\section{Preliminaries}\label{Sect=Prelim}


$\delta(x \in X)$ is the function that is 1 if $x \in X$ and 0 otherwise. Inner products are linear in the left leg. For $\xi, \eta$ vectors in a Hilbert space $H$ we write $\omega_{\xi, \eta}(x) = \langle x \xi, \eta \rangle$.

Standard theory of von Neumann algebras can be found in \cite{TakI}, \cite{TakII}. For operator spaces we refer to \cite{EffrosRuan}, \cite{Pisier}. The abbreviation ucp stands for unital completely positive.

\subsection{Finite dimensional approximations and strong solidity} See \cite{BrownOzawa} for the following notions.

\begin{dfn}
 We say that a von Neumann algebra $M$ has the {\it W$^\ast$-CBAP} (W$^\ast$ completely bounded approximation property) if there exists a  net $(\Phi_i)_i$ of normal completely bounded finite rank maps $M \rightarrow M$ such that:
\begin{enumerate}
\item There exists $\Lambda \geq 1$ such that for all $i$ we have $\Vert \Phi_i \Vert_{cb} \leq \Lambda$;
\item For every $x \in M$ we have $\Phi_i(x) \rightarrow x$ $\sigma$-weakly.
\end{enumerate}
$\Lambda$ is called the Cowling-Haagerup constant.
If $\Lambda = 1$ then we say that $M$ has the {\it W$^\ast$-CCAP} (W$^\ast$ completely contractive approximation property).
\end{dfn}

For quantum groups of Kac type the  W$^\ast$-CBAP (resp.  W$^\ast$-CCAP) is equivalent to weak amenability of the quantum group (resp. weak amenability with Cowling-Haagerup constant 1). For the Haagerup property, see also \cite{CaspersSkalskiIMRN}, \cite{OkayasuTomatsu}.

\begin{dfn}
We say that a finite von Neumann algebra with faithful normal state $(M, \tau)$ has the {\it Haagerup property} if there exists a net $(\Phi_i)_i$ of normal ucp maps $M \rightarrow M$ such that $\tau \circ \Phi_i = \tau$, such that $\Phi_i$ is compact as a map $L_2(M, \tau) \rightarrow L_2(M, \tau)$ and such that for every $x \in M$ we have $\Phi_i(x) \rightarrow x$ strongly.
\end{dfn}

We further need the notions of solidity (see \cite{BrownOzawa}, \cite{OzawaActa}) and strong solidity as in the next definition.

\begin{dfn}
A finite von Neumann algebra $M$ is called {\it strongly solid} if for every diffuse amenable von Neumann subalgebra $P \subseteq M$ we have that ${\sf Nor}_M(P)''$ is amenable, where the normalizer is defined as
\[
{\sf Nor}_M(P) = \{ u \in M \mid u \textrm{ unitary s.t. } u P u^\ast = P \}.
\]
\end{dfn}
