A note about a theorem of R. Harte

Let A and B be unital Banach algebras and T : A ? B be a unital continuous homomorphism. Put J = Ker T. Let FredT (A) = {x ? A|T (x) is invertible in B} and Fred0T(A) = {x + k|x is invertible in A, k ? J}. In this note, we prove that if T has Property (F), then FredT (A) ? GL(A) = Fred0T (A) iff ltsr (J) = 1. .