TL;DR
This paper investigates properties of Z* algebras, showing how extensions behave and applying these results to bounds in positive cones of C* algebras, revealing new structural insights.
Contribution
It demonstrates that extensions of non Z* algebras by non Z* algebras remain Z*, and provides an application to bounds in positive cones.
Findings
Extension of a Z* algebra by a Z* algebra is not necessarily Z*.
Extension of a non Z* algebra by a non Z* algebra is Z*.
Every compact subset of positive cones in a C* algebra has an upper bound.
Abstract
We study some properies of 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.
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