# Wydawnictwa / Czasopisma IMPAN / Bulletin Polish Acad. Sci. Math. / Wszystkie zeszyty

## Functionals on Banach Algebras with Scattered Spectra

### Tom 52 / 2004

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

#### Streszczenie

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 $A^* = \mathop{\overline{\rm span}} M_A$.

#### Autorzy

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

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