ be fixed. We aim to show that |Jℳ(v)|≥|v|. In view of Remark 3.8, we know that S(V)≤S(Z). Hence, applying Lemma 3.10 we obtain a partition (un)n∈ℕ of {v≠0} such that un|v|∈Z∩U and 𝜒{un⋅v≠0}∈W for every n∈ℕ. Using Corollary 3.31, we can find a sequence (𝜔n)n∈ℕ⊂ℳ* such that ⟨𝜔n,un⋅v⟩=un|v| and |𝜔n|=𝜒{un⋅v≠0} for all n∈ℕ. Then un⟨Jℳ(v),𝜔n⟩=⟨𝜔n,un⋅v⟩=un|v|=un|𝜔n||v|, for every n∈ℕ. This implies that un|Jℳ(v)|≥un|v| for every n∈ℕ, whence it follows that |Jℳ(v)|≥|v|. In view of Proposition 3.33, it is then natural to give the following definition of reflexivity: Definition 3.34 (Reflexive Banach module) Let (𝒰,U,V,W,Z) be a CSP complete dual system of metric f-structures. Suppose that S(V) = S(W). Then we say that a V-Banach U-module ℳ is reflexive provided that the embedding operator Jℳ:ℳ→ℳ** is surjective. Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 36 • D. LUˇ CI´ C AND E. PASQUALETTO 3.5. Hilbert modules In this section we investigate the properties of Hilbert modules. Among others, we will prove a Cauchy–Schwarz inequality (Lemma 3.35) and a Riesz-representation-type result (Theorem 3.38), we will study orthogonal projections (Theorem 3.36 and Proposition 3.37) and we will show that Hilbert modules are reflexive (Proposition 3.39). Let (𝒰,U,V,V,Z) be a dual system of metric f-structures, and ℋ a V-Hilbert U-module. Then v⋅w=1 4|v+w|2−1 4|v−w|2, for every v,w∈ℋ. (41) Indeed, recalling the definition of the pointwise parallelogram law, we obtain that 1 4|v+w|2=1 4|v|2+1 4|w|2+1 2v⋅w, 1 4|v−w|2=1 4|v|2+1 4|w|2−1 2v⋅w. Subtracting the second identity from the first one, we get (41). Lemma 3.35 (Cauchy–Schwarz inequality) Let (𝒰,U,V,V,Z) be a dual system of metric f-structures, and ℋ a V-Hilbert U-module. Then |v⋅w|≤|v||w|,for every v,w∈ℋ. (42) Proof. Using (41) and the fact that |v|−|w|≤|v+w|≤|v|+|w|, we obtain that v⋅w=1 4|v+w|2−1 4|v−w|2≤1 4((|v|+|w|)2− (|v|−|w|)2) =1 4(|v|2+|w|2+ 2|v||w|−|v|2−|w|2+ 2|v||w|)=|v||w|. We also have that −(v⋅w) = (−v)⋅w≤|−v||w|=|v||w|. Therefore, (eq: CS) is proved. Given a V-Hilbert U-module ℋ and a V-Hilbert U-submodule 𝒩 of ℋ, we define the orthogonal complement of 𝒩 in ℋ as 𝒩⟂:= {v∈ℋ∣v⋅w= 0, for every w∈𝒩}. One can readily check that 𝒩⟂ is a V-Hilbert U-submodule of ℋ. Theorem 3.36 (Hilbert projection theorem for Hilbert modules) Let (𝒰,U,V,V,Z) be a CSP complete dual system of metric f-structures. Let ℋ be a V-Hilbert U-module satisfying dℋ(v,0)2≤dZ(|v|2,0), for every v ∈ℋ. (43) Let C ≠∅ be a closed, convex subset of ℋ such that 𝒢(C) = C. Let v ∈ℋ be fixed. We define |v−C|:= inf{|v−w|∣w∈C}∈V+,dℋ(v,C) := inf{dℋ(v,w)∣w∈C}∈ℝ+. Then it holds that dV(|v−C|,0) = dℋ(v,C). (44) Moreover, there exists a unique PC(v)∈C, the orthogonal projection of v onto C, such that |v−C|=|v−PC(v)|. Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 AN AXIOMATIC THEORY OF NORMED MODULES VIA RIESZ SPACES • 37 Proof. Applying Lemma 3.9, we can find an element h∈V+ such that h≤1 and 𝜒{h=0}V={0}. By our assumptions, there exists a sequence (˜ wk)k∈ℕ⊂C such that |v−C|=infk∈ℕ|v−˜ wk|. One can readily check that for any n∈ℕ there exists a partition (un k)k∈ℕ of 𝜒{h≠0} such that un k|v−C|≤un k|v−˜ wk|≤un k(|v−C|+1 nh), for every k∈ℕ. In particular, it holds (un k,˜ wk)k∈ℕ∈Adm(ℋ), and thus it makes sense to set 𝜔n:= ∑k∈ℕun k⋅˜ wk∈ℋ. Given that C is closed under the gluing operation, we have that (wn)n∈ℕ⊂C. Notice also that |v−C|≤|v−wn|≤|v−C|+1 nh, for every n∈ℕ. (45) Since limndV(n−1h,0) = 0, we deduce from (45) that limndV(|v−C|,|v−wn|)= 0, and thus dℋ(v,C)≤lim n→∞dℋ(v,wn) = lim n→∞dV(|v−wn|,0) = dV(|v−C|,0). On the other hand, we have |v−C|≤|v−w|, and thus dV(|v−C|,0) ≤dℋ(v,w), for every w∈C. By taking the infimum over w∈C, we get dV(|v−C|,0) ≤dℋ(v,C). All in all, (44) is proved. The Hilbertianity of ℋ and the convexity of C ensure that for any n,m∈ℕ it holds that |wn−wm|2= 2|v−wn|2+ 2|v−wm|2− 4∣v−wn+wm 2∣2 ≤2|v−wn|2+ 2|v−wm|2− 4|v−C|2 (45) ≤2(1 n2+1 m2)h2+ 4(1 n+1 m)|v−C|h, (46) whence it follows that dV(|wn−wm|,0) ≤√dZ(|wn−wm|2,0) →0 as n,m→∞. Hence, (wn)n∈ℕ is a Cauchy sequence in ℋ, and thus limndℋ(wn,ˉ w) = 0 holds for some ˉ w∈C. In particular, we have that dV(|v−C|,|v−ˉ w|) = limndV(|v−C|,|v−wn|) = 0, so that |v−C|=|v−ˉ w|. To prove that ˉ w is the unique element of C with this property, fix any ˜ w∈C with |v−˜ w|=|v−C|. Then 0≤|˜ w−ˉ w|2= 2|v−˜ w|2+ 2|v−ˉ w|2− 4∣v−˜ w+ˉ w 2∣2≤2|v−C|2+ 2|v−C|2− 4|v−C|2= 0, which forces the identity ˜ w=ˉ w. All in all, the statement is finally achieved. The choice of the terminology ‘orthogonal projection’ is justified by the following result: Proposition 3.37 Let (𝒰,U,V,V,Z) be a CSP complete dual system of metric f-structures. Let ℋ be a V-Hilbert U-module satisfying (43). Let 𝒩 be a V-Hilbert U-submodule of ℋ. Then: (i) v −P𝒩(v)∈𝒩⟂ for every v ∈ℋ. (ii) It holds that 𝒩⊕𝒩⟂. (iii) The map P𝒩:ℋ→𝒩 belongs to Hom(ℋ,𝒩). (iv) v =P𝒩(v) + P𝒩⟂(v) and |v|2=|P𝒩(v)|2+|P𝒩⟂(v)|2 for every v ∈ℋ. Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 38 • D. LUˇ CI´ C AND E. PASQUALETTO Proof. Fix w∈𝒩. Denote ˉ w:= P𝒩(v). Pick a partition (un)n∈ℕ of 1U such that un(v−ˉ w)⋅w∈U for every n∈ℕ. Since ˉ w−u⋅w∈𝒩 for every u∈U, we have that |v−ˉ w|≤|v−ˉ w+u⋅w|, and thus |v−ˉ w|2≤|v−ˉ w+u⋅w|2=|v−ˉ w|2+u|w|2− 2u(v−ˉ w)⋅w. (47) Fix any n∈ℕ and 𝜀∈ℝ with 𝜀> 0. Substituting u= −𝜀un(v−ˉ w)⋅w into (47), multiplying both sides by 𝜀−1un and rearranging the various terms, we obtain that 2un|(v−ˉ w)⋅w|2≤𝜀un|(v−ˉ w)⋅w|2|w|2. Letting 𝜀↘0, we get un|(v−ˉ w)⋅w|= 0 for every n∈ℕ, and thus v−ˉ w∈𝒩⟂. Then (i) is proved. To prove (ii), we aim to show that 𝒩+𝒩⟂=ℋ and 𝒩∩𝒩⟂={0}. For the former, just observe that any v∈ℋ can be written as (v−P𝒩(v))+P𝒩(v), where v−P𝒩(v)∈𝒩⟂ by (i) and P𝒩(v)∈𝒩. For the latter, note that if v∈𝒩∩𝒩⟂, then |v|2=v⋅v= 0 and thus v= 0. Let us now pass to the verification of (iii). Given any v,w∈ℋ, we deduce from (i) that (v+w−P𝒩(v+w)) ⏟⏟⏟⏟⏟⏟⏟ ∈𝒩⟂ +P𝒩(v+w) ⏟ ∈𝒩 =v+w=(v−P𝒩(v) + w−P𝒩(w)) ⏟⏟⏟⏟⏟⏟⏟⏟⏟ ∈𝒩⟂ +P𝒩(v) + P𝒩(w) ⏟⏟⏟⏟⏟ ∈𝒩 , and thus accordingly (ii) implies that P𝒩(v+w) = P𝒩(v) + P𝒩(w). Similarly, for any element u∈U we have that (u⋅v−P𝒩(u⋅v)) + P𝒩(u⋅v) = u⋅v=u⋅(v−P𝒩(v)) + u⋅P𝒩(v), which forces the identity P𝒩(u⋅v) = u⋅P𝒩(v). All in all, we showed that P𝒩 is a U-linear map. We also have that |v|2=|v−P𝒩(v)|2+|P𝒩(v)|2≥|P𝒩(v)|2 for every v∈ℋ, and thus P𝒩∈Hom(ℋ,𝒩). Finally, we prove (iv). Given any w∈𝒩⟂, we have that |w−v|2=|w|2+|v|2 for every v∈𝒩 and thus |w−𝒩|2=|w|2, which implies that P𝒩(w) = 0. Using also (iii), we deduce that ∣v−(v−P𝒩(v))∣2=∣P𝒩(v)∣2=∣P𝒩(v) − P𝒩(w)∣2=∣P𝒩(v−w)∣2≤|v−w|2 for every v∈ℋ. Hence, ∣v−(v−P𝒩(v))∣=|v−𝒩⟂|, which implies P𝒩⟂(v) = v−P𝒩(v). In particular, one has |v|2=∣P𝒩(v) + P𝒩⟂(v)∣2=|P𝒩(v)|2+|P𝒩⟂(v)|2, so (iv) is also proved. Theorem 3.38 (Riesz representation theorem for Hilbert modules) Let (𝒰,U,V,V,Z) be a CSP complete dual system of metric f-structures. Let ℋ be a V-Hilbert U-module satisfying (43). We define the operator Rℋ:ℋ→ℋ* as ⟨Rℋ(w),v⟩:= v⋅w,for every v,w∈ℋ. Then Rℋ is an isomorphism of V-Banach U-modules, ℋ* is a V-Hilbert U-module and Rℋ(v)⋅Rℋ(w) = v⋅w,for every v,w∈ℋ. (48) Proof. The properties of the pointwise scalar product and the Cauchy–Schwartz inequality ensure that Rℋ(w)∈ℋ* and |Rℋ(w)|≤|w| for all w∈ℋ. Then Rℋ∈Hom(ℋ,ℋ*) and |Rℋ|≤1 (recall Example 2.26). To conclude, it remains to prove that Rℋ is surjective and that it holds that |Rℋ(w)|≥|w| for every v∈ℋ. To this aim, fix any 𝜂∈ℋ*\{0}. We know that ker(𝜂) is a V-Banach U-submodule of ℋ with ker(𝜂)≠ℋ, so that there exists ˜ w∈ker(𝜂)⟂\{0}. We can find a partition (un)n∈ℕ of 𝜒{˜ w≠0} and a sequence (an)n∈ℕ⊂U+ such that un|˜ w|,un⟨𝜂,˜ w⟩∈U and unan|˜ w|=un for every n∈ℕ. Then we define wn:= (un⟨𝜂,˜ w⟩a2 n)⋅˜ w∈ℋ for every n∈ℕ. Notice that |wn|=unan|⟨𝜂,˜ w⟩| and Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 AN AXIOMATIC THEORY OF NORMED MODULES VIA RIESZ SPACES • 39 ⟨𝜂,wn⟩=|wn|2. In particular, |wn|≤un|𝜂| for all n∈ℕ, and thus it makes sense to define w:= ∑n∈ℕun⋅wn∈ℋ. It holds that |w|≤|𝜂| and ⟨𝜂,w⟩=|w|2. Pick a partition (˜ uk)k∈ℕ of 𝜒{w≠0} and (bk)k∈ℕ⊂U+ such that ˜ uk|w|2∈U and ˜ ukbk|w|2=˜ uk for every k∈ℕ. Given any element v∈ℋ, we have that for every k∈ℕ it holds that ˜ uk⋅(v⋅w) = (˜ uk⋅v− (˜ uk⟨𝜂,v⟩bk)⋅w)⋅w+˜ uk⟨𝜂,v⟩bk|w|2. (49) Observe that ˜ uk⋅v− (˜ uk⟨𝜂,v⟩bk)⋅w∈ker(𝜂), as a consequence of the following computation: ⟨𝜂,˜ uk⋅v− (˜ uk⟨𝜂,v⟩bk)⋅w⟩=˜ uk⟨𝜂,v⟩−˜ uk⟨𝜂,v⟩bk⟨𝜂,w⟩=˜ uk⟨𝜂,v⟩−˜ uk⟨𝜂,v⟩bk|w|2= 0. It holds w∈ker(𝜂)⟂, whence it follows that (˜ uk⋅v− (˜ uk⟨𝜂,v⟩bk)⋅w)⋅w= 0, and thus (49) yields ˜ uk⋅(v⋅w) = ˜ uk⟨𝜂,v⟩bk|w|2=˜ uk⟨𝜂,v⟩. Thanks to the arbitrariness of k∈ℕ, we finally conclude that ⟨𝜂,v⟩=v⋅w for every v∈ℋ, which means that 𝜂= Rℋ(w) and |Rℋ(w)|=|𝜂|≥|w|. This completes the proof. Proposition 3.39 Let (𝒰,U,V,V,Z) be a CSP complete dual system of metric f-structures. Let ℋ be a V-Hilbert U-module satisfying (43). Then it holds that ℋ is reflexive. Proof. Given any v∈ℋ and 𝜔∈ℋ*, we have that ⟨Rℋ*(Rℋ(v)),𝜔⟩=𝜔⋅Rℋ(v) = Rℋ(R−1 ℋ(𝜔))⋅Rℋ(v)(48) = R−1 ℋ(𝜔)⋅v =⟨Rℋ(R−1 ℋ(𝜔)),v⟩=⟨𝜔,v⟩=⟨Jℋ(v),𝜔⟩. This shows that Jℋ= Rℋ*∘Rℋ. Since both Rℋ and Rℋ* are surjective by Theorem 3.38, we conclude that Jℋ is surjective, and thus ℋ is reflexive, yielding the sought conclusion. Proposition Let (𝒰1,U1,V1,V1,Z1), (𝒰2,U2,V2,V2,Z2) be CSP complete dual systems of metric f-structures. Let 𝜙:𝒰1→𝒰2 be a homomorphism of dual systems. Let ℋ be a V1-Hilbert U1-module satisfying (43). Then I𝜙:𝜙*ℋ*→(𝜙*ℋ)* is an isomorphism of V2-Banach U2-modules. Proof. By Theorem 3.27, it suffices to check that I𝜙:𝜙*ℋ*→(𝜙*ℋ)* is invertible. Recall from Remark 3.26 that 𝜙*ℋ is a V2-Hilbert U2-module and 𝜙*v⋅𝜙*w=𝜙(v⋅w) for all v,w∈ℋ. Then ⟨(I𝜙∘𝜙*∘Rℋ)(v),𝜙*w⟩=𝜙(⟨Rℋ(v),w⟩) = 𝜙(v⋅w) = 𝜙*v⋅𝜙*w=⟨(R𝜙*ℋ∘𝜙*)(v),𝜙*w⟩ for every v,w∈ℋ. Moreover, 𝜙*∘Rℋ:ℋ→𝜙*ℋ* is linear and satisfies ∣𝜙*(Rℋ(v))∣=|v| for every v∈ℋ, and thus Proposition 3.24 gives an element T∈Hom(𝜙*ℋ,𝜙*ℋ*) such that is a commutative diagram. Hence, I𝜙 is invertible and I−1 𝜙=T∘R−1 𝜙*ℋ, concluding the proof. Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 40 • D. LUˇ CI´ C AND E. PASQUALETTO 3.6. Dimensional decomposition Given a commutative ring R and a non-empty subset S of an R-module M, we denote by ⟨S⟩R the R-submodule of M generated (in the algebraic sense) by S. Namely, we define ⟨S⟩R={n ∑ i=1 ri⋅vi∣n∈ℕ, (ri)n i=1 ⊂R, (vi)n i=1 ⊂S}. Definition 3.41 (Independence, generators, local basis) Let (𝒰,U,V) be a metric f-structure, and ℳ a V-Banach U-module. Let v1,…,vn∈ℳ and u∈Idem(U) be given. Then we say that (i) v1,…,vn are independent of u if for any u1,…,un∈U it holds that n ∑ i=1 (uui)⋅vi= 0 ⟺uui= 0, for every i= 1,…,n. (ii) v1,…,vn generate ℳ on u provided 𝒢(⟨u⋅S⟩U) = u⋅ℳ, where S:= {v1,…,vn}. (iii) v1,…,vn form a local basis of ℳ on u provided that they are independent of u and they generate ℳ on u. For brevity, in the case where u=1U we do not specify ‘on 1U’ in the above terminology. In order to provide a well-defined notion of local dimension, we first need to show that two local bases on the same idempotent element must have the same cardinality: Lemma 3.42 Let (𝒰,U,V) be a Dedekind complete CSP metric f-structure, and ℳ a V-Banach U-module. Let v1,…,vn∈ℳ and w1,…,wm∈ℳ be local bases of ℳ on u ∈Idem(U). Then it holds that n=m. Proof. Observe that it suffices to check that if v1,…,vn generate ℳ on u and w1,…,wm are independent of u, then n≥m. Moreover, thanks to a finite induction argument, it is enough to show that if k≤m and w1,…,wk−1,vk,…,vn generate ℳ on some u0∈Idem(U)\{0} with u0≤u, then there exists u1∈Idem(U)\{0} with u1≤u0 such that w1,…,wk,vk+1,…,vn generate ℳ on u1 (up to reordering vk,…,vn ). First of all, we can find elements ˜ u∈Idem(U)\{0} with ˜ u≤u0 and ˜ u1,…,˜ un∈U such that ˜ u⋅wk= k−1 ∑ i=1 ˜ ui⋅wi+ n ∑ i=k ˜ ui⋅vi. (50) Since w1,…,wk are independent of ˜ u, it cannot happen that ˜ uk=…=˜ un= 0. Hence, until reordering vk,…,vn, we can assume that z:= 𝜒{˜ uk≠0}˜ u≠0. Now take a partition (zj)j∈ℕ of z and elements (˜ zj)j∈ℕ⊂U such that zj(˜ uk˜ zj− 1) = 0 for every j∈ℕ. There exists j0∈ℕ such that zj0≠0, so that multiplying both sides of (50) by zj0˜ zj0 we obtain that zj0⋅vk= (zj0˜ zj0˜ uk)⋅vk= (zj0˜ zj0)⋅wk− k−1 ∑ i=1 (zj0˜ zj0˜ ui)⋅wi− n ∑ i=k+1 (zj0˜ zj0˜ ui)⋅vi. This implies that, letting u1:= zj0, it holds that w1,…,wk,vk+1,…,vn generate ℳ on u1. In view of Lemma 3.42, the following definition is thus well-posed: Definition 3.43 (Local dimension) Let (𝒰,U,V) be a Dedekind complete CSP metric f-structure, and let ℳ be a V-Banach U-module. Then we say that ℳ has local dimension n∈ℕ on u ∈Idem(U) if there exists a local basis v1,…,vn of ℳ on u. Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 AN AXIOMATIC THEORY OF NORMED MODULES VIA RIESZ SPACES • 47 4.2.5. Some applications We list some examples of known constructions that follow from Theorems 3.19, 3.16 and 3.23: •Cotangent module. Let (X,d,𝜇) be a metric measure space and p∈(1,∞). We denote by W1,p(X) the p-Sobolev space of (X,d,𝜇) and by |Df |∈Lp(𝜇) the minimal weak upper gradient of f∈W1,p(X) (for example, in the sense of [4, 10, 38]; all these approaches are equivalent by [3]). Consider the metric f-structure (L0(𝜇),L∞(𝜇),Lp(𝜇)), as well as the map 𝜓p:W1,p( X) → Lp(𝔪)+ given by 𝜓p(f) := |Df | for every f∈W1,p(X). Then the cotangent module Lp(T*X) and the differential operator d: W1,p(X) →Lp(T*X) are (Lp(T*X),d)≅(ℳ⟨𝜓p⟩,T⟨𝜓p⟩). The cotangent module (for p= 2 ) has been introduced in [20, Definition 2.2.1] and refined in [19, Theorem/Definition 2.8]. Many other generalizations appeared later: for example, one can drop the Lp-integrability assumption (see [23, Proposition 4.18]), one can construct the capacitary tangent module on an RCD(K,∞) space (see [11, Theorem 3.6]) or one can consider the cotangent modules induced by axiomatic classes of Sobolev-type spaces (see [22, Theorem 3.2]). Concerning the latter notion, we point out that—thanks to Theorem 3.19—the strong locality assumption on the D-structure in [22, Theorem 3.2] can be removed. •Pullback module. Let (X,ΣX,𝜇X), (Y,ΣY,𝜇Y) be 𝜎-finite measure spaces and 𝜑: X → Y a map of bounded compression, that is, 𝜑 is measurable and there exists a constant C> 0 such that 𝜑#𝜇X≤C𝜇Y. Notice that 𝜑 induces a homomorphism of metric f-structures 𝝋:(L0(𝜇Y),L∞(𝜇Y),Lp(𝜇Y))→(L0(𝜇X),L∞(𝜇X),Lp(𝜇X)) for every exponent p∈[1,∞) via pre-composition. Namely, given any f∈L0(𝜇Y) we define 𝝋(f) := [ˉ f∘𝜑]𝜇X, for any ˉ f∈ℒ0(ΣY)with [ˉ f]𝜇Y=f. Let ℳ be an Lp(𝜇Y)-Banach L∞(𝜇Y)-module. Then the pullback module is given by (𝜑*ℳ,𝜑*)≅(𝝋*ℳ,𝝋*). The pullback module was introduced in [20, Definition 1.6.2], [19, Theorem/Definition 2.23] and has many generalizations: for example, for L0-Banach L0-modules and under the weaker assumption 𝜑#𝜇X≪𝜇Y (see [24, Theorem/Definition 3.2]) or for L∞-Banach L∞-modules (see [21, Theorem/Definition 4.11]). •L0-completion. Let (X,Σ,𝜇) be a 𝜎-finite measure space, and let ℳ be an Lp(𝜇)-Banach L∞(𝜇)-module, for some exponent p∈[1,∞]. Then the L0-completion of ℳ (in the sense of [19, Theorem/Definition 1.7]) is given by (ˉ ℳ,𝜄)≅(ℳ⟨𝜓ℳ⟩,T⟨𝜓ℳ⟩), where we define 𝜓ℳ:ℳ→L0(𝜇)+ as 𝜓ℳ(v) := |v| for every v∈ℳ. •von Neumann lifting. Let (X,Σ,𝜇) be a complete 𝜎-finite measure space, ℓ a von Neumann lifting of 𝜇 and ℳ an L∞(𝜇)-Banach L∞(𝜇)-module. Then the von Neumann lifting of ℳ (see [13, Theorem 3.5]) is given by (ℓℳ,ℓ)≅(ℳ⟨𝜓ℓ⟩,T⟨𝜓ℓ⟩), where we define 𝜓ℓ:ℳ→ℒ∞(Σ)+ as 𝜓ℓ(v) := ℓ(|v|) for every v∈ℳ. Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 48 • D. LUˇ CI´ C AND E. PASQUALETTO We point out that, in addition to the objects we discussed above, also the associated existence results for homomorphisms (for example, the universal property of pullback modules [20, Proposition 1.6.3] or the lifting of a homomorphism [21, Proposition 4.14]) can be deduced from Proposition 3.20. 4.3. The Realization Theorem In this conclusive section of the paper, we show (see Theorem 4.7) that any localizable f-algebra can be ‘realized’ as a space of functions. A Boolean ring is a ring (R,+,⋅) such that r2=r for every r∈R. In particular, r= −r and rs =sr for every r,s∈R. A Boolean algebra is a a Boolean ring (A,+,⋅) with a multiplicative identity 1A. A ring homomorphism 𝜙:A→B between two Boolean algebras A and B is said to be a Boolean homomorphism, provided that it is also uniferent, meaning that 𝜙(1A)=1B. Given a set X ≠∅ and an algebra Σ of subsets of X, the triple (Σ,Δ,∩) is a Boolean algebra with zero ∅ and identity X. The following fundamental result—which is known as the Stone’s Representation Theorem for Boolean algebras—states that in fact any Boolean algebra can be expressed as an algebra of sets. We will employ it in the proof of Proposition 4.6. Theorem 4.5 (Stone’s Theorem) Let A be a Boolean algebra. Then there exist a set X and an algebra Σ of subsets of X such that (A,+,⋅) and (Σ,Δ,∩) are isomorphic as Boolean algebras. Let U be a given f-algebra. Then we define the operations ⊞: Idem(U)×Idem(U)→Idem(U) and ⊠: Idem(U)×Idem(U)→Idem(U) on Idem(U) as u⊞v:= u+v− 2uv,u⊠v:= uv, for every u,v∈Idem(U). Their well-posedness follows from items (i) and (ii) of Lemma 2.10. It is easy to check that the triple (Idem(U),⊞,⊠) is a Boolean algebra with zero element 0 and multiplicative identity 1. Proposition 4.6 Let U be a Dedekind 𝜎-complete f-algebra whose multiplication map is 𝜎-order-continuous on U+×U+. Then the space (Idem(U),⊞,⊠) is Boolean isomorphic to a 𝜎-algebra. Proof. Thanks to Stone’s Representation Theorem 4.5, we can find a set X ≠∅, an algebra Σ of subsets of X and a Boolean isomorphism I: (Idem(U),⊞,⊠)→(Σ,Δ,∩). We claim that I(sup n∈ℕ un)=⋃ n∈ℕ I(un), for every (un)n∈ℕ⊂Idem(U). (52) Call u:= supn∈ℕun. Recall that u∈Idem(U) by Lemma 2.17. Given any n∈ℕ, it holds that un≤u, and thus Remark 2.11 gives I(un)∩I(u) = I(unu) = I(un), which yields ⋃n∈ℕI(un)⊂I(u). Conversely, pick any set E∈Σ with I(un)⊂E for every n∈ℕ. Calling v:= I−1(E), we have that I(unv) = I(un)∩I(v) = I(un), so that unv=vn. Hence, it holds that u=sup n∈ℕ un=sup n∈ℕ unv=vsup n∈ℕ un=uv, thus I(v)∩I(u) = I(uv) = I(u). We obtain that I(u)⊂I(v) = E, whence (52) follows. We deduce that Σ is a 𝜎-algebra, so that (X,Σ) is a measurable space. This completes the proof. Theorem 4.7 (Realization Theorem) Let U be a localizable f-algebra. Then there exists a measurable space (X,Σ) such that U is isomorphic (as an f-algebra) to an f-subalgebra of ℒ0(Σ). More precisely, the measurable space (X,Σ) can be chosen so that (Σ,Δ,∩) is isomorphic (as a Boolean algebra) to (Idem(U),⊞,⊠). Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 AN AXIOMATIC THEORY OF NORMED MODULES VIA RIESZ SPACES • 49 Proof. Proposition 4.6 yields a measurable space (X,Σ) such that (Σ,Δ,∩) and (Idem(U),⊞,⊠) are isomorphic as Boolean algebras. We introduce the mapping 𝜄:𝒮+(U)→ℒ0(Σ) as follows: given any simple element u=∑k i=1 𝜆iui∈𝒮+(U), we define the function 𝜄(u): X →[0,+∞] as 𝜄(u)(x) := k ∑ i=1 𝜆i𝟙I(ui)(x), for every x∈X, where I: Idem(U)→Σ is some fixed Boolean isomorphism. Observe that 𝜄(u) belongs to Sf(Σ). In order to extend the mapping 𝜄 to U+, we first need to prove the following two auxiliary results: (a) If u∈U+ and (un)n∈ℕ⊂𝒮+(U) is a non-decreasing sequence satisfying u=supn∈ℕun, then it holds that supn∈ℕ𝜄(un)(x)<+∞ for every x∈X. (b) If u∈U+ and (un)n∈ℕ,(vn)n∈ℕ⊂𝒮+(U) are non-decreasing sequences with u=supn∈ℕun and u=supn∈ℕvn, then it holds that supn∈ℕ𝜄(un)(x) = supn∈ℕ𝜄(vn)(x) for every x∈X. To prove (a), we argue by contradiction: suppose that supn∈ℕ𝜄(un)(x0)=+∞ for some x0∈X. For any n∈ℕ, we can find 𝜆n∈[0,+∞) and ˜ un∈Idem(U) with 𝜆n˜ un=˜ unun and x0∈I(˜ un). One has 𝜆n=𝜄(un)(x)→+∞ as n→∞. Define E:= ⋂n∈ℕI(˜ un)∈Σ and w:= I−1(E)∈Idem(U). Notice that x0∈E and w≤˜ un for every n∈ℕ. Given k∈ℕ, there is nk∈ℕ with 𝜆nk≥k, and thus kw ≤𝜆nk˜ unk=˜ unkunk≤u. Since U is Archimedean by Proposition 2.2, we deduce that w= 0 and thus E=∅. This leads to a contradiction with the fact that x0∈E so that (a) is proved. We pass to the verification of (b). Fix any point x∈X. For any n∈ℕ, we can pick 𝜆n,𝜇n∈[0,+∞) and ˜ un,˜ vn∈Idem(U) such that 𝜆n˜ un=˜ unun, 𝜇n˜ vn=˜ vnvn and x∈I(˜ un)∩I(˜ vn). Setting E:= ⋂n∈ℕI(˜ un)∩I(˜ vn)∈Σ, we have x∈E, and thus w:= I−1(E)≠0. By the 𝜎-order continuity of the multiplication, we obtain (sup n∈ℕ𝜄(un)(x))w=(sup n∈ℕ𝜆n)w=sup n∈ℕ𝜆nw=sup n∈ℕ𝜆n˜ unw=sup n∈ℕ un˜ unw=(sup n∈ℕ un)w=uw =(sup n∈ℕ vn)w=sup n∈ℕ𝜇n˜ vnw=(sup n∈ℕ𝜇n)w=(sup n∈ℕ𝜄(vn)(x))w, where we used that ˜ unw=˜ vnw=w. This yields supn∈ℕ𝜄(un)(x) = supn∈ℕ𝜄(vn)(x), proving (b). We now define the function 𝜄(u): X →[0,+∞) for any u∈U+ in the following way: given any non-decreasing sequence (un)n∈ℕ⊂𝒮+(U) such that u=supn∈ℕun—whose existence is guaranteed by the assumption that the f-algebra U is localizable—we define 𝜄(u)(x) := sup n∈ℕ𝜄(un)(x), for every x∈X. The properties (a) and (b) ensure that 𝜄(u) is well-posed. Notice that 𝜄(u)∈ℒ0(Σ), as a countable supremum of elements of ℒ0(Σ). The 𝜎-order continuity of the sum and multiplication maps gives 𝜄(u) + 𝜄(v) = 𝜄(u+v), 𝜄(uv) = 𝜄(u)𝜄(v), for every u,v∈U+. (53) Finally, we extend 𝜄 to a mapping 𝜄:U→ℒ0(Σ) by setting 𝜄(u) := 𝜄(u+) − 𝜄(u−), for every u∈U. Downloaded from https://academic.oup.com/qjmath/advance-article/doi/10.1093/qmath/haae053/7904734 by Jyvaskyla University user on 20 November 2024 50 • D. LUˇ CI´ C AND E. PASQUALETTO Using (53), one can easily show that 𝜄 is a homomorphism of f-algebras. In order to conclude, it only remains to check that 𝜄 is injective. To this aim, fix any u∈U such that 𝜄(u) = 0. We want to show that u= 0. Since u+u−= 0 by (5a), we deduce that 𝜄(u+)𝜄(u−) = 𝜄(u+u−) = 0, which yields 𝜄(u+) = 𝜄(u−) = 0. Hence, it suffices to prove the implication 𝜄(u)=0⟹u= 0 in the case where u∈U+. Choose a non-decreasing sequence (un)n∈ℕ⊂𝒮+(U) such that u=supn∈ℕun. We have that supn∈ℕ𝜄(un) = 0, whence it follows that un= 0 for every n∈ℕ and thus u= 0. Remark 4.8 Several variants of ‘realization theorems’—regarding even more general structures, such as Banach/Orlicz lattices—can be found in the literature (see [31]). An instance of such a result is [31, Theorem 3.7], which provides the following characterization: a Dedekind complete vector lattice E is isomorphic to an order-dense order-ideal in some space L0(𝜇) (see the definition of Köthe function space in [31, p. 17]) if and only if the Boolean algebra 𝔅(E) consisting of the band projections on E is a measure algebra; we refer to [31] for the precise definitions of the involved concepts. It would be interesting—but currently unclear and outside the scope of this paper—to understand whether Theorem 4.7 can be actually deduced from [31, Theorem 3.7]. ACKNOWLEDGEMENT The authors thank Nicola Gigli and Simone Di Marino for having suggested Corollary 3.22 and Theorem 3.30, respectively. 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