Functionals on Banach Algebras with Scattered Spectra

H. S. Mustafayev¹

¹ Department 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

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 52 (2004) no. 4, pp. 395-403

Résumé

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 $.

Détail

DOI : 10.4064/ba52-4-5

Keywords: complex commutative banach algebra structure space assume there exists continuous homomorphism dense range where group algebra locally compact abelian group main results note summarized follows every weakly almost periodic functional compact spectra almost periodic space scattered has nonempty perfect subset weakly almost periodic functionals compact scattered spectra almost periodic scattered algebra arens regular only * mathop overline span