TOPOLOGICAL ALGEBRAS, THEIR APPLICATIONS, AND RELATED TOPICS BANACH CENTER PUBLICATIONS, VOLUME 67 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2005 THE CENTER OF TOPOLOGICALLY PRIMITIVE GALBED ALGEBRAS MART ABEL Institute of Pure Mathematics, University of Tartu 2 Liivi St., Room 615, 50409 Tartu, Estonia E-mail: mabel@ut.ee Abstract. It is shown that every unital σ-complete topologically primitive strongly galbed Hausdorff algebra in which all elements are bounded is central. 1. Introduction 1.1. Let C be the field of complex numbers, N= {0, 1, 2,...} the set of natural numbers, Z + = {1, 2,...} the set of positive integers and l 0 the set of all C-valued sequences (α n ) where α m = 0 for only a finite number of elements α m . For every k> 0 let l k be the set of all C-valued sequences (α n ) for which the series ∞  v=0 |α v | k converges, l = l 1 \ l 0 , and l (0,1] =  k∈(0,1] l k . Let A be an associative topological algebra over C with separately continuous multipli- cation (for short, a topological algebra). Definition 1. We will say that a topological algebra A is a galbed algebra if there exists a sequence (α n ) ∈ l such that for each neighbourhood O of zero in A there is another neighbourhood U of zero in A such that  n  k=0 α k a k : a 0 ,...,a n ∈ U  ⊂ O for each n ∈ N. 2000 Mathematics Subject Classification : Primary 46H05; Secondary 46H20. Key words and phrases : galbed algebras, center of an algebra, primitive algebras, topologi- cally primitive algebras, exponentially galbed algebras. Research is partly supported by the Estonian Science Foundation grant 4514. The paper is in final form and no version of it will be published elsewhere. [45]