On closed Lie ideals of certain tensor products of C ∗ ‐algebras

For a simple C ∗ ‐algebra A and any other C ∗ ‐algebra B , it is proved that every closed ideal of A ⊗ min B is a product ideal if either A is exact or B is nuclear. Closed commutator of a closed ideal in a Banach algebra whose every closed ideal possesses a quasi‐central approximate identity is described in terms of the commutator of the Banach algebra. If α is either the Haagerup norm, the operator space projective norm or the C ∗ ‐minimal norm, then this allows us to identify all closed Lie ideals of A ⊗ α B , where A and B are simple, unital C ∗ ‐algebras with one of them admitting no tracial functionals, and to deduce that every non‐central closed Lie ideal of B ( H ) ⊗ α B ( H )contains the product ideal K ( H ) ⊗ α K ( H ) . Closed Lie ideals of A ⊗ min C ( X )are also determined, A being any simple unital C ∗ ‐algebra with at most one tracial state and X any compact Hausdorff space. And, it is shown that closed Lie ideals of A ⊗ α K ( H )are precisely the product ideals, where A is any unital C ∗ ‐algebra and α any completely positive uniform tensor norm.