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Relationship between weak matricial LLP and LLP; Ext(A) criterion

Ascertain whether the weak matricial local lifting property is strictly weaker than, or equivalent to, Kirchberg’s LLP for maximal group C*-algebras, at least for residually finite dimensional (RFD) cases; and determine whether the Brown–Douglas–Fillmore semigroup Ext(A) being a group is equivalent to LLP for such algebras.

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Background

The authors introduce a weak matricial variant of LLP tailored to lifting -homomorphisms into products of matrix algebras modulo the direct sum ideal. While general LLP and the weak matricial LLP agree for broad classes (by Ozawa’s results), the authors highlight uncertainty for group C-algebras, especially RFD ones.

They note a connection: for RFD C*-algebras, weak matricial LLP is equivalent to Ext(A) being a group. Whether this Ext condition characterizes LLP in the same generality remains open.

References

We do not know if the weak matricial LLP is genuinely weaker than the LLP for group $C*$-algebras, or even for RFD group $C*$-algebras. It was pointed out to us by Tatiana Shulman that the weak matricial LLP for $A$ is equivalent to the Brown-Douglas-Fillmore semigroup $Ext(A)$ (see for example Chapter 2) being a group whenever $A$ is an RFD $C*$-algebra, and that it is open whether $Ext(A)$ being a group is equivalent to the LLP in this level of generality.

— Conditional representation stability, classification of $*$-homomorphisms, and relative eta invariants (2408.13350 - Willett, 23 Aug 2024) in Remark \ref{weak llp}, Section “Ucp quasi-representations and the local lifting property”