Spectral Synthesis in the Multiplier Algebra of a C0(x)-Algebra

Abstract

Abstract. Let A be a C 0 (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(βX)-algebra with continuous map φ :Thus H x is strictly closed if and only if H x =J x , and the 'spectral synthesis' question asks when this happens. In this paper it is shown that, for σ-unital A, H x is strictly closed for all x ∈ Im(φ) if and only if J x is locally modular for all x ∈ Im(φ) and φ is a closed map relative to its image. Various related results are obtained.

Separation properties in the primitive ideal space of a multiplier algebra

“…Combining this with the characterization of spectral synthesis in [12,Corollary 3.10], it follows that Orc(M (A)) = 1 if and only if Orc(A) = 1 and the Glimm ideals of A are locally modular (Corollary 4.7).…”

“…Then J x ⊆ H x ⊆J x , so H x is strictly closed if and only if H x =J x . When H x =J x we say that spectral synthesis holds at x [12]. For further details, see [10] and [12].…”

“…Thus we see that if A is separable then necessary conditions for Orc(M (A)) = 1 are that Orc(A) = 1 and that every Glimm ideal of M (A) which does not contain A should be strictly closed. This latter property has been called 'global spectral synthesis' in [12].…”

“…Since Prim(A) is canonically homeomorphic to an open subset of Prim(M (A)) [37, 4.1.10], it is easily seen that such algebras necessarily have Orc(A) = 1. Regarding A as a C 0 (X)-algebra in a natural way, we show in Theorem 4.2 that a necessary condition for Orc(M (A)) = 1 is that M (A) should have spectral synthesis as a C(βX)-algebra in the sense of [12] (where βX is the Stone-Čech compactification of X).…”

Norms of inner derivations for multiplier algebras of C⁎-algebras and group C⁎-algebras, II

“…[14, Lemma 4.1] Let X be a locally compact space and let Y and Z be subsets of X which are Lindelöf in the relative topology. Then the following are equivalent:…”