Maps preserving semi-Fredholm operators on Hilbert C*-modules

Let M be a Hilbert C*-module over the C*-algebra A, and L-A(M) the C*-algebra of all adjointable operators on M. Let phi: L-A(M) -> L-A(M) be a unital linear map preserving semi-A-Fredholm operators in both directions and assume that phi is surjective up to compact operators. We will show that phi preserves compact operators and in certain cases, the induced map (phi) over tilde : L-A(M)/K-A(M) -> L-A(M)/K-A(M) is an automorphism or an anti-automorphism. (C) 2009 Elsevier Inc. All rights reserved.