Invariant algebras on compact groups
Sbornik. Mathematics, Tome 10 (1970) no. 2, pp. 165-172
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The main result is a description of homogeneous (i.e. invariant relative to left and right shifts) algebras with uniform convergence on a compact group. As a corollary we obtain a generalization of a theorem of Rider: let the real annihilater $A^\perp$ of a homogeneous antisymmetric algebra $A$ be separable in the topology of the definite norm in the conjugate space. Then the connected component of the identity $G_0$ of the group $G$ is commutative and $\dim A^\perp\geq{\operatorname{card}}(G/G_0)$. Rider proved that if $A^\perp=\{0\}$, then $G$ is commutative and connected. Bibliography: 3 titles.
@article{SM_1970_10_2_a1,
author = {A. L. Rozenberg},
title = {Invariant algebras on compact groups},
journal = {Sbornik. Mathematics},
pages = {165--172},
year = {1970},
volume = {10},
number = {2},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1970_10_2_a1/}
}
A. L. Rozenberg. Invariant algebras on compact groups. Sbornik. Mathematics, Tome 10 (1970) no. 2, pp. 165-172. http://geodesic.mathdoc.fr/item/SM_1970_10_2_a1/
[1] D. Rider, “Translation invariant Dirichlet algebras on compact groups”, Proc. Amer. Math. Soc., 17:5 (1966), 977–983 | DOI | MR | Zbl
[2] A. Veil, Integrirovanie v topologicheskikh gruppakh i ego primeneniya, IL, Moskva, 1950
[3] E. Bishop, “A generalization of the Stone–Weierstrass theorem”, Pacific. J. Math., 10:3 (1961), 777–783 | MR