$L(\bigoplus_{i\in I} H_i)\cong \prod_{i\in I}L(H_i)$ as Banach spaces? Let $\{H_i:i\in I\}$ be a system of complex Hilbert spaces, $L(\bigoplus_{i\in I} H_i)$ the set of bounded linear maps $T:\bigoplus_{i\in I} H_i\to \bigoplus_{i\in I} H_i$, equipped with the operator norm. 
Let $\prod_{i\in I}L(H_i)$ the direct product of linear maps endowed with the norm $\|(T_i)_{i\in I}\|=sup_{i\in I}\|T_i\|$.
Is $$L(\bigoplus_{i\in I} H_i)\cong \prod_{i\in I}L(H_i)$$ as banach spaces?
The map which comes to my mind constructed as follows: Is $$\iota_i: H_i\to \bigoplus_{i\in I} H_i$$ the canonical injection, we define for each $i\in I$ a map $$\varphi_i:L(\bigoplus_{i\in I} H_i)\to L(H_i),\; \alpha\mapsto \alpha\circ \iota_i$$ Then extend it to a map $$L(\bigoplus_{i\in I} H_i)\to \prod_{i\in I}L(H_i).$$ But I'm not sure if it's possible here or if the construction makes sense in this context. 
 A: Note that isomorphically there is only one Hilbert space in each dimension. Moreover, $$d( \bigoplus_{i\in I} H_i)=\sum_{i\in I}  d(H_i),$$
where $d$ stands for the dimension, that is, cardinality of any orthonormal basis. Thus if all $H_i$ have the same dimension $\lambda$ say, then $d(\bigoplus_{i\in I} H_i) = \max\{\lambda, |I|\}$. Presumably, by $\prod_{i\in I} E_i$ you mean the $\ell_\infty(I)$-direct sum of $E_i$ ($i\in I$). 
In general then the answer is no. 
Indeed, if each $H_i$ is two-dimensional and $I$ is infinite, then $(\bigoplus_{i\in I} L(H_i))_{\ell_\infty(I)}$ is Banach-space isomorphic to $\ell_\infty(I)$ but $L( (\bigoplus_{i\in I}H_i)_{\ell_2(I)})$ is the space of operators on an infinite dimensional Hilbert space $H$ which is known to be not isomorphic to $\ell_\infty(I)$ for any infinite index set $X$. To see this, note that $L(H)$ contains a complemented subspace isomorphic to $\ell_2$. Pick a non-zero vector $x\in H$ and consider 
$$H^\prime=\{x\otimes y\colon\quad y\in H\}.$$
Then $H^\prime$ is complemented and $H\cong H^\prime$. As $\ell_\infty$ has the Dunford–Pettis property, it contains no infinite-dimensional, complemented reflexive subspaces.
In some cases the answer is yes. For example, it is an unpublished result of Lindenstrauss and Haagerup which says that
$$L(\ell_2)\quad \text{and }\quad(\bigoplus_{n\in \mathbb{N}} L(\ell_2^n))_{\ell_\infty}$$
are isomorphic as Banach spaces. The proof is highly non-constructive and uses the Pełczyński decomposition method. It is explained in 

E. Christensen and A. Sinclair, Completely bounded isomorphisms of injective von Neumann algebras, Proc. Edingurgh Math. Soc. 32 (1989), 317–327.

