The Serre–Swan theorem for normed modules

Abstract

The aim of this note is to analyse the structure of the $L^0$-normed $L^0$-modules over a metric measure space. These are a tool that has been introduced by Gigli to develop a differential calculus on spaces verifying the Riemannian Curvature Dimension condition. More precisely, we discuss under which conditions an $L^0$-normed $L^0$-module can be viewed as the space of sections of a suitable measurable Banach bundle and in which sense such correspondence can be actually made into an equivalence of categories.

Appendices

Appendix A: Comparison with the Serre–Swan theorem for smooth manifolds

We point out the main analogies and differences between our work and the Serre–Swan theorem for smooth manifolds, for whose presentation we refer to [20, Chapter 11].

The result in the smooth case can be informally stated as follows: the category of smooth vector bundles over a connected manifoldMis equivalent to the category of finitely-generated projective$C^\infty (M)$-modules.

In our non-smooth setting we had to replace ‘smooth’ with ‘measurable’, in a sense, and this led to these discrepancies with the case of manifolds:

• i) The fibers of a measurable Banach bundle need not have the same dimension (still, they are finite dimensional), while on a connected manifold any smooth vector bundle must have constant dimension by topological reasons.

• ii) In the definition of measurable Banach bundle we do not speak about the analogous of the ‘trivialising diffeomorphisms’, the reason being that one can always patch together countably many measurable maps still obtaining a measurable map. Hence there is no loss of generality in requiring the total space of the bundle to be of the form $\bigsqcup _{n\in \mathbb {N}}E_n\times \mathbb {R}^n$ and its measurable subsets to be those sets whose intersection with each $E_n\times \mathbb {R}^n$ is a Borel set.

• iii) Given that we want to correlate the measurable Banach bundles with the $L^0(\mathfrak m)$-normed $L^0(\mathfrak m)$-modules, which are naturally equipped with a pointwise norm $|\cdot |$, we also require the existence of a function ${{\mathbf {\mathsf{{n}}}}}$ that assigns a norm to (almost) every fiber of our bundle. A similar structure is not treated in the smooth case.

• iv) The Serre–Swan theorem for smooth manifolds deals with modules that are finitely-generated and projective. In our context, any finitely-generated module is automatically projective, as seen in Proposition 1. Moreover, the flexibility of $L^0(\mathfrak m)$ actually allowed us to extend the result to all proper modules, that are not necessarily ‘globally’ finitely-generated but only ‘locally’ finitely-generated, in a sense.

Appendix B: A variant for $L^p$-normed $L^\infty$-modules

The original presentation of the concept of $L^0$-normed $L^0$-module, which has been proposed in [6], follows a different line of thought with respect to the one presented here. In [6] it is first given the notion of $L^p$-normed$L^\infty$-module, then by suitably completing such objects one obtains the class of $L^0$-normed $L^0$-modules. The role of this completion is to ‘remove any integrability requirement’. On the other hand, the axiomatisation of $L^0$-normed $L^0$-modules that we presented in Subsection 1.2 is taken from [8].

Our choice of using the language of $L^0$-normed $L^0$-modules, instead of $L^p$-normed $L^\infty$-modules, is only a matter of practicity and is not due to any theoretical reason. Indeed, in this appendix we show that all the results we obtained so far can be suitably reformulated for $L^p$-normed $L^\infty$-modules.

Let ${{\mathbb {X}}}=(\mathrm{X},\textsf {d},\mathfrak m)$ be a given metric measure space. Fix an exponent $p\in [1,\infty ]$. In order to keep a distinguished notation, we shall indicate by ${\mathscr {M}}^p$ the $L^p(\mathfrak m)$-normed $L^\infty (\mathfrak m)$-modules, for whose definition and properties we refer to [6] or [8], while the $L^0(\mathfrak m)$-normed $L^0(\mathfrak m)$-modules will be denoted by ${\mathscr {M}}^0$. The category of $L^p(\mathfrak m)$-normed $L^\infty (\mathfrak m)$-modules is denoted by $\mathbf {NMod}^p({{\mathbb {X}}})$ and that of $L^0(\mathfrak m)$-normed $L^0(\mathfrak m)$-modules by $\mathbf {NMod}^0({{\mathbb {X}}})$. Moreover, the subcategories of $\mathbf {NMod}^p({{\mathbb {X}}})$ and $\mathbf {NMod}^0({{\mathbb {X}}})$ that consist of all proper modules will be called $\mathbf {NMod}^p_\mathrm{pr}({{\mathbb {X}}})$ and $\mathbf {NMod}^0_\mathrm{pr}({{\mathbb {X}}})$, respectively. Observe that we added the exponent 0 to the notation of Definition 6. Similarly, we shall denote by $\varGamma _0$ the section functor $\varGamma$ that has been introduced in Definition 11.

The following results look upon the relation that subsists between the class of $L^p(\mathfrak m)$-normed $L^\infty (\mathfrak m)$-modules and that of $L^0(\mathfrak m)$-normed $L^0(\mathfrak m)$-modules. First of all, it has been proved in [8, Theorem/Definition 1.7] that

Theorem 5

($L^0$-completion) Let ${\mathscr {M}}^p$ be an $L^p(\mathfrak m)$-normed $L^\infty (\mathfrak m)$-module. Then there exists a unique couple $({\mathscr {M}}^0,\iota )$, called $L^0$-completion of ${\mathscr {M}}^p$, where ${\mathscr {M}}^0$ is an $L^0(\mathfrak m)$-normed $L^0(\mathfrak m)$-module and $\iota :\,{\mathscr {M}}^p\rightarrow {\mathscr {M}}^0$ is a linear map with dense image that preserves the pointwise norm. Uniqueness has to be intended up to unique isomorphism.

It can be easily seen that the local dimension of a module is invariant under taking the $L^0$-completion, namely for any Borel set $E\subseteq \mathrm{X}$ with $\mathfrak m(E)>0$ and for any $n\in \mathbb {N}$ it holds

$$\begin{aligned} {\mathscr {M}}^p\text { has dimension }n\text { on }E \quad \Longleftrightarrow \quad {\mathscr {M}}^0\text { has dimension }n\text { on }E. \end{aligned}$$

(B.1)

Given two $L^p(\mathfrak m)$-normed $L^\infty (\mathfrak m)$-modules ${\mathscr {M}}^p$, $\mathscr {N}^p$ and a module morphism $\varPhi :\,{\mathscr {M}}^p\rightarrow \mathscr {N}^p$, there exists a unique module morphism $\widetilde{\varPhi }:\,{\mathscr {M}}^0\rightarrow \mathscr {N}^0$ extending $\varPhi$, where ${\mathscr {M}}^0$ and $\mathscr {N}^0$ denote the $L^0$-completions of ${\mathscr {M}}^p$ and $\mathscr {N}^p$, respectively.

Definition 14

($L^0$-completion) The $L^0$-completion functor is the functor $\textsf {C}^p:\,\mathbf {NMod}^p({{\mathbb {X}}})\rightarrow \mathbf {NMod}^0({{\mathbb {X}}})$ that assigns to any ${\mathscr {M}}^p$ its $L^0$-completion ${\mathscr {M}}^0$ and to any module morphism $\varPhi :\,{\mathscr {M}}^p\rightarrow \mathscr {N}^p$ its unique extension $\widetilde{\varPhi }:\,{\mathscr {M}}^0\rightarrow \mathscr {N}^0$.

Conversely, given any $L^0(\mathfrak m)$-normed $L^0(\mathfrak m)$-module ${\mathscr {M}}^0$, one has that

$$\begin{aligned} {\mathscr {M}}^p:=\big \{v\in {\mathscr {M}}^0\;\big |\;|v|\in L^p(\mathfrak m)\big \} \quad \text { has a structure of }L^p(\mathfrak m)\text {-normed }L^\infty (\mathfrak m)\text {-module.} \end{aligned}$$

(B.2)

Moreover, it holds that the $L^0$-completion of ${\mathscr {M}}^p$ is the original module ${\mathscr {M}}^0$.

Definition 15

($L^p$-restriction) The $L^p$-restriction functor is the functor $\textsf {R}^p:\,\mathbf {NMod}^0({{\mathbb {X}}})\rightarrow \mathbf {NMod}^p({{\mathbb {X}}})$ that assigns to any ${\mathscr {M}}^0$ its ‘restriction’ ${\mathscr {M}}^p$, as in (B.2), and to any module morphism $\widetilde{\varPhi }:\,{\mathscr {M}}^0\rightarrow \mathscr {N}^0$ its restriction , which turns out to be a morphism of $L^p(\mathfrak m)$-normed $L^\infty (\mathfrak m)$-modules.

We can finally collect all of the properties described so far in the following statement:

Theorem 6

($\mathbf {NMod}^p({{\mathbb {X}}})$ is equivalent to $\mathbf {NMod}^0({{\mathbb {X}}})$) Both the functors $\textsf {C}^p$ and $\textsf {R}^p$ are equivalence of categories, one the inverse of the other.

Property (B.1) ensures that ${\mathscr {M}}^p$ and $\textsf {C}^p({\mathscr {M}}^p)$ have the same dimensional decomposition, thus in particular the above functors naturally restrict to $\textsf {C}^p_\mathrm{pr}:\,\mathbf {NMod}^p_\mathrm{pr}({{\mathbb {X}}})\rightarrow \mathbf {NMod}^0_\mathrm{pr}({{\mathbb {X}}})$ and $\textsf {R}^p_\mathrm{pr}:\,\mathbf {NMod}^0_\mathrm{pr}({{\mathbb {X}}})\rightarrow \mathbf {NMod}^p_\mathrm{pr}({{\mathbb {X}}})$. Therefore:

Corollary 1

($\mathbf {NMod}^p_\mathrm{pr}({{\mathbb {X}}})$ is equivalent to $\mathbf {NMod}^0_\mathrm{pr}({{\mathbb {X}}})$) The functors $\textsf {C}^p_\mathrm{pr}$ and $\textsf {R}^p_\mathrm{pr}$ are equivalence of categories, one the inverse of the other.

Now fix a measurable Banach bundle ${{\mathbb {T}}}$ over ${{\mathbb {X}}}$. Then let us define

$$\begin{aligned} \varGamma _p({{\mathbb {T}}}):=\big \{s\in \varGamma _0({{\mathbb {T}}})\;\big |\;|s|\in L^p(\mathfrak m)\big \}. \end{aligned}$$

(B.3)

The space $\varGamma _p({{\mathbb {T}}})$ can be viewed as an $L^p(\mathfrak m)$-normed $L^\infty (\mathfrak m)$-module. Moreover, given any two measurable Banach bundles ${{\mathbb {T}}}_1$, ${{\mathbb {T}}}_2$ over ${{\mathbb {X}}}$ and a bundle morphism $\varphi \in \mathrm{Mor}({{\mathbb {T}}}_1,{{\mathbb {T}}}_2)$, let us define $\varGamma _p(\varphi )\in \mathrm{Mor}\big (\varGamma _p({{\mathbb {T}}}_1),\varGamma _p({{\mathbb {T}}}_2)\big )$ as

$$\begin{aligned} \varGamma _p(\varphi ):={}^\varGamma _0(\varphi )|_{\varGamma _p({{\mathbb {T}}}_1)}:\,\varGamma _p({{\mathbb {T}}}_1)\rightarrow \varGamma _p({{\mathbb {T}}}_2). \end{aligned}$$

(B.4)

Hence such construction induces an $L^p$-section functor $\varGamma _p:\,\mathbf {MBB}({{\mathbb {X}}})\rightarrow \mathbf {NMod}^p_\mathrm{pr}({{\mathbb {X}}})$. Then

is a commutative diagram. We can finally conclude that

Theorem 7

(Serre–Swan for $L^p$-normed $L^\infty$-modules) The section functor $\varGamma _p:\,\mathbf {MBB}({{\mathbb {X}}})\rightarrow \mathbf {NMod}^p_\mathrm{pr}({{\mathbb {X}}})$ on ${{\mathbb {X}}}$ is an equivalence of categories.

Proof

It follows from Theorem 3, from Corollary 1 and from the fact that the diagram in (B.5) commutes. $\square$