2009/08/25 by Esslamzadeh, G. H., Shojaee, B.
#46H20 #46H25 #46H35 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.0908.3577
In this paper we deal with four generalized notions of amenability which are called approximate, approximate weak, approximate cyclic and approximate n-weak amenability. The first two were introduced and studied by Ghahramani and Loy in [9]. We introduce the third and fourth ones and we show by means of some examples, their distinction with their classic analogs. Our main result is that under some mild conditions on a given Banach algebra \A, if its second dual \A** is (2n-1)-weakly [respectively approximately/ approximately weakly/ approximately n-weakly] amenable, then so is \A. Also if \A is approximately (n+2)-weakly amenable, then it is approximately n-weakly amenable. Moreover we show the relationship between approximate trace extension property and approximate weak [respectively cyclic] amenability. This answers question 9.1 of [9] for approximate weak and cyclic amenability.