On the Lifting Property for $C^*$-algebras

TL;DR

This paper characterizes the lifting property of separable $C^*$-algebras through tensor product conditions, linking it to maximal and normal tensor products with von Neumann algebras, providing a new characterization.

Contribution

It provides a novel characterization of the lifting property for separable $C^*$-algebras via maximal tensor product conditions with other $C^*$-algebras and von Neumann algebras.

Findings

Lifting property characterized by tensor product isometry conditions.

Equivalence between maximal and normal tensor products for $A$ with von Neumann algebras.

New criterion for the lifting property in terms of tensor products.

Abstract

We characterize the lifting property (LP) of a separable $C^*$-algebra $A$ by a property of its maximal tensor product with other $C^*$-algebras, namely we prove that $A$ has the LP if and only if for any family $(\{D_i\mid i\in I\}$ of $C^*$-algebras the canonical map $$ {\ell_\infty(\{D_i\}) \otimes_{\max} A}\to {\ell_\infty(\{D_i \otimes_{\max} A\}) }$$ is isometric. Equivalently, this holds if and only if $M \otimes_{\max} A= M \otimes_{\rm nor} A$ for any von Neumann algebra $M$.