Spectral synthesis for Banach algebras, II

Abstract

This paper continues the study of spectral synthesis and the topologies τ ∞ and τ r on the ideal space of a Banach algebra, concentrating particularly on the class of Haagerup tensor products of C * -algebras. For this class, it is shown that spectral synthesis is equivalent to the Hausdorffness of τ ∞ . Under a weak extra condition, spectral synthesis is shown to be equivalent to the Hausdorffness of τ r .

Nontrivially Noetherian $C^*$-algebras

“…An important result, which we use frequently in Section 4, is that Prim(A) is sober when A is separable [3, 3.9.1(b)], a property that is equivalent to every prime ideal being primitive. 6 Note that Prim(A) is not sober in general, as shown in both [17] and [12, Proposition 31], as each author exhibits a (nonseparable) prime C * -algebra that is not primitive. Proof.…”

“…By assumption, this family is nonempty, and since X is Noetherian it has a minimal element, call it Y . So Y is not irreducible, and hence can be written as 6 We say that A is primitive if the zero ideal is so. See, e.g., [8, Corollary 1.5 & Appx.…”

Spectral Synthesis and Topologies on Ideal Spaces for Banach*-Algebras

“…Let P rime(A) be the space of proper closed prime ideals of A, and let P rim s (A) be the space of semisimple prime ideals of A (such ideals are automatically closed), both spaces also being equipped with the hull-kernel topology. The notation, and the importance of P rim s (A), is explained in [13]. The paper [33] contained a definition of 'spectral synthesis', but unfortunately that definition was slightly too restrictive, and was replaced in [13] by the following definition.…”

“…The notation, and the importance of P rim s (A), is explained in [13]. The paper [33] contained a definition of 'spectral synthesis', but unfortunately that definition was slightly too restrictive, and was replaced in [13] by the following definition.…”

“…Definition of spectral synthesis A Banach algebra A has spectral synthesis if it has the following properties: Remarks (a) It was noted in [13], see [33; 1.1], that if A has spectral synthesis then τ w has the normality property on the whole of Id(A).…”