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# Holomorphic Generation of Continuous Inverse Algebras

Published:2007-02-01
Printed: Feb 2007
• Harald Biller
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## Abstract

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
 MSC Classifications: 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]