A Note on Z^∗ Algebras

We study some properties of Z∗ algebras, those C∗ algebras whose all positive elements are zero divisors. Using an example, we show that an extension of a Z∗ algebra by a Z∗ algebra is not necessarily a Z∗ algebra. However, we prove that the extension of a non-Z∗ algebra by a non-Z∗ algebra is a non-Z∗ algebra. We also prove that the tensor product of a Z∗ algebra by a C∗ algebra is a Z∗ algebra. As an indirect of our methods, we prove the following inequality type results: (i) Let an be a sequence of positive elements of a C∗ algebra A which converges to zero. Then, there are positive sequences bn of real numbers and cn of elements of A which converge to zero such that an+k ≤ bnck . (ii) Every compact subset of the positive cones of a C∗ algebra has an upper bound in the algebra.