Let A be a complex, commutative Banach algebra and let be the structure space of A. Assume that there exists a continuous homomorphism h:L¹(G) → A with dense range, where L¹(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 is scattered (i.e., has no nonempty perfect subset). (b) Weakly almost periodic functionals on A with compact scattered spectra are almost periodic. (c) If is scattered, then the algebra A is Arens regular if and only if .
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba52-4-5,
author = {H. S. Mustafayev},
title = {Functionals on Banach Algebras with Scattered Spectra},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {52},
year = {2004},
pages = {395-403},
zbl = {1099.43004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba52-4-5}
}
H. S. Mustafayev. Functionals on Banach Algebras with Scattered Spectra. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 52 (2004) pp. 395-403. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba52-4-5/