Let $A$ be a unital C*-algebra. Is there a unital positive self-map $F:A\to A$ which is invertible (i.e. injective and surjective) but not a $*$-automorphism? If yes, how does appear its Gelfand-Naimark-Segal covariant representation associated to a state $\varphi$ on $A$ which is invariant under $F$? Are there examples of such maps which are involutive (i.e. such that $F^2=id_A$, or equivalently $F=F^{-1}$?
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3$\begingroup$ There are many unital positive bijection of the form $a\mapsto S^*aS+(1-S^*S)\varphi(a)$, where $S$ invertible and contractive. However, if you are asking unital positive $F$ with positive inverse, then $F$ is a Jordan isomorphism. $\endgroup$– Narutaka OZAWACommented Feb 26 at 0:07
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$\begingroup$ Many thanks, I wanted also to exclude anti-automorphisms. It seems to me that, for such maps, the GNS covariant representation doesn't exist in general $\endgroup$– fidaleoCommented Feb 26 at 8:48
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$\begingroup$ It is inconsiderate to cross-post without indicating so: math.stackexchange.com/questions/4870592/… $\endgroup$– JP McCarthyCommented Feb 26 at 10:39
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