Anoussis, M.; Bisbas, A. Continuous measures on compact Lie groups. (English) Zbl 0969.43001 Ann. Inst. Fourier 50, No. 4, 1277-1296 (2000). An analog of Wiener’s classical characterization of continuous measures on the circle group is obtained for compact semisimple Lie groups (and symmetric spaces of compact type): \[ \lim_{n\to\infty}|A_n|^{-1}\sum_{\lambda\in A_n}d_\lambda^{-1}\|\widehat\mu(\pi_\lambda)\|^2_2= \sum_{g\in G}|\mu(\{g\})|^2. \] Here \(A_n\) is an “admissible” sequence of subsets of the set of dominant weights parametrizing the irreducible unitary representations of \(G\) and \(\mu\) is a measure on \(G\). For a compact set \(C\) of continuous measures on \(G\) the existence of a central singular measure \(\nu\) is shown, such that \(\mu*\nu\) is absolutely continuous for all \(\mu\in C\). This multiplier theorem is obtained by first constructing a singular measure using Riesz products on the maximal torus of \(G\). Furthermore it is proved that any finite linear combination of characters on \(G\) \(f\) can be represented as \(f=\mu*\nu\), where \(\mu\) and \(\nu\) are singular central measures. Reviewer: M.Blümlinger (Wien) Cited in 2 Documents MSC: 43A05 Measures on groups and semigroups, etc. 43A80 Analysis on other specific Lie groups 22E30 Analysis on real and complex Lie groups 43A85 Harmonic analysis on homogeneous spaces Keywords:compact semisimple Lie group; continuous measures; multipliers × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] [1] and , On the continuity of measures, Applicable Analysis, 48 (1993), 23-35. · Zbl 0792.42014 [2] [2] , Rajchman measures on compact groups, Math. Ann., 284 (1989), 55-62. · Zbl 0673.43005 [3] [3] , Intégration, Hermann, Paris, 1963. [4] [4] and , Representations of compact Lie groups, Springer-Verlag, New York, 1985. · Zbl 0581.22009 [5] [5] and , Continuous singular measures with absolutely continuous convolution squares, Proc. Amer. Math. Soc., 124 (1996), 3115-3122. · Zbl 0861.43006 [6] [6] and , Helson sets in compact and locally compact groups, Mich. Math. J., 19 (1971), 65-69. [7] [7] and , 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 0652.46051 [8] [8] and , A multiplier theorem for continuous measures, Studia Math., LXVII (1980), 213-225. [9] [9] and , Essays in Commutative Harmonic Analysis, Springer-Verlag, New York, 1979. · Zbl 0439.43001 [10] [10] , The size of characters of compact Lie groups, Studia Math., 129 (1998), 1-18. · Zbl 0946.43006 [11] [11] , Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978. · Zbl 0451.53038 [12] [12] , Groups and Geometric Analysis, Academic Press, London, 1984. · Zbl 0543.58001 [13] [13] and , Abstract Harmonic Analysis II, Springer-Verlag, Berlin, 1970. · Zbl 0213.40103 [14] [14] and , A remark on Fourier-Stieltjes transforms, An. da Acad. Brasileira de Ciencias, 34 (1962), 175-180. · Zbl 0117.09601 [15] [15] and , A Wiener theorem for orthogonal polynomials, J. Funct. Anal., 133 (1995), 395-401. · Zbl 0843.42011 [16] [16] , Introduction to Lie algebras and representation theory, Springer-Verlag, New York, 1972. · Zbl 0254.17004 [17] D. L. RAGOZIN, Central measures on compact simple Lie groups, J. Funct. Anal., 10 (1972), 212-229.0286.4300249 #5715 · Zbl 0286.43002 [18] D. L. RAGOZIN, Zonal measure algebras on isotropy irreducible homogeneous spaces, J. Funct. Anal., 17 (1974), 355-376.0297.4300251 #1297 · Zbl 0297.43002 [19] D. RIDER, Central lacunary sets, Monatsh. Math., 76, (1972), 328-338.0258.4300851 #3801 · Zbl 0258.43008 [20] R. S. STRICHARTZ, Wavelet Expansions of Fractal Measures, The Journal of Geometric Analysis, 1 (1991), 269-289.0734.2801193c:42036 · Zbl 0734.28011 [21] V. S. VARADARAJAN, Lie Groups, Lie Algebras and their Representations, Springer-Verlag, New-York, 1984.0955.2250085e:22001 · Zbl 0955.22500 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.