Continuous measures on compact Lie groups
Annales de l'Institut Fourier, Tome 50 (2000) no. 4, pp. 1277-1296.

On étudie des mesures continues sur un groupe de Lie compact semi-simple G en utilisant la théorie des représentations. À la section 2 nous donnons une caractérisation des mesures continues sur G analogue à celle de Wiener pour le groupe du cercle. Puis on construit des mesures sur G qui sont liées aux produits de Riesz sur les groupes abéliens localement compacts. En utilisant cette mesure on montre à la section 3 que si C est une partie compacte des mesures continues sur G, il existe une mesure singulière ν telle que la mesure ν*μ soit absolument continue par rapport à la mesure de Haar sur G pour toute mesure μ dans C. À la section 4 on montre que si f est une combinaison linéaire finie de caractères, il existe deux mesures singulières μ et ν sur G telles que f=μ*ν. À la section 5 on donne une caractérisation des mesures continues sur un espace symétrique compact G/K.

We study continuous measures on a compact semisimple Lie group G using representation theory. In Section 2 we prove a Wiener type characterization of a continuous measure. Next we construct central measures on G which are related to the well known Riesz products on locally compact abelian groups. Using these measures we show in Section 3 that if C is a compact set of continuous measures on G there exists a singular measure ν such that ν*μ is absolutely continuous with respect to the Haar measure on G for every μ in C. In Section 4 we show that if f is a finite linear combination of characters then there exist two singular measures μ and ν on G such that f=μ*ν. In the final section we obtain a Wiener-type characterization of a continuous measure on a symmetric space of compact type G/K.

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     title = {Continuous measures on compact {Lie} groups},
     journal = {Annales de l'Institut Fourier},
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     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {50},
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Anoussis, M.; Bisbas, A. Continuous measures on compact Lie groups. Annales de l'Institut Fourier, Tome 50 (2000) no. 4, pp. 1277-1296. doi : 10.5802/aif.1793. http://archive.numdam.org/articles/10.5802/aif.1793/

[1] A. Bisbas and C. Karanikas, On the continuity of measures, Applicable Analysis, 48 (1993), 23-35. | MR | Zbl

[2] M. Blümlinger, Rajchman measures on compact groups, Math. Ann., 284 (1989), 55-62. | MR | Zbl

[3] N. Bourbaki, Intégration, Hermann, Paris, 1963.

[4] T. Bröcker and T. Tom Dieck, Representations of compact Lie groups, Springer-Verlag, New York, 1985. | Zbl

[5] A. H. Dooley and S. K. Gupta, Continuous singular measures with absolutely continuous convolution squares, Proc. Amer. Math. Soc., 124 (1996), 3115-3122. | MR | Zbl

[6] C. F. Dunkl and D. E. Ramirez, Helson sets in compact and locally compact groups, Mich. Math. J., 19 (1971), 65-69. | MR | Zbl

[7] J. M. G. Fell and R. S. Doran, Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles: Volume 2, Banach *-Algebraic Bundles, Induced Representations, and the Generalized Mackey Analysis, Academic Press, London, 1988. | Zbl

[8] C. C. Graham and A. Maclean, A multiplier theorem for continuous measures, Studia Math., LXVII (1980), 213-225. | MR | Zbl

[9] C. C. Graham and O. C. Mcgehee, Essays in Commutative Harmonic Analysis, Springer-Verlag, New York, 1979. | MR | Zbl

[10] K. E. Hare, The size of characters of compact Lie groups, Studia Math., 129 (1998), 1-18. | MR | Zbl

[11] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978. | MR | Zbl

[12] S. Helgason, Groups and Geometric Analysis, Academic Press, London, 1984. | Zbl

[13] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis II, Springer-Verlag, Berlin, 1970. | Zbl

[14] E. Hewitt and K. Stromberg, A remark on Fourier-Stieltjes transforms, An. da Acad. Brasileira de Ciencias, 34 (1962), 175-180. | MR | Zbl

[15] V. Hösel and R. Lasser, A Wiener theorem for orthogonal polynomials, J. Funct. Anal., 133 (1995), 395-401. | MR | Zbl

[16] J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York, 1972. | MR | Zbl

[17] D. L. Ragozin, Central measures on compact simple Lie groups, J. Funct. Anal., 10 (1972), 212-229. | MR | Zbl

[18] D. L. Ragozin, Zonal measure algebras on isotropy irreducible homogeneous spaces, J. Funct. Anal., 17 (1974), 355-376. | MR | Zbl

[19] D. Rider, Central lacunary sets, Monatsh. Math., 76, (1972), 328-338. | MR | Zbl

[20] R. S. Strichartz, Wavelet Expansions of Fractal Measures, The Journal of Geometric Analysis, 1 (1991), 269-289. | MR | Zbl

[21] V. S. Varadarajan, Lie Groups, Lie Algebras and their Representations, Springer-Verlag, New-York, 1984. | MR | Zbl

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