Semi-orthogonal projections in unital C∗-algebras

Let A sem = { A ∈ A : Re A = A ∗ A } be the set of all semiorthogonal projections of the unital C ∗ -algebra A , I be the identity of A . The formula U = 2 A − I ( A ∈ A sem ) defines a bijection between the set A sem and the set of all isometries from A . For any natural number n ≥ 2, there exists a non-commutative polynomial of degree n that yields a semi-orthogonal projection when substituted for an arbitrary set A 1 , . . . , A n ∈ A sem . Each element A ∈ A sem is hyponormal and lies in the unit ball of the C ∗ -algebra A . If A ∈ A sem, then A 2 is hyponormal. If A, A 2 ∈ A sem, then A is a projection. If A ∈ A sem and A = A n for some n ∈ N, n ≥ 2, then A is a normal element, and A is a projection for n = 2.