Introduction

Let $X$ be a locally compact and Hausdorff space, and consider $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. The initial motivation for this work is the question of whether we can recover $X$ (up to homeomorphism) from $C_c(X,\mathbb{K})$, the set of continuous, compactly supported $\mathbb{K}$-valued functions on $X$. This is a problem initiated mostly after Stone's seminal work [MR1501865] on the representations of Boolean algebras, and has proven to be a rich area of study with several important applications in Logic, Functional Analysis, and Operator Algebras.

By Milutin's Theorem ([MR0206695], or [MR1999613, Chapter 36, Theorem 2.1]), just the topological vector space structure of $C_c(X,\mathbb{K})$, when endowed with the supremum norm, is not enough to recover $X$. On the other hand, throughout the last century several authors have proved that by considering additional algebraic structures on $C_c(X,\mathbb{K})$ ‒ such as that of a ring, a C*-algebra if $\mathbb{K}=\mathbb{C}$, a lattice if $\mathbb{K}=\mathbb{R}$, etc. ‒ we can in fact recover $X$. See Banach and Stone [MR1357166], MR1501905], Gelfand and Kolmogorov [gelfandkolmogorov1939], Milgram [MR0029476], Gelfand and Naimark [MR0009426], Kaplansky [MR0020715], Jarosz [MR1060366], Li and Wong [MR3162258], Hernández and Ródenas [MR2324919], Kania and Rmoutil [MR3813611].

In fact, the results of [MR0020715], MR2324919] also hold for certain spaces of non-$\mathbb{R}$ or $\mathbb{C}$-valued functions. In a similar manner, Stone's duality for Boolean algebras ([MR1501865]) can also be seen as a result on spaces of functions: The Boolean algebra of clopen sets of a topological space $X$ is order-isomorphic to the lattice of functions $C(X,\left\{0,1\right\})$, and if $X$ is a Stone (zero-dimensional, compact Hausdorff) space, then it completely determines $X$.

Our goal in this paper is to provide a unified and elementary approach to all these results, under hypotheses that can be easily verified in different settings. For this, we use a stronger version of the “disjointness” relation for (supports of) functions as considered by Jarosz in [MR1060366]. As we will see in Section 3, all of the results above immediately fall in this more general setting.

Let us describe the main idea in the case real-valued functions: Two functions $f,g\in C_c(X,\mathbb{R})$ are strongly disjoint if $\supp(f)\cap\supp(g)=\varnothing$, in which case we write $f\perpp g$. Then it is possible to describe, purely in terms of the relation $\perpp$, the subsets of $C_c(X,\mathbb{R})$ of the form $\mathbf{I}(U)=\left\{f:\supp(f)\subseteq U\right\}$, where $U\subseteq X$ is open. These sets are called $\perpp$-ideals, and we have a bijection $x\mapsto \mathbf{I}(X\setminus\left\{x\right\})$ between $X$ and the set $\widehat{C_c(X,\mathbb{R})}$ of maximal $\perpp$-ideals. This bijection can be made into a homeomorphism, by endowing $\widehat{C_c(X,\mathbb{R})}$ with a topology, described again only in terms of $\perpp$. Therefore, any $\perpp$-isomorphism $T\colon C_c(X,\mathbb{R})\to C_c(Y,\mathbb{R})$ will induce a homeomorphism $Y\cong\widehat{C_c(Y,\mathbb{R})}\cong \widehat{C_c(X,\mathbb{R})}\cong X$. In all of the previously-proven theorems listed above, the algebraic isomorphisms under considerations happen to be also $\perpp$-isomorphisms, and thus those results follow from this.

As supports of functions are central to the result above, and we wish also to look at a theory involving non-scalar maps, we will need to extend the notion of support, which is the first problem tackled in Section 1.

This article is organized as follows: In the first section we introduce all necessary terminology and prove our main recovery theorem (Theorem 1.17). In Section 2, we study an important class of maps, called “basic”, between spaces of functions, and which will appear in most applications. These are the maps which are “classifiable” in a certain manner.

Due to the level of generality we seek, the first two sections are rather abstract, so the reader is invited to read Definition 1.1 in order to get familiarized with the notation, read the main Theorem 1.17, and proceed directly to the applications in Section 3, referring back to previous parts of this article as necessary (or desired).

In Section 3 we obtain classifications of isomorphisms for different algebraic structures on spaces of continuous functions, including the ones mentioned at the beginning of this introduction. The new applications consist of a classification of linear isomorphisms which are isometric with respect to $L^1$-norms (Theorem 3.20), classifications of classes of isomorphisms of algebras associated to groupoids (Theorems 3.25, 3.27 and 3.41), and a classification of (uniform-metric) isometric isomorphisms between groups of circle-valued functions.