composition of bijective map and expolential map of an operator

Question

Let $B(H)_{pi}$ be the set of all positive invertible operators on $H$,let $\phi$ be a bijective map from $B(H)_{pi}$ to $B(H)_{pi}$,$A\in B(H)_{pi}$,can we have $\phi(e^A)=e^{\phi (A)}$?if $\phi$ is a $*$-homomorphism,the above conclusion holds.If $\phi$ does not preserve $*$-operation,is the conclusion also correct?

Answer 1

The map $\phi:B(H)_{pi}\to B(H)_{pi}$ given by $x\mapsto x^{-1}$ is continuous and bijective, but $$\phi(e^2)=e^{-2}\neq e^{1/2}=e^{\phi(2)}.$$