Background. I think I can prove something about a certain construction definition for Fourier algebras of discrete groups, under the assumption that the group is exact (well, really I use Yu's Property A). When the group has the AP of Haagerup and Kraus the result can be improved further. However, the improved conclusion would follow very quickly from known ideas if the following question had a positive answer! So I would rather find out if my argument is unnecessary before I start spending time expanding the result into a short paper.

Question. Let $M$ be a von Neumann algebra which is weakly exact in the sense of Kirchberg. Let $f:M_* \to M$ be a completely bounded map that satisfies the following two conditions: $\newcommand{\veps}{\varepsilon}$

(SCC) for every $\veps>0$ there is a finite-dimensional subspace $W\subset M$ such that $\Vert q_W\circ f\Vert_{cb} < \veps$, where $q_W:M\to M/W$ is the quotient map of operator spaces.

(GCC) for every $\veps>0$ there is a finite-codimension subspace $V\subset M_*$ such that $\Vert f\vert_V \Vert_{cb} < \veps$.

Does it follow that we can find finite-rank maps $f_n: M_* \to M$ such that $\Vert f_n-f\Vert_{cb} \to 0$?

Vague thoughts. By Ozawa's local characterization of weak exactness (see this paper for the precise statement), given $W$ as in the SCC condition we know that the inclusion $W \hookrightarrow M$ approximately factorizes through finite-dimensional matrix algebras, with ucp maps in the factorization. If we knew that there was actually an intermediate $W \subset M_k \subset M$ then I think the answer to my question is yes, because we just compose $f$ with an appropriate projection of $M$ onto $M_k$ (note that we don't use the GCC condition in this argument). However it is not clear to me right now if the approximate factorization given by Ozawa's characterization can be dragged back into $M$.

On the other hand, we are in a fairly special situation: we can interpret $f$ as an element of $M\overline{\otimes} M$, which may give us extra leverage when trying to build approximating elements in the algebraic tensor product $M\odot M$.


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