Examples of not completely bounded maps Let $\phi:\mathcal{A}\longrightarrow\mathcal{B}$ be a bounded map between $C^*$ algebras. $\phi$ is said to be completely bounded if the natural extension map
\begin{eqnarray}
\phi_n:M_n(\mathcal{A})&\longrightarrow & M_n(\mathcal{B})\\
((a_{i,j}))&\longmapsto & ((\phi(a_{ij}))
\end{eqnarray}
is also bounded for all $n$. ($M_n(\mathcal{A})$ denotes $n\times n$ matrices whose entries are elements of $\mathcal{A}$.) This bound defines a norm as well which is known as completely bounded norm on the the set of maps. 
The standard example of a 'not' completely bounded bounded map  is transpose. I could not construct any other example which does not involve transpose. Unfortunately I could not locate any other example from the literature. Please help. 
 A: Let $H$ be a Hilbert space. I will use the following standard notations for quantizations


*

*$MAX(H)$ - maximal quantization of $H$.

*$MIN(H)$ - minimal quantization of $H$.

*$C(H)$ - column quantization of $H$.

*$R(H)$ - row quantization of $H$.

*$OH(H)$ - operator quatization of $H$.


and spaces of operators


*

*$B(H)$ - bounded operators on $H$.

*$S_p(H)$ - Schatten $p$-class operators on $H$.

*$CB(X,Y)$ - completely bounded operators between operators spaces $X$ and $Y$.


Then we have the following table of spaces of completely bounded operators between different quantizations of $H$ up to isometric ($\simeq_1$) or usual ($\simeq$) isomorphism:
$$
\begin{array}{cccccc}
CB(\downarrow,\rightarrow) & MIN(H) & C(H) & OH(H) & R(H) & MAX(H)\\
MIN(H) & \simeq_1 B(H) & \simeq_1 S_2(H) & \simeq   S_2(H) & \simeq_1 S_2(H) & \simeq S_1(H)\\
C(H)   & \simeq_1 B(H) & \simeq_1 B(H)   & \simeq_1 S_4(H) & \simeq_1 S_2(H) & \simeq_1 S_2(H)\\
OH(H)  & \simeq_1 B(H) & \simeq_1 S_4(H) & \simeq_1 B(H)   & \simeq_1 S_4(H) & \simeq   S_2(H)\\
R(H)   & \simeq_1 B(H) & \simeq_1 S_2(H) & \simeq_1 S_4(H) & \simeq_1 B(H)   & \simeq_1 S_2(H)\\
MAX(H) & \simeq_1 B(H) & \simeq_1 B(H)   & \simeq_1 B(H)   & \simeq_1 B(H)   & \simeq_1 B(H)\\
\end{array}
$$
This characterization was taken from this page.
From this table we see that for $13$ of $25$ cases the spaces of completery bounded operators are proper subspaces of $B(H)$. Thus we have quite a lot of non-completely bounded operators.
A: (The original version of this answer was in response to a slightly more general question.)

I'm not sure if the following is what you are really looking for, but you could always take an arbitrary Banach space $E$ and consider the identity map $E\to E$; this is almost never completely bounded from ${\rm MIN}(E)$ to ${\rm MAX(E)}$.
update: see also Jon Bannon's comment and link here https://mathoverflow.net/questions/86550/positive-but-not-completely-positive/88309#88309
