on odd duals of a banach algebra as a banach algebra

It is known that even duals of a banach algebra a with one of arens products are banach algebras, these products are natural multiplications extending the one on a. but the essence of a ∗ is completely different. by defining new products, we investigate some algebraic and spectral properties of odd duals of a. we will show relations between these products and arens products, weak or weak-star continuity, commutativity and unit elements of these algebras. we also determine the spectrum and multiplier algebra for a ∗ , and we calculate the quasi-inverses, spectrum and spectral radius for elements of these kinds of algebras