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

Gilles Pisier

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arXiv · 2009.12911

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

Gilles Pisier

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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 $M$.

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Gilles Pisier. 2020-09-27. On the Lifting Property for $C^*$-algebras. https://arxiv.org/abs/2009.12911

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