Invariant algebras on compact groups

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.
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