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We study complex commutative Banach algebras (and, more generally, continuous inverse algebras) in which the holomorphic functions of a fixed 
$n$-tuple of elements are dense. In particular, we characterize the compact subsets of 
${{\mathbb{C}}^{n}}$ which appear as joint spectra of such $n$
-tuples. The characterization is compared with several established notions of holomorphic convexity by means of approximation conditions.
