Maps on the positive definite cone of a C⁎-algebra preserving certain quasi-entropies

Abstract

Abstract. We describe the structure of those bijective maps on the cone of all positive invertible elements of a C * -algebra with a normalized faithful trace which preserve certain kinds of quasi-entropy. It is shown that essentially any such map is equal to a Jordan *-isomorphism of the underlying algebra multiplied by a central positive invertible element.

Quantum Rényi relative entropies on density spaces of C⁎-algebras: Their symmetries and their essential difference

“…In Theorem 1 in [11] we described the structure of all bijective maps between the positive definite cones of C * -algebras which preserve the Umegaki relative entropy. (In fact, there we considered an even more general numerical quantity, the so-called quasi-entropy that involves a parameter, namely an invertible element of the underlying algebra which is the identity in our present case.)…”

“…The proof of that result is very much different from the proof of our Theorem 1 here. One can easily see that the method of the proof of Corollary 2 above can be used to derive the following result from Theorem 1 in [11] on maps respecting the Umegaki relative entropy between density spaces of C * -algebras. The structure of maps preserving the Belavkin-Staszewski relative entropy is again the same as we can see in the following theorem.…”

“…However, one can easily see that those restrictions in [11] are not crucial, and we could apply an appropriately modified version of the result there to prove Theorem 4. Let us also mention that the approach in [11] was completely different from what we follow here, not relied on structural theorems of Thompson isometries and related maps. However, we can give also a direct argument following the general approach of the present paper.…”

Maps on positive definite operators preserving the quantum $$\chi _\alpha ^2$$ χ α 2 -divergence

“…We note that the computation rule (9) Tr R X RY = TrR X · Tr RY can be verified by easy computation for any operators X , Y ∈ L (H ) and for any rank one projection R ∈ P 1 (H ). This will be used in the sequel several times.…”

“…In [18] the same conclusion was obtained for the same kind of preservers which are bijective and defined not on the state space but on the whole set L + (H ) of positive semidefinite operators. (We also remark that in the very recent paper [9] we have made some steps towards the description of quasi-entropy preservers on positive definite cones in the setting of C * -algebras but the level of generality of the considered quasi-entropies falls far from what we could consider sufficient. )…”