𝛅- homomorphism maps into dual Banach algebras

Subject Areas : Statistics

B. Hayati ^{1 *} , H. Khodaei ²

1 - Department of mathematics, Malayer University
2 - Department of mathematics, Malayer University, Malayer, Iran

Received: 2019-01-27 Accepted : 2019-08-19 Published : 2019-12-22

Keywords: جبرهای باناخ دوگان, نگاشت δ-همریختی, تابعک δ-ضربی,

Abstract :

Let A be a Banach algebra and (B,B_*) be a dual Banach algebra. A linear map φ:A⟶B is said to be a δ - homomorphism map if ‖‖φ(a_1 a_2)-φ(a_1)φ(a_2)‖‖≤δ‖‖a_1‖‖ ‖‖a_2 ‖‖ for every a_1,a_2∈A. In this paper, we study the δ - homomorphism maps from A into B. Among other things, we prove that if φ:A⟶B is a δ - homomorphism map and B_* is multiplicative on the algebra generated by φ(A), then φ is bounded and ‖‖φ‖‖≤1+δ. Let A be a Banach algebra and (B,B_*) be a dual Banach algebra. A linear map φ:A⟶B is said to be a δ - homomorphism map if ‖‖φ(a_1 a_2)-φ(a_1)φ(a_2)‖‖≤δ‖‖a_1‖‖ ‖‖a_2 ‖‖ for every a_1,a_2∈A. In this paper, we study the δ - homomorphism maps from A into B. Among other things, we prove that if φ:A⟶B is a δ - homomorphism map and B_* is multiplicative on the algebra generated by φ(A), then φ is bounded and ‖‖φ‖‖≤1+δ.

References:

[1] F. F. Bonsal and J. Duncan, Complete normed algebras, Springer Verlag (1973).

 

[2] M. Daws, Multipliers, Self-induced and dual Banach algebras, Dissertations Mathematicae 470 (2010), 62 pp.

 

[3] H. G. Dales, Automatic continuity: a survey, Bulletin of The London Mathematical Society 10 (1987), 129-183.

 

[4] H. G. Dales, Banach algebras and automatic continuity. Oxford University press (2001).

 

[5] B. E. Johnson, The uniqueness of the (complete) norm topology, Bulletin of The American Mathematical Society 73 (1967), 537-539.

 

[6] B. E. Johnson, Continuity of generalized homomorphisms, Bulletin of The London Mathematical Society 19 (1987), 67-71.

 

[7] B. Hayati and M. Amini, Connes-amenability of multiplier Banach algebras, Kyoto Journal of Mathematics   50 (2010), 41-50.

 

[8] B. Hayati and M. Amini, Dual multiplier Banach algebras and Connes-amenability, Publications  Mathematicae Debrecen 86 (2015), 169-182.

 

[9] K. Jarosz, Perturbations of Banach algebras, Lectures Notes in Mathematics, 1120 Springer, Berlin (1985).

 

[10] V. Runde, Lectures on amenability, Lecture Notes in Mathematics, 1774 Springer, Berlin (2002).