Local lifting property for Banach Spaces

Question

I am reading a paper in which the introduction of the paper talks about lifting property which I have never seen.

Can someone explain or give me some reference for the local lifting property of Banach spaces and of $C^*$-algebra? Intuition behind the definition would be much appreciated.

Thanks in advance

Answer 1

A C$^*$-algebra $A$ has the local lifting property (LLP) if whenever you have:

• a C$^*$-algebra $B$,

• $J\subset B$ an ideal,

• $F\subset A$ a finite-dimensional operator system,

• $\phi:A\to B/J$ ucp,

there exist $\psi:A\to B$ ucp such that $\psi|_F=\phi|_F$. So $\psi$ lifts $\phi$ "locally".

The property was introduced by Kirchberg, who showed that $A$ having LLP is equivalent to the cone/suspension of $A$ having Ext a group.

It is known that the WEP implies the LLP (very non-trivial: Kirchberg showed that this is equivalent with $B(H)\odot B(H)$ admiting a single C$^*$-norm). The implication LLP $\implies$ WEP is equivalent to Connes Embedding Problem.