Functionals on Banach Algebras with Scattered Spectra

Tom 52 / 2004

H. S. Mustafayev Bulletin Polish Acad. Sci. Math. 52 (2004), 395-403 MSC: 43A20, 43A60, 46J99. DOI: 10.4064/ba52-4-5


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 $.


  • H. S. MustafayevDepartment of Mathematics
    Faculty of Arts and Sciences
    Yuzuncu Yil University
    65080 Van, Turkey
    Institute of Mathematics and Mechanics
    National Academy of Sciences of Azerbaijan
    F. Agaev St. 9
    Baku, Azerbaijan

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