example of positive but not completely positive operator

Question

I was looking for some example of a positive operator which is not completely positive on a banach algebra. if I consider my banach algebra to be $\text{M}_n(\mathbb{C})$ of matrices over complex numbers. And consider my operator as $A$ going to transpose of $A$. So this operator is coming out to be positive. Is this map completely positive as well or its just positive?

Answer 1

Transpose is the obvious example. That it is not completely positive, can be quickly checked by using Choi–Jamiołkowski isomorphism.

Following Woronowicz (who used techniques of St{\o}rmer) you can show that any positive map $\phi:M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ can be decomposed as a completely positive $+$ transpose of a completely positive map.

If you consider $M_n(\mathbb{C})$ for $n\geq3$, then there are a few examples which can not be written in the above form. The first and the most famous example was constructed by Choi. For a quick review and a list of such maps, see the paper by Chruscinski and Kossakowski.

The problem is more complicated if you consider sub-algebras of $\mathcal{B}(\mathbb{C}^n)$ or when the $C^*$ algebra is infinite dimensional.

Answer 2

Transposition is not completely positive. For the proof see example 3.6 page 24 of this notes.