Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X38320120915varphi-amenability of Banach algebras725738237ENAliGhaffariSemnan UniversityAhmadAlinejadSemnan UniversityJournal Article20101106Let $A$ be an arbitrary Banach algebra and $varphi$ a homomorphism from $A$ onto $Bbb C$. Our first purpose in this paper is to give some equivalent conditions under which <br /> guarantees a $varphi$-mean of norm one. Then we find some conditions under which there exists a $varphi$-mean in the weak$^*$ cluster of ${ain A; |a|=varphi(a)=1}$ in $A^{**}$.http://bims.iranjournals.ir/article_237_2d7117eadedb5265aee5e2e415a2034f.pdf