Abstract
We study extensions of Wermer’s maximality theorem to several complex variables. We exhibit various smoothly embedded manifolds in complex Euclidean space whose hulls are non-trivial but contain no analytic disks. We answer a question posed by Lee Stout concerning the existence of analytic structure for a uniform algebra whose maximal ideal space is a manifold.
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Notes
Throughout the paper, by an analytic set we mean a subset of \(\mathbb {C}^n\) that is locally the common zero set of finitely many holomorphic functions. Such sets are often referred to as analytic varieties or holomorphic varieties.
When every point of \(b\Omega \) is known to be a peak point for \([z_1\ldots z_n]_{{\overline{\Omega }}}\) [e.g., for strictly pseudoconvex domains] one can invoke a weaker result of Anderson, Izzo, and Wermer [4] in place of Stout’s theorem.
Although this approximation theorem often is formulated in the \(\mathcal {C}^2\)-category, it holds also in \(\mathcal {C}^1\).
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The first author was partially supported by NFR Grant 209751/F20. The second author was partially supported by the Swedish Research Council. The third author was supported by NFR Grant 209751/F20.
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Izzo, A.J., Samuelsson Kalm, H. & Wold, E.F. Presence or absence of analytic structure in maximal ideal spaces. Math. Ann. 366, 459–478 (2016). https://doi.org/10.1007/s00208-015-1330-9
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DOI: https://doi.org/10.1007/s00208-015-1330-9