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A spectral synthesis property for \(C_b(X,\beta)\). (English) Zbl 1182.46042

Let \(X\) be a nonempty completely regular Hausdorff space, and let \(\mathbb F\) be either \(\mathbb R\) or \(\mathbb C\). Denote by \((C_b(X),\beta)\) the algebra over \(\mathbb F\) of all bounded continuous \(\mathbb F\)-valued functions on \(X\), equipped with the strict topology \(\beta\), determined by the seminorms \(\|f\|_\varphi=\sup_{x\in X}|f(x)||\varphi(x)|\), where \(\varphi\) ranges over all bounded \(\mathbb F\)-valued functions vanishing at infinity. For an \(f\in(C_b(X),\beta)\), denote by \(Z(f)\) the zero set of \(f\). Let \(A\) be a commutative topological algebra with unit and let \(\mathcal M(A)\) be the space of nontrivial continuous linear multiplicative functionals of \(A\) under the weak\(^*\)-topology. For an ideal \(I\subset A\), the set \(h(I)=\{\phi\in\mathcal M(A): I\subset Z(\phi)\}\) is the hull of \(I\), and for an \(E\subset\mathcal M(A)\), the closed ideal \(k(E)=\bigcap_{\phi\in E}Z(\phi)\) (\(k(\emptyset)=A\)) is the kernel of \(E\).
In this paper, it is shown that a proper ideal \(I\) of \((C_b(X),\beta)\) is closed if and only if \(I=k(h(I))\). Therefore, \((C_b(X),\beta)\) has the spectral synthesis property, i.e., each of its closed ideals is an intersection of closed maximal ideals of codimension 1, and \(cl(I)=k(h(I))\) for any ideal \(I\), namely, \(g\in cl(I)\) if and only if \(h(I)\subset Z(g)\). It is also shown that, if \((C_b(X),\beta)\) has no proper nonzero closed principal ideals, then \(X\) is connected. Conversely, if \(X\) is connected and either is locally connected, or, is a Fréchet-Urysohn space (i.e., if \(S\subset X\) then \(x\in cl(S)\) if and only if there is a sequence in \(S\) that converges to \(x\)), then \((C_b(X),\beta)\) has no proper nonzero closed principal ideals.

MSC:

46J20 Ideals, maximal ideals, boundaries
46J05 General theory of commutative topological algebras
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