The Moore–Penrose inverses of products and differences of projections in a C∗-algebra

For two given projections p and q in a C∗-algebra, we investigate how to express Moore–Penrose inverses of products pq and differences p-q and pq-qp. Moreover, it is shown that pq-qp is Moore–Penrose invertible if and only if pq and p-q are Moore–Penrose invertible. In addition, some related topics are considered.