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⊗minB 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⊗αB, 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)⊗αB(H) contains the product ideal K(H)⊗αK(H). Closed Lie ideals of A⊗minC(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⊗αK(H) are precisely the product ideals, where A is any unital C∗†algebra and α any completely positive uniform tensor norm.