Unital completely positive maps and their operator systems

A vector subspace $\text{cls}$ of ${\text{IM}}_n(\text{IC})$ is called unital operator system if $x\in \text{cls}$ if and only if $x^{\ast }\in \text{cls}$ and the identity operator $I_n \in \cls$,where$n$isanyfixedpositiveinteger\mathrm{.}Let$C^(\cls)$bethe$C^$sub-algebraof$\IM_n(\IC)$generatedbytheoperatorsystem$\cls$. We prove that a unital complete order isomorphism $\text{cli}:\text{cls}\text{raro}{\text{cls}}^{\mathrm{\prime }}$ between two such operator systems $\text{cls}$ and ${\text{cls}}^{\mathrm{\prime }}$ of ${\text{IM}}_n(\text{IC})$ has a unique extension to a $C^{\ast }$-isomorphism $\text{cli}:C^{\ast }(\text{cls})\text{raro}{C}^{\ast }({\text{cls}}^{\mathrm{\prime }})$ if and only if $\text{cls}$ and ${\text{cls}}^{\mathrm{\prime }}$ are having equal set of complete ranks. The operator system $\cls = \mbox{span}{v_iv_j^*:1 \le i,j \le d }$ is uniquely determined for a unital completely positive map $\tau (x)={\sum }_{1\le k\le d}{v}_{k}x{v}_k^{\ast }$ of index $d \ge 1$. As an application of our main result, we explore this correspondence and characterize up to co-cycle conjugacy all extreme points in the convex set of unital completely positive maps on ${\text{IM}}_n(\text{IC})$. Using the main result, we also characterize up to co-cycle conjugacy all extreme elements in the convex set of normalized trace preserving unital completely positive maps on ${\text{IM}}_n(\text{IC})$.