×

A constructive proof of the Peter-Weyl theorem. (English) Zbl 1099.03052

The authors present a new proof of the famous Peter-Weyl theorem on the representation of compact groups. This proof is based on the Gel’fand representation theorem for commutative \(C^*\)-algebras and may be seen as a direct generalization of Burnside’s algorithm to compute the characters of finite groups. Thus, unlike the original proof by F. Peter and H. Weyl [Math. Ann. 97, 737–755 (1927; JFM 53.0387.02)] or the one by I. Segal [“Algebraic integration theory”, Bull. Am. Math. Soc. 71, 419–489 (1965; Zbl 0135.17402)], the authors do not use the spectral theory of compact operators. The authors’ proof is also different from the one presented by L. H. Loomis [An introduction to abstract harmonic analysis. Toronto-New York-London: D. Van Nostrand Company (1953; Zbl 0052.11701)], which uses the representation theorem for \(H^*\)-algebras due to W. Ambrose [“Structure theorems for a special class of Banach algebras”, Trans. Am. Math. Soc. 57, 364–386 (1945; Zbl 0060.26906)].
The paper opens with an outline of the theory of the characters of finite groups. It then gives the said (non-constructive) proof of the Peter-Weyl theorem. Finally the authors show that their proof can in fact be transformed (by avoiding an application of a non-constructive variant of the least upper bound principle) into a proof of the Peter-Weyl theorem in the context of Bishop’s constructive mathematics.

MSC:

03F60 Constructive and recursive analysis
22C05 Compact groups
43A77 Harmonic analysis on general compact groups
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ambrose, Trans. Amer. Math. Soc. 57 pp 364– (1945)
[2] and Constructive Analysis (Springer-Verlag, Berlin et al. 1985).
[3] Theory of Groups of Finite Order, 2. ed. (Dover Publications Inc., New York 1955).
[4] Schur’s Lemma and Irreducibility from a Constructive Point of View. Manuscript June 2000. Available under: http://zaphod.uchicago.edu/mfrank.
[5] An Introduction to Abstract Harmonic Analysis (van Nostrand, New York 1953).
[6] and Theory of Group Representations (Springer-Verlag, Berlin et al. 1982).
[7] Peter, Math. Annalen 97 pp 737– (1927)
[8] Segal, Bull. Amer. Math. Society 71 pp 419– (1965)
[9] Constructive and intuitionistic integration theory and functional analysis. PhD thesis, University of Nijmegen 2002.
[10] Zum Haarschen Mass in topologischen Gruppen. In: Collected Works, vol. II: Operators, ergodic theory and almost periodic functions in a group (A. H. Taub, ed.), pp. 445-453 (Pergamon Press, New York 1961).
[11] L’intègration dans les groupes topologiques et ces applications. Actual. Sci. Ind., no. 869 (Hermann et Cie., Paris 1940). [This book has been republished by the author at Princeton, N. J., 1941.].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.