Abstract
It is shown under certain conditions that a uniform algebra on the unit sphere S in C 2 that is invariant under the action of the 2-torus must be C(S). Contrasting with this, an example is presented showing that the statement becomes false when 2 is replaced by n > 2. It is also shown that C(M) is the only uniform algebra on a smooth manifold M that is invariant under a transitive Lie group action on its maximal ideal space. The results presented answer a question raised by Ronald Douglas in connection with a conjecture of William Arveson.
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Izzo, A.J. Uniform algebras on the sphere invariant under group actions. Math. Ann. 344, 989–995 (2009). https://doi.org/10.1007/s00208-009-0349-1
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DOI: https://doi.org/10.1007/s00208-009-0349-1