positive functionals on the full group C*-algebra

Question

It it true that every positive linear functional on the full group C*-algebra is Completely positive? I am reading Brown and Ozawa's book and they seem to use this at some point, yet I'm not sure how to show it.

Answer 1

That's true for any C$^*$-algebra (for any operator system, actually): any positive linear functional is completely positive.

Assume $A\subset B(H)$. If $x\in M_n(A)$ and $f:A\to\mathbb C$ is positive, then for any $\xi\in \mathbb C^n$ we have $$ \langle f^{(n)}(x)\xi,\xi\rangle =\sum_{k,j=1}^n f(x_{kj})\xi_j\overline{\xi_k}=f((y^*xy)_{11})\geq0, $$ where $y\in M_n(\mathbb C)$ is given by $$ y=\begin{bmatrix} \xi_1&0&\cdots&0\\ \xi_2&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots\\ \xi_n&0&\cdots&0\end{bmatrix} $$