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. We investigate some algebraic and spectral properties of odd duals of A, by defning the products ⃝a, ⃝F as in [12]. We will show relations between these products and Arens products, weak or weak-starcontinuity, commutativity and unit elements of these algebras. Also we 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.