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

Let A be a C0(X)-algebra with continuous map φ from Prim(A), the primitive ideal space of A, to a locally compact Hausdorff space X. Then the multiplier algebra M(A) is a C(β X)-algebra with continuous map Graphic: Prim(M(A)) → β X extending φ. For x ∈ Im(φ), let Jx = ⋂{P ∈ Prim(A): φ(P) = x} and Graphic. Then Graphic, the strict closure of Jx in M(A). Thus, Hx is strictly closed if and only if Graphic, and the ‘spectral synthesis’ question asks when this happens. In this paper, it is shown that, for σ-unital A, Hx is strictly closed for all x ∈ Im(φ) if and only if Jx is locally modular for all x ∈ Im(φ) and φ is a closed map relative to its image. Various related results are obtained.