Is $\mathcal{B}(H)$ amenable for any Hilbert space H? This question is motivated by the following, given $M$ an amenable Von Neumann algebra and a Hilbert space H, is $M \overline{\otimes} \mathcal{B}(H)$ amenable? 
I have two main questions:
1) Given any finite dimensional Hilbert space H, is $\mathcal{B}(H)$ amenable?
2) If the above is true, after proving that given two amenable VNA's M,N then $M \overline{\otimes} N$ is amenable, then can you conclude by approximation that $M \overline{\otimes} \mathcal{B}(H)$ is amenable? 
 A: A working definition of "amenable" is AFD. This answers  your questions trivially:


*

*$B(H)$ is AFD if $H$ is separable (it is the wot limit of the "left upper corners"). 

*When $H$ is not separable, $B(H)$ cannot be AFD, because it is not separable as a von Neumann algebra. In particular, $M\bar\otimes B(H)$ is not amenable when $H$ is not separable and $M$ is any von Neumann algebra. 

*any finite-dimensional von Neumann algebra is obviously approximately finite-dimensional.

*If $M$ and $N$ are AFD, then $M\bar\otimes N$ is AFD, by tensoring the corresponding increasing sequences of finite-dimensional subalgebras. 

For a proof that $B(H)$ is amenable when $H$ is separable, let $\{p_n\}$ be an increasing sequence of projections with $p_n\nearrow I$. So $p_n\to I$ sot. For any $x\in B(H)$, $\xi\in H$, 
$$
\langle p_nxp_n\xi,\xi\rangle\to\langle x\xi,\xi\rangle,
$$
so
$$
B(H)=\overline{\bigcup_n p_nB(H)p_n},
$$
and $p_nB(H)p_n\simeq M_{k_n}(\mathbb C)$ where $k_n=\text{Tr}\,(p_n)$. 
