Equivalence Classes of Linear Mappings on B(M)

Let M be a Hilbert C*-module over the C*-algebra A , B(M) the C*-algebra of all adjointable operators on M, L(B(M)) the algebra of all linear operators on B(M). For a property P on B(M) and ϕ1, ϕ2 ϵ L(B(M)) we say that ϕ1~P ϕ2, whenever for every T 2 B(M), ϕ1(T) has property P if and only if ϕ2(T) has this property. Each property P produces an equivalence relation on L(B(M)). If I denotes the identity map on B(M) it is clear that ϕ~PI means that ϕ preserves and reflects property P. We are going to study the equivalence classes with respect to different properties such as being A -Fredholm, semi-A -Fredholm, compact and generalized invertible.