This is an exercise from Wegge Olsen's K-theory and C*-algebras. In the exercise 5.A (b) he states that the product of an isometry by a partial isometry in a C*-algebra is a partial isometry. Taking $u$ a partial isometry (this means $uu^*u=u$) and $v$ isometry (this means $v^*v=1$) I managed to show that $vu$ is a partial isometry. I also tried to show that $uv$ is a partial isometry without success, is this even true? Doing some research, I found this question that states that $vv^*$ and $u^*u$ should commute for this to happen, I don't think having $v$ be an isometry is a sufficient condition for this.
There's a big chance that the book is only asking me to show that $vu$ is the partial isometry but I'd like to confirm that.