# On closed Lie ideals of certain tensor products of $C^*$-algebras

## TL;DR

This paper characterizes closed Lie ideals and product ideals in tensor products of simple $C^*$-algebras under various norms, revealing their structure and conditions for their identification.

## Contribution

It provides a comprehensive description of closed Lie ideals in tensor products of simple $C^*$-algebras with specific norms, extending previous results and identifying conditions for product ideals.

## Key findings

- Closed ideals of $A ensor^{ ext{min}} B$ are product ideals if $A$ is exact or $B$ is nuclear.
- All closed Lie ideals of $A ensor^{ ext{alpha}} B$ are identified for certain norms and algebra conditions.
- Every non-central closed Lie ideal of $B(H) ensor^{ ext{alpha}} B(H)$ contains the product ideal $K(H) ensor^{ ext{alpha}} K(H)$.

## Abstract

For a simple $C^*$-algebra $A$ and any other $C^*$-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 $C^*$-minimal norm, then this allows us to identify all closed Lie ideals of $A \otimes^{\alpha} B$, where $A$ and $B$ are simple, unital $C^*$-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 $C^*$-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 $C^*$-algebra and $\alpha$ any completely positive uniform tensor norm.