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1 Bulletin of the Polish Academy of Sciences. Mathematics

1 2004

1 Mustafayev H.

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Functionals on Banach Algebras with Scattered Spectra

100%

Mustafayev H.

nr 4

395-403

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: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 $M_A$ 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 $A* = \overline{span} M_A$.