I was recently studying this paper by Man-Duen Choi about inequalities for positive maps on C*-algebras. He demonstrates that
Let $\phi : \mathcal{A} \to \mathcal{B}$ be a unital positive linear map between the two C*-algebras $\mathcal{A}, \mathcal{B}$. Then
$$
\phi(A^* A) \ge \phi(A)^* \, \phi(A)
$$
and$$
\phi(A^* A) \ge \phi(A) \, \phi(A)^*
$$for every subnormal $A \in \mathcal{A}$.
He then conjectures that the same result might apply also for hyponormal opertators, i.e. operators such that $A^* A \ge A A^*$ (this conjecture is equivalent to the Woronowicz's conjecture).
I was wondering whether this conjecture has been better investigated or not. Also, is there a characterization of hyponormal operators?
Best Answer
If you scroll to the bottom of the PDF (e.g. the one in https://www.jstor.org/stable/24714007 ), it says:
So, the conjecture is false.