  We study some properies of $Z^{*}$ algebras, thos C^* algebra which all
positive elements are zero divisors. We show by means of an example that an
extension of a Z* algebra by a Z* algebra is not necessarily Z* algebra.
However we prove that an extension of a non Z* algebra by a non Z* algebra is
again a Z^* algebra. As an application of our methods, we prove that evey
compact subset of the positive cones of a C* algebra has an upper bound in the
algebra.
