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Extremely strong boundary points and real-linear isometries. (English) Zbl 1352.46009

Summary: Let \(X,Y\) be locally compact Hausdorff spaces, \(A\) be a complex subspace of \(C_0(X)\) and \(T: A \to C_0(Y)\) be a real-linear isometry, whose range is not assumed to be a complex subspace of \(C_0(Y)\). In this paper, using the set \(\Theta(A)\) and \(\tau(A)\) consisting of all extremely strong boundary points and strong boundary points of \(A\), respectively, we introduce appropriate subsets \(Y_0\) and \(Y_1\) of \(Y\) and give a description of \(T\) on these sets. More precisely, we show that there exist continuous functions \(\Phi:Y_0\to \Theta(A)\), \(\alpha:Y_0\to [-1,1]\) and \(w:Y_0\to \mathbb {T}\), where \(\mathbb {T}\) is the unit circle, such that \[ Tf(y)=w(y) \cdot (\operatorname{Re}(f({\Phi}(y)))+{\alpha}(y) i \operatorname{Im}(f({\Phi}(y))) \] for all \(f\in A\) and \(y\in Y_0\). The result is improved in the case where either = 6mm
(i)
\(T(A)\) is a complex subspace of \(C_0(Y)\) and \(\Theta(A)= \mathrm{ch}(A)\), where \(\mathrm{ch}(A)\) is the Choquet boundary of \(A\), or
(ii)
\(T(A)\) satisfies a certain separating property.
In the first case, we show that there exists a clopen subset \(K\) of \(Y_0\) such that \[ (Tf)(y)= w(y)\left\{ \begin{aligned} &f(\Phi(y)) \quad y \in K,\\ &\overline{f(\Phi(y))} \quad y \notin K,\end{aligned} \right. \] for each \(f\in A\) and \(y\in Y_0\). In the second case, we obtain similar results for \(\tau(A)\cap \mathrm{ch}(A)\) and \(Y_1\) instead of \(\Theta(A)\) and \(Y_0\).

MSC:

46B04 Isometric theory of Banach spaces
47B48 Linear operators on Banach algebras
46J05 General theory of commutative topological algebras
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References:

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