On closed Lie ideals of certain tensor products of $C^{\ast }$-algebras

Abstract

For a simple $C^{\ast }$-algebra $A$ and any other $C^{\ast }$-algebra $B$, it is proved that every closed ideal of $A{\otimes }^{\min }B$ 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 $\alpha$ is either the Haagerup norm, the operator space projective norm or the $C^{\ast }$-minimal norm, then this allows us to identify all closed Lie ideals of $A{\otimes }^{\alpha }B$, where $A$ and $B$ are simple, unital $C^{\ast }$-algebras with one of them admitting no tracial functionals, and to deduce that every non-central closed Lie ideal of $B(H){\otimes }^{\alpha }B(H)$ contains the product ideal $K(H){\otimes }^{\alpha }K(H)$. Closed Lie ideals of $A{\otimes }^{\min }C(X)$ are also determined, $A$ being any simple unital $C^{\ast }$-algebra with at most one tracial state and $X$ any compact Hausdorff space. And, it is shown that closed Lie ideals of $A{\otimes }^{\alpha }K(H)$ are precisely the product ideals, where $A$ is any unital $C^{\ast }$-algebra and $\alpha$ any completely positive uniform tensor norm.

16pages. Few observations have been included in Section 2 and, Section 4 has been revised in 2nd version

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