Functionals on Banach Algebras with Scattered Spectra
Volume 52 / 2004
Bulletin Polish Acad. Sci. Math. 52 (2004), 395-403
MSC: 43A20, 43A60, 46J99.
DOI: 10.4064/ba52-4-5
Abstract
Let $A$ be a complex, commutative Banach algebra and let $M_A $ be the structure space of $A$. Assume that there exists a continuous homomorphism $h:L^1(G) \to A$ with dense range, where $L^1(G)$ is a group algebra of the locally compact abelian group $G$. The main results of this note can be summarized as follows:
(a) If every weakly almost periodic functional on $A$ with compact spectra is almost periodic, then the space $M_A $ is scattered (i.e., $M_A $ has no nonempty perfect subset).
(b) Weakly almost periodic functionals on $A$ with compact scattered spectra are almost periodic.
(c) If $M_A $ is scattered, then the algebra $A$ is Arens regular if and only if $A^* = \mathop{\overline{\rm span}} M_A $.
