IOSR Journal of Mathematics (IOSRJM) ISSN: 2278-5728 Volume 2, Issue 6 (Sep-Oct. 2012), PP 36-47 www.iosrjournals.org www.iosrjournals.org 36 | P a ge K-derivation and symmetric bi-k-derivation on Gamma Banach Algebras 1 P. Rajkhowa, 2 Md. Shahidul Islam Khan Department of Mathematics, Gauhati University, Guwahati-781014 Abstract: In this paper, we define and study k-derivations and symmetric bi-k-derivations on a -Banach algebra. We also define and study hk-derivation d on the projective tensor product / V V p  for the h- and k-derivations 1 d and 2 d on -Banach Algebras ) ( F V and ) ( / F V  respectively. AMS subject classification Code: 17D20 (γ, δ) Key words: k-derivation, symmetric bi-k-derivation, centroid. I. Introduction: N. Nobusawa [7] introduced the notion of a -ring, more general than a ring. W. E. Barnes [11] weakened slightly the condition in definition of -ring in the sense of Nobusawa. W. E. Barnes [11], J. Luh [4] and S. Kyuno [10] studied the structure of -rings and obtained various generalizations analogous to corresponding parts in ring theory. Bhattacharya and Maity [2] introduced the notion of a -Banach algebra. In recent times, many far reaching results of general algebras have been extended to -algebras by many outstanding research workers. In this paper, we study k-derivation on -Banach algebras V and kh- deviation on -Banach algebra / V V p  . We define symmetric bi-k-derivation on -Banach algebras in which k:    is an additive map such that k k n  , where n is a positive integers. Some important results relating to this concepts are proved. For example we show that (a) Let   F V and   / / F V be two -Banach algebra and /  -Banach algebra respectively with   V x x e x x e       ,     , V e and y e y y e   / / / /   (yV),        , V e . If 1 d and 2 d are k- and h-inner derivation on   F V and   / / F V respectively implemented by    , a and   / , b  respectively then d is a kh- inner deviation on / V V p  implemented by   / / , b e e a       , (b) Let V be a 2-torsion free prime -Banach algebra, 1 D (.,.) , 2 D (.,.) and 3 D (.,.) and 4 D (.,.) the symmetric bi-k-derivations on V and 1 d , 2 d , 3 d and 4 d traces of 1 D (.,.), 2 D (.,.), 3 D (.,.) and 4 D (.,.) respectively. If 1 d (x) 2 d (y)= 3 d (x) 4 d (y), for all x, yV and  and 1 d 0 4 d , then there exists   C such that 2 d (x)= 1 d (x) for all , where  C is the extended centroid of V, (c) Let V be a 2-torsion free prime Gamma Banach algebra and U be a non zero ideal of V. Suppose there exist symmetric bi-k-derivations V V V D   : 1 and V V V D   : 2 such that    0 x , x d D 2 1  holds for all xU where 2 d denotes the trace of 2 D . In this case 0 D 1  or 2 D =0, (d) Let V be a 2- and 3-torsion free prime -Banach algebra. Let U be a non zero ideal of V and 1 D :VVV and 2 D :VVV be symmetric bi-k- derivations. Suppose further that there exists a symmetric bi-additive mapping B: VVV such that     x f x d d 2 1  holds, for all x U, where 1 D and 2 D are the traces of 1 D and 2 D respectively and f is the trace of B. Then either 1 D =0 and 2 D =0. II. Preliminaries Let V and  be two additive abelian groups. If for all      , ; , , V z y x , the following conditions are satisfied, (a) V y x   , (b)   z y z x z x        ,   y x y x y x        ,