What is an example of a C$^{*}$ algebra such that the span of nilpotent elements is not a sub vector space of the span of commutator elements?
Obviously any such C$^{*}$ algebra would be a counter example to the question $2$ of the following MO post:
https://mathoverflow.net/questions/231328/the-saturation-of-murray-von-neumann-relation
I know that every stable C$^{*}$ algebra or every properly infinite C$^{*}$ algebra can be generated by nilpotent elements. But it seems that such algebras do not admit "nice" trace. However for my purpose an algebraic trace (a linear tracial functional which vanishes on commutators, not necessarily positive or bounded) is sufficient.