Let A be a Banach algebra with a bounded approximate identity. Let and be respectively, the topological centers of the algebras A** and (AA*)* with respect to the second Arens multiplication. In this paper, we show that is isometrically isomorphic to LM(A) , where is a closed subalgebra of and LM(A) is the set of left multipliers operators of the Banach algebra A. Key words: Topological center, Arens multiplication, Banach algebra, Left multiplier operator