On the lifting property for C ∗ C^* C ∗ -algebras

[摘要] We characterize the lifting property (LP) of a separable C∗C^*C∗-algebra AAA by a property of its maximal tensor product with other C∗C^*C∗-algebras, namely we prove that AAA has the LP if and only if for any family {Di∣i∈I}\{D_i\mid i\in I\}{Di​∣i∈I} of C∗C^*C∗-algebras the canonical map ℓ∞({Di})⊗max⁡A→ℓ∞({Di⊗max⁡A}){\ell_\infty(\{D_i\}) \otimes_{\max} A}\to {\ell_\infty(\{D_i \otimes_{\max} A\}) }ℓ∞​({Di​})⊗max​A→ℓ∞​({Di​⊗max​A}) is isometric. Equivalently, this holds if and only if M⊗max⁡A=M⊗norAM \otimes_{\max} A= M \otimes_\mathrm{nor} AM⊗max​A=M⊗nor​A for any von Neumann algebra MMM.

[学科分类] 神经科学

[关键词] C -algebras;von Neumann algebras;lifting property;tensor products