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On the Banach-Stone theorem for algebras of holomorphic germs. (English) Zbl 1373.46037

Let \(K\) and \(L\) be balanced compact subsets of Banach spaces \(X\) and \(Y\), respectively. Suppose that for every \(m \in \mathbb N\), every \(m\)-homogeneous polynomial on \(X^{\ast\ast}\) is approximable, and similarly for \(Y^{\ast\ast}\). Under these conditions, the authors prove a Banach-Stone type of result, characterizing when the algebras of holomorphic germs \(\mathcal H(K)\) and \(\mathcal H(L)\) are algebra-isomorphic in terms of properties of \(K\) and \(L.\) (Here, by \(P\) being an \(m\)-homogeneous approximable polynomial on a Banach space \(Z\) we mean that \(P\) is in the norm closure of the finite type polynomials on \(Z.\)) The result is given in terms of compact subsets of the bidual, \(K^{\prime\prime}\) and \(L^{\prime\prime}\): For a fixed bounded subset \(A \subset Z\), let \(\hat{A^{\prime\prime}}_{\mathcal P(Z)} :\equiv \{w \in Z^{\ast\ast} \;| \;|\tilde{P} (w)| \leq \sup_{a \in A} |P(a)| \) for all continuous polynomials \(P\) on \( Z\}.\) (Here, \(\tilde P\) is the canonical extension of \(P\) to the bidual \(Z^{\ast\ast}\).)
The main result is the following Theorem: For \(X, Y, K\), and \(L\) satisfying the above hypotheses, the following conditions are equivalent:
(i)
\(\mathcal H(K)\) and \(\mathcal H(L)\) are isomorphic as topological algebras,
(ii)
\(\hat{K^{\prime\prime}}_{\mathcal P(X)}\) and \(\hat{L^{\prime\prime}}_{\mathcal P(Y)}\) are biholomorphically equivalent,
(iii)
\(\mathcal H(\hat{K^{\prime\prime}}_{\mathcal P(X)})\) and \(\mathcal H(\hat{L^{\prime\prime}}_{\mathcal P(Y)})\) are isomorphic as topological algebras.
Several examples are provided to illustrate situations in which the above theorem applies.

MSC:

46G20 Infinite-dimensional holomorphy
46E25 Rings and algebras of continuous, differentiable or analytic functions
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