A generalization of amenability for topological semigroups and semigroup algebras

In this paper for two topological semigroups $S$ and $T$, and a continuous homomorphism $\varphi$ from $S$ into $T$, we introduce and study the concept of $(\varphi, T)$-derivations on $S$ and $\varphi$-amenability of $T$ and investigate the relations between these two concepts. For two Banach algebras $A$ and $B$ and a continuous homomorphism $\varphi$ from $A$ into $B$ we also introduce the notion of $(\varphi, B)$-amenability of $A$ and show that a foundation semigroup $T$ with identity is $\varphi$-amenable whenever the Banach algebra $M_a(S)$ is $(\tilde{\varphi},M_a(T))$-amenable, where $\tilde{\varphi}:M(S)\to M(T)$ denotes the unique extension of $\varphi$. An example is given to show that the converse is nottrue.

Keywords:

Continuous homomorphism, semigroup, Banach algebra $(\varphi; T)$-derivation, $\varphi$-amenable,

PDF

___

A. C. Baker, J. W. Baker, Algebra of measures on a locally compact semigroup III, J. London Math. Soc. (4), 685-695, 1972.
H. G. Dales, Banach slgebras and automatic continuity, Clarendon Press, Oxford, 2000.
M. M. Day, Ergodic theorems for Abelian semigroups, Trans. Amer. Math. Soc., 51, 399-412, 1972.
J. Duncan, I. Namioka, Amenability of inverse semigroups and thier semigroup algebras, Proceeding of the Royal Society of Edinburgh 80 A, 309-321, 1998.
J. Duncan, A. L. T. Paterson, Amenability for discrete convolution semigroup algebras, Math. Scand. 66, 141-146, 1990.
H. A. M. Dzinotyiweyi, The analoge of the group algebra for topological semigroups, Research Notes in Mathematics, 98, Pitman, NewYork, 1984.
Z. Ghorbani, M. Lashkarizadeh Bami, $\varphi$-Approximate biat and $\varphi$-amenabel Banach alge- bras, Proceedings of the Romanian Academy, Series A, 13 (1), 3-10, 2012.
Z. Ghorbani, M. Lashkarizadeh Bami, $\varphi$-Amenable and $\varphi$-biflat Banach algebras, Bull. Iranian Math. Soc. 39 (3), 507-515, 2013.
B. E. Johnson, Cohomology in Banach algebras, harmonic problems, Memoirs Amer. Math. Soc. 127, 1972.
E. Kaniuth, A. Lau, J. Pym, On $\varphi$-amenability of Banach algebras, Math. Proc. Camp. Phil. Soc., 144, 85-96, 2008.
M. Lashkarizadeh Bami, Ideals of M(S) as ideals of $LUC(S)^*$ of a compactly cancellative semigroup S, Math. Japon., 48, 363-366, 1998.
M. Lashkarizadeh Bami, Representations of foundation semigroups and their algebras, Cana- dian J. Math, 37, 29-47, 1985.
A. T. M. Lau, Amenability of semigroups, the analytical and topological theory of semi- groups, trends and developments K. H. Hofman, J. D. Lawson and J. S. Pym, eds., Walter de Gruyter and Co., 1990.
M. Lashkarizadeh Bami, B. Mohammadzadeh and H. Samea, Derivations on certain semi- group algebras, Journal of Sciences Islamic Republic of Iran, 18 (4), 339-345, 2007.
M. Mirzavaziri and M. S. Moslehian, $\sigma$-derivations in Banach algebras, Bull. Iranian Math. Soc. 32 (1), 65-78, 2006.
M. Mirzavaziri and M.S. Moslehian, Automatic continuity of $\sigma$-derivations in $C^*$-algebras, Proc. Amer. Math. Soc. 134 (11), 3319-3327, 2006.
I. Namioka, On certain actions of semigroups on L-spaces, Studia Math, 29, 63-77, 1967.
V. Runde, Lectures on Amenability, Lecture Notes in Mathematics 1774, Springer-Verlag, Berlin-Heidelberg-New York, 2002.

Hacettepe Journal of Mathematics and Statistics-Cover

Yayın Aralığı: Yılda 6 Sayı
Başlangıç: 2002
Yayıncı: Hacettepe Üniversitesi Fen Fakultesi

Arşiv