In groups and rings, we usually define inner automorphisms to be those of the form $\phi(x) = axa^{-1}$ for some invertible element $a$.
But in operator algebras, an inner automorphism is almost always defined as something of the form $\phi(x) = uxu^* = uxu^{-1}$ for some unitary element $u$.
Is there a good reason why the definition in an operator algebra setting is different/unexpected? What do we call an automorphism in a $C^*$-algebra when it is a conjugation by some non-unitary invertible element?
If $A$ is a $*$-algebra, then by definition an automorphism $\phi: A \rightarrow A$ commutes with the $*$, that is, $\phi(x)^{*} = \phi(x^{*})$.
If we define $\phi$ to be conjugation by some unitary element of $A$, then this property holds automatically.
If we conjugate by a non-unitary element then this property need not hold.
I am guessing that an inner automorphism commutes with the $*$-operation if and only if it is given by conjugation with a unitary element, but I don't see how to prove this right now.