LIE PRODUCT ON ALGEBRAS GENERATED BY A VECTOR SPACE an‎d A LINEAR FUNCTIONAL

Let V be a vector space on C and be an element in V, the dual space of V. The linear space V accompanied by the product a ، b = (a)b; a; b 2 V, converts into an associative algebra. We investigate some properties of commutator bracket such as, bilinearity, antisymmetry, Jacobi identity on these algebras. Also we investigate the strongly Lie zero-product preserving maps on these algebras in the case where V is a non-zero normed vector space and is a non-zero bounded linear functional with ∥∥ ⩽ 1