On closed Lie ideals of certain tensor products of C*-algebras II

TL;DR

This paper characterizes all closed Lie ideals in various tensor products of C*-algebras, extending previous results and covering multiple tensor product types and algebra classes.

Contribution

It provides a comprehensive identification of closed Lie ideals for several tensor products of C*-algebras, generalizing earlier work and including new cases.

Findings

Complete classification of closed Lie ideals for Haagerup, projective, and operator space tensor products.

Extension of previous results to tensor products involving simple C*-algebras with specific trace properties.

Identification of Lie ideals in minimal tensor products with commutative C*-algebras.

Abstract

We identify all closed Lie ideals of $A \otimes^{\alpha} B$ and $B(H) \otimes^{\alpha} B(H)$, where $\otimes^{\alpha}$ is either the Haagerup tensor product, the Banach space projective tensor product or the operator space projective tensor product, $A$ is any simple C*-algebra, $B$ is any C*-algebra with one of them admitting no tracial states, and $H$ is an infinite dimensional separable Hilbert space. Further, generalizing a result of Marcoux, we also identify all closed Lie ideals of $A\otimes^{\min} B$, where $A$ is a simple C*-algebra with at most one tracial state and $B$ is any commutative C*-algebra.