non-equivalent norms on c^b(k)

Let a be a non-zero normed vector space and let k = b (0) 1 be the closed unit ball of a. also, let φ be a non-zero element of a ∗ such that ∥φ∥ ≤ 1. we first define a new norm ∥ · ∥φ on c b (k), that is a non-complete, non-algebraic norm and also nonequivalent to the norm ∥ · ∥∞. we next show that for 0 ̸= ψ ∈ a ∗ with ∥ψ∥ ≤ 1, the two norms ∥ · ∥φ and ∥ · ∥ψ are equivalent if and only if φ and ψ are linearly dependent. also by applying the norm ∥ · ∥φ and a new product “ · ” on c b (k), we present the normed algebra ( c bφ(k), ∥ · ∥φ ) . finally we investigate some relations between strongly zero-product preserving maps on c b (k) and c bφ(k).