Completely positive matrix numerical index on matrix regular operator spaces

In the article, we compute the completely positive matrix numerical index of matrix regular operator spaces and show that they take values in the interval $[\frac {1}{2},1]$. Moreover, we show that the dual of a unital operator system has the completely positive matrix numerical index $\frac {1}{2}$ if its dimension is greater than $1$. Furthermore, both $S_p(\mathbf {H})$ and $L_p(\mathbf {M})$ have the completely positive matrix numerical index $2^{-\frac {1}{p}}$ if their dimensions are greater than $1$, where $p\in [1, + \infty )$, $\mathbf {H}$ is a Hilbert space and $\mathbf {M}$ is a finite von Neumann algebra.