Fibrewise compactifications and generalised limits in commutative and noncommutative topology

2025/04/28 by Mundey, Alexander
#46L85 (Primary) 55R70 #46M40 (Secondary) #FOS: Mathematics #General Topology (math.GN) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2504.19425

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.

Citations

• Recovering the boundary path space of a topological graph using\n pointless topology
• The Path Space of a Directed Graph
• Functoriality of Rieffel's Generalised Fixed-Point Algebras for Proper Actions
• Groupoid models for the C*-algebras of topological higher-rank graphs
• A new look at the Crossed-Product of a C*-Algebra by a Semigroup of Endomorphisms
• Topological higher-rank graphs and the C^*-algebras of topological 1-graphs
• Tychonoff's theorem for locally compact space and an elementary approach to the topology of path spaces
• Ideal structure of C^*-algebras associated with C^*-correspondences
• On C^*-algebras associated with C^*-correspondences
• Ideal structure and simplicity of the C*-algebras generated by Hilbert bimodules

Related

• A Brief Survey on Fibrewise General Topology
• Limit sets as examples in noncommutative geometry
• Hausdorff compactifications in ZF
• A Stone-Čech Theorem for C₀(X)-algebras
• Fibrewise nullification and the cube theorem