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Locally compact unimodular groups with atomic duals

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Un gruppo localmente compatto commutativo è compatto se e solo se il suo gruppo duale è discreto, in altri termini se la sua rappresentazione regolare è atomica. Esistono però gruppi unimodulari noncommutativi che non sono compatti, ma la cui rappresentazione regolare è atomica. Il presente articolo è una rassegna di esempi critici di tali gruppi, e delle loro principali proprietà di struttura. Si discutono infine alcuni esempi e risultati nuovi sull'esistenza di rappresentazioni fedeli di gruppi unimodulari con duale atomico.

Abstract

A locally compact commutative group is compact if and only if its dual group is discrete, in other words if its regular representation is atomic. Nevertheless, there exist noncommutative unimodular groups with atomic regular representation which are not compact. The present paper is a survey of significant examples of such groups and of their main structure properties. Moreover, new examples and results are presented concerning the existence of faithful representations of unimodular groups with atomic duals.

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Bibliography

  1. Anh N. H.,Classification des groupes de Lie connexes unimodulaires possédant une série discrète. C. R. Acad. Sc. Paris 287 (1978), 847–849.

    MATH  Google Scholar 

  2. Auslander L., Brezin J.,Almost-algebraic Lie algebras. J. of Algebra 8 (1968), 295–313.

    Article  MATH  MathSciNet  Google Scholar 

  3. Auslander L., Moore C. C.,Unitary representations of solvable Lie groups. Mem. Amer. Math. Soc. 62 (1966).

  4. Baggett L.,A separable group having a discrete dual space is compact. J. Functional Anal. 10 (1972), 131–148.

    Article  MATH  MathSciNet  Google Scholar 

  5. Baggett L., Taylor K.,Groups with completely reducible regular representations. Proc. Am. Math. Soc. 72 (1978), 593–600.

    Article  MATH  MathSciNet  Google Scholar 

  6. Baggett L., Taylor K.,A sufficient condition for the complete reducibility of the regular representation. To appear.

  7. Blackadar B.,The regular representation of restricted direct product groups. J. Functional Anal., 25 (1977), 267–274.

    Article  MATH  MathSciNet  Google Scholar 

  8. Cecchini C., Figà-Talamanca A.,Projections of uniqueness for L'(G). Pacific J. Math. 51 (1974), 37–47.

    MATH  MathSciNet  Google Scholar 

  9. Dixmier J.,Les C *-algèbres et leurs représentations. Gauthier-Villars, Paris, 1969.

    Google Scholar 

  10. Fell J. M. G.,The dual spaces of C *-algebras. Trans. Amer. Math. Soc. 94 (1960), 365–403.

    Article  MATH  MathSciNet  Google Scholar 

  11. Figà-Talamanca A.,Positive definite functions vanishing at infinity. Pacific. J. Math. 69 (1977), 355–363.

    MATH  MathSciNet  Google Scholar 

  12. Figà-Talamanca A., Gaudry G. I.,Multipliers and sets of uniqueness of L p. Michigan Math. J. 17 (1970), 179–191.

    Article  MATH  MathSciNet  Google Scholar 

  13. Gelfand I. M., Graev M. I., Piatesky-Shapiro I. I.,Representation theory and automorphic functions. Saunders, Philadelphia, 1969.

    Google Scholar 

  14. Katznelson Y.,Sets of uniqueness for some classes of trigonometric series. Bull. Amer. Math. Soc. 70 (1964), 722–723.

    Article  MATH  MathSciNet  Google Scholar 

  15. Kirillov A. A.,Unitary representations of nilpotent Lie groups. Russ. Math. Surveys 17 (1962), 53–103.

    Article  MATH  MathSciNet  Google Scholar 

  16. Kleppner A., Lipsman R. L.,The Plancherel formula for group extensions, I, Ann. Sci. Ec. Norm. Sup. 5 (1972), 459–516, and II, Ann. Sci. Ec. Norm. Sup. 6 (1973), 103–132.

    MATH  MathSciNet  Google Scholar 

  17. Mackey G.,Borel structure in groups and their duals. Trans. Amer. Math. Soc. 85 (1957), 134–165.

    Article  MATH  MathSciNet  Google Scholar 

  18. Mauceri G.,Square integrable representations and the Fourier algebra of a unimodular group. Pacific J. Math. 73 (1977), 143–154.

    MATH  MathSciNet  Google Scholar 

  19. Mauceri G., Picardello M. A.,Noncompact unimodular groups with purely atomic Plancherel measures. To appear in Proc. Amer. Math. Soc.

  20. Menchoff M. D.,Sur l'unicité du dévelopement trigonométrique. C. R. Acad. Sc. Paris 163 (1916), 443–446.

    Google Scholar 

  21. Moore C. C.,Decomposition of unitary representations defined by discrete subgroups of nilpotent groups. Ann. of Math. 82 (1965), 146–182.

    Article  MathSciNet  Google Scholar 

  22. Moore C. C.,Representations of solvable and nilpotent groups and harmonic analysis on nil and solvmanifolds. Proc. Symp. Pure Math. 26 (1974), 1–44.

    Google Scholar 

  23. Mostow G. D.,Fully reducible subgroups of algebraic groups. Am. J. Math. 68 (1956), 200–221.

    Article  MathSciNet  Google Scholar 

  24. Picardello M. A.,A non type I group with purely atomic regular representation. Boll. U.M.I. 16-A (1979), 331–334.

    MathSciNet  Google Scholar 

  25. Picardello M. A.,Unimodular Lie groups without discrete series. To appear in Boll. U.M.I.

  26. Rickert N. W.,Amenable groups and groups with the fixed point property. Trans. Amer. Math. Soc. 127 (1967), 221–232.

    Article  MATH  MathSciNet  Google Scholar 

  27. Robert A.,Examples de groupes de Fell. C. R. Acad. Sc. Paris 287 (1978), 603–606.

    MATH  Google Scholar 

  28. Stern A. I.,Separable locally compact groups with discrete support for the regular representation. Soviet Math. Dokl. 12 (1971), 994–998.

    MathSciNet  Google Scholar 

  29. Takesaki M., Tatsuuma N.,Duality and subgroups. Ann. of Math. 93 (1971), 344–364.

    Article  MathSciNet  Google Scholar 

  30. Varadarajan V. S.,Lie groups, Lie algebras and their representations. Prentice-Hall, Englewood Cliffs, N.J., 1974.

    MATH  Google Scholar 

  31. Wang P. S.,On isolated points in the dual spaces of locally compact groups. Math. Ann. 218 (1975), 19–34.

    Article  MATH  MathSciNet  Google Scholar 

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  1. dell' Università di Roma, Roma, Italia

    Massimo A. Picardello

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  1. Massimo A. Picardello
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(Conferenza tenuta il 15 dicembre 1978)

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Picardello, M.A. Locally compact unimodular groups with atomic duals. Seminario Mat. e. Fis. di Milano 48, 197–216 (1978). https://doi.org/10.1007/BF02925571

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