Local lifting property - Imported Knowlpedia Content

Definition

A unital $C^*$ -algebra $A$ has the local lifting property (LLP) if the following holds. For every unital $C^*$ -algebra $B$ , closed two-sided ideal $J\triangleleft B$ , unital completely positive map $\phi:A\to B/J$ , and finite-dimensional operator system $E\subseteq A$ (that is, a unital self-adjoint linear subspace), there is a unital completely positive map

\widetilde\phi_E:E\longrightarrow B

such that $q\circ\widetilde\phi_E=\phi|_E$ , where $q:B\to B/J$ is the quotient map. The lift may depend on $E$ ; no single lift on all of $A$ is required. Thus LLP is local in the finite-dimensional domain, while the quotient and target remain arbitrary.

Local versus global lifting

The lifting property (LP) requires one unital completely positive lift $\widetilde\phi:A\to B$ for the whole map. Hence LP implies LLP. The converse is false in general: compatible local lifts need not assemble into a global lift.

Every separable nuclear $C^*$ -algebra has LP by the Choi–Effros lifting theorem and therefore has LLP. The full group algebra $C^*(\mathbb F_\infty)$ is an important nonnuclear example with LLP, so LLP is not a form of nuclearity.

Tensor-product characterization

Kirchberg's characterization says that $A$ has LLP exactly when, for an infinite-dimensional Hilbert space $H$ , the canonical map identifies

A\otimes_{\max}B(H)=A\otimes_{\min}B(H)

isometrically. In other words, $A$ and $B(H)$ form a nuclear pair even though neither algebra need be nuclear. This connects a local quotient-lifting condition to the comparison of maximal and minimal $C^*$ -tensor norms Brown–Ozawa, §13.2.

Nonunital convention

For a nonunital $C^*$ -algebra, LLP is normally defined by requiring its unitization to have LLP. Equivalent formulations use completely positive contractive maps on finite-dimensional self-adjoint subspaces. Stating the unitization convention prevents ambiguity about where the order unit in the local operator system comes from.

References

Nathanial P. Brown and Narutaka Ozawa, $C^*$ -Algebras and Finite-Dimensional Approximations, Graduate Studies in Mathematics 88, American Mathematical Society, 2008. AMS chapter record. Relevant: Chapter 13, especially §13.1 on LLP and §13.2 on tensorial characterizations.
Gilles Pisier, Introduction to Operator Space Theory, London Mathematical Society Lecture Note Series 294, Cambridge University Press, 2003. Publisher record. Relevant: Chapter 16 on the local lifting property.