As the  comment of  Andreas Thom indicated [here](http://mathoverflow.net/questions/188344/simple-z-algebra), a  separable $C^\star$ algebra $A$ can  not contain a $Z^\star$ algebra.(A $Z^\star$ algebra is  a $C^\star$ algebra which all elements are zero  divisor). So  separability is  an obstruction for  $A$ to  **contain** a $Z^\star$  algebra.

Now  we  ask:

>Is it the only  obstruction? What type  of other obstructions can be introduced? In particular is it true that every non separable  algebra contains a $Z^\star$ algebra?