Let $u:A \to \prod\_{\mathcal U} M\_n$ be a unital completely positive map (ucp) from a unital separable $C^\*$algebra into the von Neumann algebra ultraprodut $\prod\_{\mathcal U} M\_n$. 


Here $\mathcal U$ is an ultrafilter on $\mathbb N$ and $\prod\_{\mathcal U} M\_n$ is the quotient of $B=\{(x\_n)\_{n \in \mathbb N}, x\_n \in M\_n(\mathbb C) , \sup\_n \|x\_n\|<\infty\}$ by the ideal $I\_{\mathcal U}=\{(x\_n)\_n, \lim\_{\mathcal U} Tr(x\_n^\* x\_n)/n = 0\}$.



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> Does there exist a ucp lifting of $u$, i.e. a sequence $u\_n:A \to M\_n(\mathbb C)$ of ucp maps such that $u=q \circ (u\_n)$, where $q:B\to B/I\_{\mathcal U}$ is the quotient map?
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If not, is $u$ locally liftable? That is: given a finite dimensional operator system $E\subset A$ (= a subspace $E$ of $A$ containing $1$ and stable under $a \mapsto a^\*$), does there exist a ucp lifting of the restriction of $u$ to $E$?


Some comments: as in [my related question](https://mathoverflow.net/questions/68367/), I know that the answer is no in general if one replaces the von Neumann algebra ultraproduct by the $C^\*$-algebra ultraproduct. But I hope that again, the situation might be much simpler in the von Neumann algebra setting. (I even have the feeling that I have known the answer to this question, but that I have forgotten it).



Lifting of a ucp map with values in a von Neumann algebra ultraproduct of matrix algebras
The answer is no, in general there is no lifting. A lifting exists if the $C^{\ast}$-algebra has the so-called lifting property (LP), and local liftings exist if it has the local lifting property (LLP).


I constructed in 


Andreas Thom, *Examples of hyperlinear groups without factorization property*, Groups Geom. Dyn. 4, no. 1 (2010) 195-208.


an example of a group $G$, such that the universal group $C^{\ast}$-algebra of $G$ does not have the LLP. The idea is that $G$ is hyperlinear, but cannot have Kirchberg's factorization property. The hyperlinearity is shown by a concrete construction of micro-states, Kirchberg's factorization property has to fail since $G$ has property (T), but is not residually finite. Note that Kirchberg showed that Kazhdan groups with factorization property are residually finite. See also


Narutaka Ozawa, *About the QWEP conjecture*, Internat. J. Math. 15 (2004), no. 5, 501–
530.


where all these concepts are explained.


