Abstract
We investigate uniform algebras of bounded analytic functions on the unit ball of a complex Banach space. We prove several cluster value theorems, relating cluster sets of a function to its range on the fibers of the spectrum of the algebra. These lead to weak versions of the corona theorem for ℓ 2 and for c 0. In the case of the open unit ball of c 0, we solve the corona problem whenever all but one of the functions comprising the corona data are uniformly approximable by polynomials in functions in \({c_0^*}\).
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Aron, R.M., Carando, D., Gamelin, T.W. et al. Cluster values of analytic functions on a Banach space. Math. Ann. 353, 293–303 (2012). https://doi.org/10.1007/s00208-011-0681-0
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DOI: https://doi.org/10.1007/s00208-011-0681-0