On functional representation of normed algebras

Let A be a commutative algebra over complex numbers with a space norm ‖⋅‖ making the multiplication on A separately continuous. We will study the Gelfand representation of this type of normed algebra. In particular, we look at the cases where the standard Gelfand representation (i.e., the use of supremum-norm on the Gelfand transform algebra ) gives different properties from the original algebra (A,‖⋅‖). We show that there are even Banach algebras for which this type of difficulty may happen. We will provide with some weighted supremum-norm and by using these weights we can avoid the difficulties mentioned above. For the definition of these weights we adopt the ideas of Cochran represented in [A.C. Cochran, Representation of A-convex algebras, Proc. Amer. Math. Soc. 30 (1973) 473-479].