Generalized symmetric *-rings and Jacobson's Lemma for Moore-Penrose inverse

It is well known as Jacobson's Lemma that 1 - ba is invertible in a ring if so is 1 ab. Moreover, if c = (1 - ab)(-1), then (1 - ba)(-1) = 1 + bca. However, the analogous statement for Moore-Penrose inverse in a *-ring is not true in general. Note that Jacobson's Lemma for Moore Penrose inverse holds true in a symmetric *-ring. In this paper, we study symmetric *-rings and introduce the notion of a generalized symmetric *-ring. A *-ring R is called generalized symmetric if 1 (u* - u)(2) is invertible for all units u in R. When 1 ab is Moore Penrose invertible in such a ring, we provide sufficient and necessary conditions under which 1 ba has a Moore Penrose inverse (1 - ba)(dagger) and give a formula for (1 - ba)(dagger).