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Homogeneous spaces of compact connected Lie groups which admit nontrivial invariant algebras. (English) Zbl 1023.22013
Let \(M=G/H\) be a homogeneous space of a compact Lie group \(G\), \(C(M)\) be the Banach algebra of all complex valued continuous functions on \(M\) endowed with the \(\sup\)-norm. An invariant algebra \(A\) on \(M\) is a closed \(G\)-invariant subalgebra of \(C(M)\). If \(A\) is self-adjoint with respect to complex conjugation then, according to the Stone-Weierstrass theorem, there exists a closed subgroup \(H'\supseteq H\) such that \(A\cong C(M')\), where \(M'=G/H'\). Let us consider \(G\) as the homogeneous space of the group \(G\times G\), with \(G\) acting on itself by left and right translations. In this setting, J. A. Wolf [Pac. J. Math. 15, 1093-1099 (1965; Zbl 0141.31802)] characterized compact groups \(G\) which admit only self-adjoint invariant algebras. Among the connected Lie groups, this property distinguishes semisimple groups (this was independently proved by R. Gangolli [Bull. Am. Math. Soc. 71, 634-637 (1965; Zbl 0137.31602)]). In this note, under the assumption that \(G\) and \(H\) are connected, it is proved that each invariant algebra on \(M\) is self-adjoint if and only if the decomposition of the isotropy representation of \(H\) does not have a trivial component. Another equivalent condition is the following: the group \(N/H\), where \(N\) is the normalizer of \(H\), is finite. This generalizes the result of Wolf and Gangolli. Reviewer’s remark. It was noted by M. Raïs that the result remains true if \(H\) and \(G\) are not assumed to be connected. He proved this using his earlier results [C. R. Acad. Sci., Paris, Sér. I 305, 713-716 (1987; Zbl 0696.22008)]. Another proof is contained in the joint paper of the author and the reviewer [Transformation Groups 6, 321-331 (2001; Zbl 0999.22020)].
Reviewer: V.Gichev (Omsk)

MSC:
22F30 Homogeneous spaces
43A85 Harmonic analysis on homogeneous spaces
22E46 Semisimple Lie groups and their representations
46J10 Banach algebras of continuous functions, function algebras
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