What is an example of a simple $C^{\*}$ algebra which all elements are (two sided or equivalently one sided) zero divisor?



Simple $Z^{*}$ algebra
Take compact operators on a non-separable Hilbert space. For each such an operator you'll find a rank-one projection with range in the kernel of that operator.


~~I guess that by some Löwenheim–Skolem-type argument, this gives you a separable example too.~~ (As observed by Andreas Thom.)


