Schwarz inequality for unital positive maps on C*-algebras

Question

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?

Answer 1

If you scroll to the bottom of the PDF (e.g. the one in https://www.jstor.org/stable/24714007 ), it says:

Added in proofs. E. Kirchberg has recently shown by a non-constructive proof that there exists a counter-example to Woronowicz's Conjecture. Hence, Conjecture 3.4 is also false by virtue of Proposition 3

So, the conjecture is false.