A $Z^{*}$ algebra is a $C^{*}$ algebra which all elements are(two sided or equivalently one sided) zero divisor.
Are there two $Z^{*}$ algebras $A,B$ such that for every short exact sequence of $C^{*}$ algebras: $$0\to A \to C \to B \to 0$$ $C$ is a $Z^{*}$ algebra, too?
In particular is there an extension of $C_{0}(X)$ by itself which is not a $Z^{*}$ algebra where $X$ is the long line?
This is already aked here as a comment but we ask it here as an independent question.
Edit:When a commutative algebra $A$ is in the form of $A=C_{0}(X)$, we get a formulation of our question in terms of general topology as follows:
Assume that $X,Y,Z$ are three locally compact Hausdorff spaces. We say that $Z$ is an extension of $X$ by $Y$ if $X$ can be embedded as an open set in $Z$ which remainder is homeomorphic to $Y$. We say that $X$ is approximately sigma compact (briefly A$\sigma$C) if there is a sequence of compact sets $K_{n}\subset X$ such that $\bigcup K_{n}$ is dense in $X$. Our question is the following:
Are there two NON $A\sigma C$ spaces $X$ and $Y$ such that every extension of $X$ by $Y$ is NON $A\sigma C$?
Remark: An alternative terminology: "Extension of $Y$ by $X$" can be replaced by "Extension of $X$ by $Y$"
