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1. .,t ,/.2'- /,- ( -..--. M atlrrcrnatical S cie'nce s Vol. 5, No. 7 (2011) 25-32 Approximate cyclic ararenability of Banach algebras B. Sh oj aeeo,l , A. B odaghi b " Dep:rrtrnent of N{a,thematics, I(araj l3ranch, Islamic Azad Univcr-sity, I(ara.j, Iran. t' Departtnertt of \ilell,hcrnntics, Galrns:rr Br:urch, Islarnic Aza<l Universil,y, Calntsa,r, L'an. Abstract In thc currcnt papcr, we dcal lvilh gcneralizcd notion ol arncnabilily which is calied approximate cyclic arnenabilitS'. Wc lntroducc this concept ancl wc shor,i,by meaus of att cxatnple . its distinction rvilh ils classic a1ralogs. l,Ior-covcr rvc s]row thc relationship bctu,een a,pproxiruaLctrirce cxtension property and approxirnatc cyclic amcnabiliiv. This answers pa,rl,ially,qucstion 9.1 of [3] lbr approximate c.yclic arnenabilitS'. I{eywords: Bana,ch algebra, Cyclic amcnability. Approximatc cyclic arnenability. O 2011 Published by Islamic Azad Urriversity-l{araj Branch. Introduction The concept of amenability for Banach algcbras rvas initiai,ed by B. tr. Johnson in [6]. Later, Gronbael<in [5] invcstigated properbies of c1,gll6amenable Banach algebras. He also ident,ified the rclai,ionship betu'ccn hereditary ploperties of cr,rsllsamenability and the trace extension property. Thc conccpt of apploximately amenable Banach algcbras rvas introclucccl and stud- iccl for thc first timc by Ghahra,rnani and Loy in [3]. In i,hc nrerrtioncd papcr', they characterizcd tire structu-re of apploxirnately arncnable llauach algcbras tirrough scv- cral wavs. At the begirrnirrg,authors asl<cd vrhicir of the standard r-csultson amcnabilitv ' Corresponding Author'. U-nrail Arldress: shouja,ei@liiau.ac.ir