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

TL;DR

This paper investigates spectral synthesis in the multiplier algebra of a C_0(X)-algebra, focusing on when certain ideals coincide with their strict closures, extending understanding of the algebraic structure in this context.

Contribution

It characterizes conditions under which the ideals H_x and the strict closures of J_x are equal within the multiplier algebra of a C_0(X)-algebra.

Findings

Identifies when H_x equals the strict closure of J_x

Provides criteria for spectral synthesis in multiplier algebras

Enhances understanding of ideal structures in C_0(X)-algebras

Abstract

Let A be a C_0(X)-algebra. Then the multiplier algebra M(A) is a C(Y)-algebra in a natural way, where Y is the Stone-Cech compactification of X. Each x in X gives rise to an ideal J_x of A and an ideal H_x of M(A). The ideal J_x is contained in H_x, and H_x is contained in the strict closure of J_x in M(A). This paper studies the problem of determining when H_x is equal to the strict closure of J_x. THis can be interpreted as a problem of spectral synthesis type.