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Fibrewise compactifications and generalised limits in commutative and noncommutative topology

Alexander Mundey

TL;DR

This work develops a fibrewise compactification framework to address non-closure of inverse limits in locally compact spaces and in the $C^*$-algebraic setting. By extending maps to proper morphisms and employing Whyburn's unified space and its sub-/quotient-perfections, the authors define regulated limits that preserve local compactness and yield new, robust limit constructions. In the topological realm, regulated limits generalize inverse limits and give concrete realizations for spaces such as directed-graph path spaces; in the noncommutative realm, they correspond to universal constructions like the unified algebra and quotient perfections, with cores of relative Cuntz–Pimsner algebras realized as regulated limits. The framework provides a coherent bridge between classical topological limits and noncommutative $C^*$-algebraic limits, and clarifies how Pimsner and Katsura ideals govern the noncommutative boundary data. Overall, regulated limits offer a systematic, locally compact-compatible approach to limit procedures across both commutative and noncommutative topology, with implications for graph algebras and Cuntz–Pimsner constructions.

Abstract

We introduce fibrewise compactifications in both the setting of locally compact Hausdorff spaces and continuous maps, and the parallel setting of $C^*$-algebras and nondegenerate multiplier-valued $*$-homomorphisms. In both situations, we use fibrewise compactifications to define regulated limits. In the topological setting, regulated limits extend classical inverse limits so that the resulting limit space remains locally compact; examples include the path spaces of directed graphs. In the operator-algebraic setting, regulated limits realise a direct-limit construction for multiplier-valued $*$-homomorphisms; examples include the cores of relative Cuntz-Pimsner algebras.

Fibrewise compactifications and generalised limits in commutative and noncommutative topology

TL;DR

This work develops a fibrewise compactification framework to address non-closure of inverse limits in locally compact spaces and in the -algebraic setting. By extending maps to proper morphisms and employing Whyburn's unified space and its sub-/quotient-perfections, the authors define regulated limits that preserve local compactness and yield new, robust limit constructions. In the topological realm, regulated limits generalize inverse limits and give concrete realizations for spaces such as directed-graph path spaces; in the noncommutative realm, they correspond to universal constructions like the unified algebra and quotient perfections, with cores of relative Cuntz–Pimsner algebras realized as regulated limits. The framework provides a coherent bridge between classical topological limits and noncommutative -algebraic limits, and clarifies how Pimsner and Katsura ideals govern the noncommutative boundary data. Overall, regulated limits offer a systematic, locally compact-compatible approach to limit procedures across both commutative and noncommutative topology, with implications for graph algebras and Cuntz–Pimsner constructions.

Abstract

We introduce fibrewise compactifications in both the setting of locally compact Hausdorff spaces and continuous maps, and the parallel setting of -algebras and nondegenerate multiplier-valued -homomorphisms. In both situations, we use fibrewise compactifications to define regulated limits. In the topological setting, regulated limits extend classical inverse limits so that the resulting limit space remains locally compact; examples include the path spaces of directed graphs. In the operator-algebraic setting, regulated limits realise a direct-limit construction for multiplier-valued -homomorphisms; examples include the cores of relative Cuntz-Pimsner algebras.
Paper Structure (14 sections, 39 theorems, 107 equations)

This paper contains 14 sections, 39 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. The topological setting
  3. Fibrewise compactifications and perfections
  4. The unified space
  5. Sub-fibrewise compactifications of the unified space
  6. Composing sub-fibrewise compactifications
  7. Locally compactifying inverse limits
  8. The C*-algebraic setting
  9. Noncommutative fibrewise compactifications and perfections
  10. The unified algebra
  11. Quotient fibrewise compactifications of the unified algebra
  12. Composing quotient perfections of the unified algebra
  13. Generalised limits of C*-algebras
  14. Cores of Cuntz--Pimsner algebras as regulated limits

Key Result

Lemma 2.2

Suppose that $f \colon X \to Y$ is a continuous map between locally compact Hausdorff spaces. If $y \in \overline{f(X)} \setminus f(X)$, then for any precompact open neighbourhood $W \subseteq Y$ of $y$, the preimage $f^{-1}(\overline{W})$ is not compact in $X$. In particular, if $f$ is proper, then

Theorems & Definitions (136)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Example 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Definition 2.7
  • Remark 2.8
  • ...and 126 more