Invertible positive maps which are not automorphisms

Question

Let be a unital C*-algebra. Is there a unital positive self-map :→ which is invertible (i.e. injective and surjective) but neither a ∗-automorphism, nor a *-antiautomorphism (i.e. a *-isomorphism with the opposite algebra)? If yes, how does appear its Gelfand-Naimark-Segal covariant representation associated to a state on which is invariant under ? Are there examples of such maps which are involutive (i.e. such that ^2=_, or equivalently =^{−1})?

Answer 1

$A = \mathbb{M}_2(\mathbb{C})$ and $F(X) = X^T$.