Spectral synthesis in the multiplier algebra of a \(C_0(X)\)-algebra (Q2922406)

Let \(A \) be a \(C^* \)-algebra, let \(\varphi \) be a continuous mapping from the primitive ideal space \(\mathrm{Prim}(A) \) with the hull-kernel topology of \(A \) into a locally compact topological Hausdorff space \(X \). We say that \(A \) is a \(C_0 \)-algebra with base map \(\varphi \). Then there exists a continuous extension \(\tilde \varphi \) from the primitive ideal space \(\mathrm{Prim}(M(A)) \) of the multiplier algebra \(M(A) \) of \(A \) into the Stone-Ĉech compactification \(\beta X \) of \(X \) extending \(\varphi \). For \(x \) in the image \(X_\varphi \) of \(\varphi \), let \(J_x \) be the closed ideal of \(A \) defined by \(J_x = \{P \in \mathrm{Prim}(A) : \varphi(P ) = x\} \) and let \(H_x \) be the closed ideal of \(M(A) \) defined by \( H_x = \{Q \in \mathrm{Prim}(M(A)) : \tilde\varphi(Q) =x\} \). Let \(\tilde J_x \) be the strict closure of \(J_x \) in \(M(A) \). Then \(J_x \subset H_x \subset \tilde J_x \) and hence \(H_x \) is strictly closed if and only if \(H_x = \tilde J_x \).NEWLINENEWLINEThe ``spectral synthesis'' question asks for conditions on \(A \) and \(X \) characterizing when \(H_x \) is strictly closed. It was shown in [\textit{R. J. Archbold} and \textit{D. W. B. Somerset}, J. Lond. Math. Soc., II. Ser. 85, No. 2, 365--381 (2012; Zbl 1245.46043)] that if \(A \) is stable and \(\sigma \)-unital, then for \(x \) in \(X_\varphi \), \(H_x \) is strictly closed if and only if \(x \) is a \(P \)-point in the image of \(\varphi \). The main result of the paper is that if \( A \) is a \(\sigma \)-unital \(C_0 (X) \)-algebra with base map \(\varphi \), then \(H_x \) is strictly closed for all \(x \in X_\varphi \) if and only if \(J_x \) is locally modular for all \(x \in X_\varphi \) and \(\varphi \) is a closed map relative to its image. Here \(J_x \) is said to be locally modular if, whenever \(Q \) lies in the boundary in \(\mathrm{Prim}(A) \) of \(H (x) = \{P \in \mathrm{Prim}(A) : P \supset J_x \} \), then there exists a neighbourhood \(V \) of \(Q \) in \(\mathrm{Prim}(A) \setminus U (x) \) (where \(U (x) \) is the interior of \(H (x) \)) such that \(A/ \text{ker} (V) \) is a unital \( C^* \)-algebra.NEWLINENEWLINEIf \(A \) is separable, then the same characterization is valid for spectral synthesis at a point \(x \in X_\varphi \), namely, \(H_x \) is strictly closed if and only if \(J_x \) is locally modular and \(\varphi \) is a locally closed map at \(x \). For general \(\sigma \)-unital \(C_0 (X) \)-algebras, this condition is close to characterizing spectral synthesis at a point, but not completely, due to an example in the paper. The authors also study the important special case when \(\varphi \) is the complete regularization map for \(\mathrm{Prim}(A) \) and the connecting order \(\mathrm{Orc}(A) \) is finite and they prove a result in the case \(J_x \) is locally modular. (Review based on the authors' introduction.)