Tripotents in Algebras: Invertibility and Hyponormality

Let A be a unital algebra over complex field C, I be the unit of A. An element A is an element of A is called tripotent if A(3) = A. Let A(tri) = {A is an element of A : A(3) = A}. We show that A is an element of A(tri) if and only if I +/- A - A(2) is an element of A(tri). We study invertibility properties of elements I + lambda A with A is an element of A(tri) and lambda is an element of C \{-1,1}. Let X be a Banach space with the approximation property and A, B is an element of B(X)(tri). If A- B is a nuclear operator then tr( A - B) is an element of Z. We show that if H is a Hilbert space and an operator A is an element of B(H)(tri) is hyponormal or cohyponormal then A = A*. We also prove that every A is an element of B(H)(tri) similar to a Hermitian tripotent.