We study some properties of Z∗\documentclass[12pt]{minimal}
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\begin{document}$$Z^{*}$$\end{document} algebras, those C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^{*}$$\end{document} algebras whose all positive elements are zero divisors. Using an example, we show that an extension of a Z∗\documentclass[12pt]{minimal}
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\begin{document}$$Z^{*}$$\end{document} algebra by a Z∗\documentclass[12pt]{minimal}
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\begin{document}$$Z^{*}$$\end{document} algebra is not necessarily a Z∗\documentclass[12pt]{minimal}
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\begin{document}$$Z^{*}$$\end{document} algebra. However, we prove that the extension of a non-Z∗\documentclass[12pt]{minimal}
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\begin{document}$$Z^{*}$$\end{document} algebra by a non-Z∗\documentclass[12pt]{minimal}
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\begin{document}$$Z^{*}$$\end{document} algebra is a non-Z∗\documentclass[12pt]{minimal}
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\begin{document}$$Z^{*}$$\end{document} algebra. We also prove that the tensor product of a Z∗\documentclass[12pt]{minimal}
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\begin{document}$$Z^{*}$$\end{document} algebra by a C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^{*}$$\end{document} algebra is a Z∗\documentclass[12pt]{minimal}
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\begin{document}$$Z^{*}$$\end{document} algebra. As an indirect consequence of our methods, we prove the following inequality type results: (i) Let an\documentclass[12pt]{minimal}
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\begin{document}$$a_{n}$$\end{document} be a sequence of positive elements of a C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^{*}$$\end{document} algebra A which converges to zero. Then, there are positive sequences bn\documentclass[12pt]{minimal}
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\begin{document}$$b_{n}$$\end{document} of real numbers and cn\documentclass[12pt]{minimal}
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\begin{document}$$c_{n}$$\end{document} of elements of A which converge to zero such that an+k≤bnck.\documentclass[12pt]{minimal}
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\begin{document}$$a_{n+k}\le b_{n}c_{k}.$$\end{document} (ii) Every compact subset of the positive cones of a C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^{*}$$\end{document} algebra has an upper bound in the algebra.
