Every Lie ideal in a unital, properly infinite C*-algebra is commutator equivalent to a unique two-sided ideal, and the same uniqueness holds in von Neumann algebras without a commutative summand.
Bre s ar, Zero product determined algebras, Frontiers in Mathematics
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Lie ideals in properly infinite C*-algebras
Every Lie ideal in a unital, properly infinite C*-algebra is commutator equivalent to a unique two-sided ideal, and the same uniqueness holds in von Neumann algebras without a commutative summand.