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Functionals on Banach algebras with scattered spectra

Mustafayev H. S.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, 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 : L^1(G) --> 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* = span M[A].