On asymmetry of topological centers of the second duals of Banach algebras

Abstract

Let A be a Banach algebra with a bounded approximate identity and let Z1(2**) and Z2(2**) be the left and right topological centers of W**. It is shown that i) W*2 = AZ* is not sufficient for ZI(f**) = Z2(f**); ii) the inclusion 2fZj(2**) C Af is not sufficient for Z2(2**)2f C Af; iii) Zl(2**) = Z2(2**) = 2f is not sufficient for A to be weakly sequentially complete. These results answer three questions of Anthony To-Ming Lau and Ali Ulger. Suppose that St is a Banach algebra and let Wf* be the dual space of QI. Then Wf* can be made into a Banach Q-bimodule as follows: for f E W,a E Q, fa and af are defined by (fa, b) = (f, ab), (af, b) = (f, ba) (b E A) where (,) is used for the dual pairing between elements of W* and A. Let Q*%2 = {fa: f E t*,a E A} and %2%2* = {af :-a E Q(,f E :%*}. The dual space W(* is said to factor on the left (resp. right) if Wf* = Qf*Qf (resp. Wf* = At*). In a recent paper [5] Anthony To-Ming Lau and Ali Ulger have obtained various necessary and sufficient conditions for factoring of W.** Here we answer three questions left open in [5]. The second dual space (** of a Banach algebra St admits two Banach algebra products known as first (left) and second (right) Arens products. Each of these products extends the product of St as canonically embedded in (** (see [1] or [3]). We briefly recall the definition of these products. For m, n E Q**, their first (left) Arens product indicated by m O n is given by (m l n,f) =(m,nf) (fEQ*) where nf E W* is defined by (nff,a) = (nr,fa) ;(a E Q, where fa is as defined earlier. Received by the editors December 5, 1996. 1991 Mathematics Subject Classification. Primary 46H99.