Non-normal elements in Banach *-algebras

Let A be a Banach *-algebra with an identity, continuous involution, center Z and set of self-adjoint elements Sigma. Let h is an element of Sigma. The set of v is an element of Sigma such that (h + iv)(n) is normal for no positive integer n is dense in Sigma if and only if h is not an element of Z. The case where A has no identity is also treated.