Let $u: H \to H$ be an isometric operator on a Hilbert space. Let $\ast$ be an involution.
I was wondering if $u^\ast$ is also an isometry. I tried to prove it but didn't quite manage. Then I started to wonder whether it was false. But to come up with a counter example I would have to come up with a bunch of isometries which is already difficult for me. The only Hilbert space I can think of off the top of my head is $\ell^2$. Then the identity operator is an isometry but that's not interesting. The shift operators and the projection operators come to mind next but this is also unhelpful since both cannot be isometries since they are not invertible.
Can someone give an example of an isometry with non isometric adjoint?