The dual

The dual⁹ of a matricially normed space $X$ is defined as $X^*=CB(X,{\mathbb{C}})$ [Ble92a].¹⁰Its first matrix level is the dual of the first matrix level of $X$: $M_1(X^*)=(M_1(X))^*$.

The canonical embedding $X \hookrightarrow X^{**}$ is completely isometric [BP91, Thm. 2.11].

Footnotes

...dual⁹

In the literature, this dual was originally called standard dual [Ble92a].

...Blecher92b.¹⁰

The norm of a matricially normed space $X$ is given by the unit ball $\mathrm{Ball}X\subset M(X)$. Here, $\mathrm{Ball}X^*=\left\{\Phi:X\to M_n\;\vert\;n\in {\mathbb{N}},\;\Phi\mbox{ completely contractive}\right\}$.