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~$\C^n$ 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.

Keywords:

holomorphic functional calculus, commutative continuous inverse algebra, holomorphic convexity, Stein compacta, meromorphic convexity, holomorphic approximation holomorphic functional calculus, commutative continuous inverse algebra, holomorphic convexity, Stein compacta, meromorphic convexity, holomorphic approximation

MSC Classifications:

46H30, 32A38, 32E30, 41A20, 46J15 show english descriptions Functional calculus in topological algebras [See also 47A60]
Algebras of holomorphic functions [See also 30H05, 46J10, 46J15]
Holomorphic and polynomial approximation, Runge pairs, interpolation
Approximation by rational functions
Banach algebras of differentiable or analytic functions, $H^p$-spaces [See also 30H10, 32A35, 32A37, 32A38, 42B30] 46H30 - Functional calculus in topological algebras [See also 47A60]
32A38 - Algebras of holomorphic functions [See also 30H05, 46J10, 46J15]
32E30 - Holomorphic and polynomial approximation, Runge pairs, interpolation
41A20 - Approximation by rational functions
46J15 - Banach algebras of differentiable or analytic functions, $H^p$-spaces [See also 30H10, 32A35, 32A37, 32A38, 42B30]

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