Constructive Peter--Weyl Theory: What is Known and What Remains Open

This survey-style note reviews constructive versions of the Peter–Weyl theorem in the Bishop–Coquand–Spitters line. Its main purpose is to clarify which parts of the classical Peter–Weyl package admit constructive reformulations, which parts survive only in weaker or reorganized form, and which questions still appear to remain open. The term ``constructive’’ is used here primarily in the Bishop-style sense, together with the related locale-theoretic and formal-topological developments that occur in the work of Coquand and Spitters. We review the constructive compact-group results of Coquand and Spitters, the later role of almost periodic functions and compact completions, and the interaction with constructive Gelfand representation and locale-theoretic compactness. The guiding theme is that the constructive theory exists, but it is often most naturally expressed not as a literal transcription of the classical theorem in terms of irreducible decompositions alone, but rather through finite-rank approximation, characters, and compactifications attached to functions or groups. For orientation and comparison, we also include an appendix giving a standard classical form of the Peter–Weyl theorem together with a pedagogical Haar-measure-based proof, followed by comments indicating where the classical argument relies on steps that are not automatically constructive. Possible later extensions to topological loops and quasigroups are included as a programmatic direction rather than as part of the currently established core.