Lifting of a ucp map with values in a von Neumann algebra ultraproduct of matrix algebras

2011-06-29T20:05:09Z

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).

Answer by Andreas Thom for Lifting of a ucp map with values in a von Neumann algebra ultraproduct of matrix algebras

2011-07-01T18:15:41Z

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.

Lifting of a ucp map with values in a von Neumann algebra ultraproduct of matrix algebras - MathOverflow

Lifting of a ucp map with values in a von Neumann algebra ultraproduct of matrix algebras

Answer by Andreas Thom for Lifting of a ucp map with values in a von Neumann algebra ultraproduct of matrix algebras