Invertible idempotent in a C-star algebra question

Question

Let $J$ be an idempotent element in a unital $C^*$ algebra. Why is $I+(J-J^*)(J^*-J)$ invertible?

I have been trying to show that $\|(J-J^*)(J^*-J)\|<1$, but I could not do this.

Answer 1

Well, I do not see why the idempotence of $ J $ is required.

Proof: Let $ x = J - J^{*} $. Then $ x^{*} = J^{*} - J $. Your expression thus becomes $ \mathbf{1} + x x^{*} $. According to a basic C*-algebraic theorem, the spectrum of $ x x^{*} $ is contained in $ [0,\infty) $ (this theorem is proven on Page 5 of Marc Rieffel's Spring 2008 notes on C*-algebras). It follows readily that the spectrum of $ \mathbf{1} + x x^{*} $ is contained in $ [1,\infty) $. As $ \mathbf{1} + x x^{*} $ is self-adjoint, apply the Continuous Functional Calculus to conclude that $ \mathbf{1} + x x^{*} $ is invertible.