R. Aron, B.J. Cole,T . Gamelin, S ectra of algebras of analytic functions on a Banach space, J. reine angew. Math. 415 (1991), 51--93.  Home/Search   Document Not in Database   Summary   Related Articles

....all # # Sn . Now recall (e.g. from [DF, 1.6. that for # #L(E,F) theexten#MD n # is separately weak # con tin uous if an don ly ifL # : E # F # is weakly compact.Sin#F every permutation # # Sn is a product of tran# osition s o n# obta in# (b) # (a) of the well kn# wn (see e.g. ACG, sect. 8] Corollary. Let E be a normed space and n # 3. Then the following are equivalent: a) For every # #L( n E) the Arens extension # is separately weak # continuous. b) The same as (a) with n =2. c) Every T #L(E;E # ) is weakly compact. Proof. For the remain# implication (a) # ....

....verges to (1, 0, 1, 0, 1, 0, x ## ;ifb # # # # is a Ban ach limiton the odd compon en ts, then #b, xm # =0, but #b, x ## # = 1. It follows thatL # (B #1 )isn ot#(##,# # # ) compact a n# the Are n# exten#F]U # is n# t symmetric. An# ther but related example was given in [ACG . 6.4. Usin g the same ideas as in 6.2. it is straightforward to verify that the followin g holds true: Proposition ( ACG ) For every normed space E the following statements are equivalent: a) For every n # 2 and every # #L s ( n E) the Arens extension # is symmetric (equivalently: ....

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R. Aron, B.J. Cole,T . Gamelin, S ectra of algebras of analytic functions on a Banach space, J. reine angew. Math. 415 (1991), 51--93.

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