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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 $I_{\mathcal U}=\{(x_n)_n, \lim_{\mathcal U} Tr(x_n^* x_n)/n = 0\}$.

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?

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, 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).

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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.

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  • $\begingroup$ Thanks Andreas for this answer. I was aware of Ozawa's survey, and I have read your paper during my phd thesis. The point that I seem to be missing is why the lack of LP (or LLP) for $A$ is necessarily witnessed by a ucp map with values in $\prod_{\mathcal U} M_n$ (in my understanding if $A$ does not have the (L)LP, there exists a C*algebra B with an ideal $J$ and a ucp map $A\to B/J$ without (local) lifting). $\endgroup$
    – Mikael de la Salle
    Commented Jul 1, 2011 at 19:16
  • $\begingroup$ Mikael, the way I show that the universal group $C^{\ast}$-algebra does not have LLP is precisely by showing that there are no local lifts from some matrix ultraproduct. $\endgroup$
    – Andreas Thom
    Commented Jul 1, 2011 at 20:51
  • $\begingroup$ Ok, everything is clear now. I had completely forgotten Kirchberg's factorization property, but it was exactly what I was asking for. $\endgroup$
    – Mikael de la Salle
    Commented Jul 2, 2011 at 12:15
  • $\begingroup$ Andreas, are there any other examples of groups (except Abel's type groups from your paper) that are known to fail LLP? $\endgroup$
    – Kate Juschenko
    Commented Jul 4, 2011 at 20:16
  • $\begingroup$ Any hyperlinear, non-residually finite, Kazhdan group. There are only few examples, I guess. $\endgroup$
    – Andreas Thom
    Commented Jul 5, 2011 at 11:02

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