As the comment of Andreas Thom indicated here, 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?

As the comment of Andreas Thom indicated here, 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?