On closed Lie ideals of certain tensor products of ‐algebras

For a simple ‐algebra A and any other ‐algebra B, it is proved that every closed ideal of 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 ‐minimal norm, then this allows us to identify all closed Lie ideals of , where A and B are simple, unital ‐algebras with one of them admitting no tracial functionals, and to deduce that every non‐central closed Lie ideal of contains the product ideal . Closed Lie ideals of are also determined, A being any simple unital ‐algebra with at most one tracial state and X any compact Hausdorff space. And, it is shown that closed Lie ideals of are precisely the product ideals, where A is any unital ‐algebra and α any completely positive uniform tensor norm.