On the lifting property for $C^{\ast }$-algebras

Abstract

We characterize the lifting property (LP) of a separable $C^{\ast }$-algebra $A$ by a property of its maximal tensor product with other $C^{\ast }$-algebras, namely we prove that $A$ has the LP if and only if for any family $\{D_i\mid i\in I\}$ of $C^{\ast }$-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 $M{\otimes }_{\max }A=M{\otimes }_{\mathrm{nor}}A$ for any von Neumann algebra $M$.