On σ-cyclic amenability for Banach algebras

Authors
Eghbal Ghaderi 1
Mehdi Nemati 2
Saber Naseri 1
1 University of Kurdistan
2 Isfahan University of Technology
Abstract
Suppose that σ is homomorphism on Banach algebra A. Then in this paper we introduce and study the new two notions σ-cyclic derivation and σ-cyclic amenability for A. We investigate the relation between trace extension property and σ-cyclic amenability; indeed we show that the σ-cyclic amenability of AI implies that I has the trace extension property. Next, prove that the it’s converse can be true under the special conditions. One of the important result is that every σ-cyclic amenable is essential. Furthermore, for every closed two-sided ideal I of A, the relation between of σ-cyclic amenability of A and σ-cyclic amenability of AI has been studied. Also, we show that the σ-cyclic amenability of A and AI is equivalent. Finally, we study this notion on θ-Lau algebras and we investigate its relation with the similar concept on algebras A and B.
Keywords
Banach algebras
trace extension property
σ-cyclic derivation
σ-cyclic amenability
unitization
θ-Lau product
[1] Bodaghi A., Eshaghi Gordji M., Medghalchi A.R., “A generalization of the weak amenability of Banach algebras”, Banach J. Math. Anal., 3(1) (2009) 131-142.

[2] Gronbaek N., “Weak and cyclic amenability for non-commutative Banach algebras”, Proc. Edinburgh. Math. Soc., 35 (1992) 315-328.

[3] Johnson B.E., “Cohomology in Banach algebras”, Mem. Amer. Math. Soc., 127 (1972).

[4] Lau A. T.-M., “Analysis on a class of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups”, Fund. Math. 118 (1983) 161-175.

[5] Mirzavaziri M., Moslehian M.S., “Automatic continuity of σ-derivations on C^*-algebras”, Proc. Amer. Math. Soc., 134(11) (2006) 3319-3327.

[6] Mirzavaziri M., Moslehian M.S., “σ-amenability of Banach algebras”, Southeast Asian Bull., 33 (2009) 89-99.

[7] Monfared M. S., “On certain products of Banach algebras with applications to Harmonic analysis”, Studia Math. 178 (2007) 277-294.

[8] Moslehian M.S., Niazi Motlagh A., “Some notes on (σ, τ )-amenability of Banach algebras”, Stud. Univ. Babes-Bolyai Math., 53(3) (2008) 57-68.

[9] Shojaee B., Bodaghi A., “Aproximate cyclic amenability of Banach algebras”, Mathematical Sciences, 5(1) (2011) 25-32.