I am interested in the $\ell^1$ analogue of direct sums for Operator spaces, e.g. Operator Space Dictionary. Briefly, and operator space is either a concrete subspace of $B(H)$, the operators on a Hilbert space, or equivalently a normed space $E$ with norms on the matrix spaces $M_n(E)$, satisfying Ruan's axioms. The analogue of the $\ell^\infty$ direct sum can be stated as $M_n(E\oplus F) = M_n(E)\oplus M_n(F)$, which is compatible with the diagonal embedding $B(H)\oplus B(K)\rightarrow B(H\oplus K)$.

As a Banach space, the dual of $E\oplus F$ is $E^*\oplus_1 F^*$ the $\ell^1$ sum. We use this idea to define an operator space structure on $E\oplus_1 F$ be embedding it into $(E^*\oplus F^*)^*$. One can then show, in increasing order of difficulty (IMHO):

- $E\oplus_1 F$ has the universal property that if $u:E\rightarrow X, v:F\rightarrow X$ are complete contractions, then $u\oplus v: E\oplus_1 F\rightarrow X$ is a complete contraction.
- $E\rightarrow E\oplus_1 F$ (and for $F$) is a complete isometry, and $E\oplus_1 F\rightarrow E$ (and for $F$) is a complete quotient map.
- $(E\oplus_1 F)^* = E^* \oplus F^*$
- $(E\oplus F)^* = E^*\oplus_1 F^*$. This in particular had me stumped for a bit; I needed to use the fact that $M_n(E)^{**} = M_n(E^{**})$.

I am struggling to find references. Points (2) and (3) above are covered in these notes of Blecher. The books by Paulsen and Effros & Ruan seem not to consider $\oplus_1$. The book of Blecher & Le Merdy leaves all the proofs to the reader. The original paper of Blecher also does not give details for points (3) and (4). Pisier's book instead defines $\oplus_1$ using point (1) (the universal property) but leaves (3) and (4) as exercises.

I would like a reference to a clear proof of (3) and (4).

Alternatively, am I missing some genuinely "easy" argument?

In particular, just using the universal properties, I can show that $(E\oplus_1 F)^* = E^*\oplus F^*$. How can one give an analogous proof that $(E\oplus F)^* = E^*\oplus_1 F^*$?