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Central Idempotent Measures on Unitary Groups

  • Daniel Rider (a1)

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Let G be a locally compact group and M(G) the space of finite regular Borel measures on G. If μ and v are in M(G), their convolution is defined by

Thus, if f is a continuous bounded function on G,

μ is central if μ(Ex) = μ(xE) for all xG and all measurable sets E. μ is idempotent if μ * μ = μ.

The idempotent measures for abelian groups have been classified by Cohen [1]. In this paper we will show that for a certain class of compact groups, containing the unitary groups, the central idempotents can be characterized. The method consists of showing that, in these cases, the central idempotents arise from idempotents on abelian groups and applying Cohen's result.

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References

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1. Cohen, P. J., On a conjecture of Littlewood and idempotent measures, Amer. J. Math. 82 (1960), 191212.
2. Helgason, S., Lacunary Fourier series on noncommutative groups, Proc. Amer. Math. Soc. 9 (1958), 782790.
3. Ito, T. and Amemiya, I., A simple proof of the theorem of P. J. Cohen, Bull. Amer. Math. Soc. 70 (1964), 774776.
4. Kelley, J. L., Averaging operators on C°°(Z), Illinois J. Math. 2 (1958), 214223.
5. Parthasarathy, K. R., A note on idempotent measures in topological groups, J. London Math. Soc. 42 (1967), 534536.
6. Rudin, W., Fourier analysis on groups (Interscience, New York, 1962).
7. Weyl, H., The classical groups (Princeton Univ. Press, Princeton, N.J., 1946).
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Central Idempotent Measures on Unitary Groups

  • Daniel Rider (a1)

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