Projection operators nearly orthogonal to their symmetries

For any order 2 automorphism α of a C*-algebra A (a symmetry of A), we prove that for each projection e   such that ‖eα(e)‖≤920, there exists a projection q   with qα(q)=0qα(q)=0 satisfying the norm estimate‖e−q‖≤12‖eα(e)‖+4‖eα(e)‖2. In other words, if e   is a projection that is “nearly orthogonal” to its symmetry α(e)α(e) in the sense that the norm ‖eα(e)‖‖eα(e)‖ is no more than 920, then e can be approximated by a projection q   that is exactly orthogonal to its symmetry in a fairly optimal fashion. (Optimal in the sense that the first term in the estimate satisfies 12‖eα(e)‖≤‖e−q‖ for any such q.)