Weak Centrality of unital C(X)-algebras

Abstract

This paper establishes a fibrewise characterization of weak centrality for unital $C(X)$-algebras whose defining homomorphism maps $C(X)$ onto the center: such an algebra is weakly central if and only if each of its nonzero fibres has a unique maximal ideal. This yields a corresponding characterization for arbitrary unital $C^{\ast }$-algebras through their canonical fibres over the spectra of their centers. The resulting criterion explains Vesterstrøm's AF-algebra counterexample, whose obstruction is also interpreted through its Bratteli diagram. A parallel criterion characterizes centrality by simplicity of the fibres. Applications to full group $C^{\ast }$-algebras give an alternative proof for the discrete Heisenberg group and show that the full group $C^{\ast }$-algebra of every countable non-abelian torsion-free nilpotent group is not weakly central.

Title changed, erroneous Theorem 2.1 removed, and Bratteli diagrams analysed in greater detail