Lifting invertibles in von Neumann algebras

Abstract

Given B ( H ) \mathcal {B}(\mathcal {H}) , the algebra of bounded operators on a separable Hilbert space H \mathcal {H} , and K \mathcal {K} , the ideal of compact operators, it is a well-known fact that T T in B ( H ) \mathcal {B}(\mathcal {H}) is a Fredholm operator if and only if π ( T ) \pi (T) is invertible in B ( H ) \mathcal {B}(\mathcal {H}) where π \pi is the canonical quotient map. A natural question arises: When can a Fredholm operator be perturbed by a compact operator to obtain an invertible operator? Equivalently, when does the invertible element π ( T ) \pi (T) lift to an invertible operator? The answer is well known: T T may be perturbed by a compact operator to obtain an invertible operator if and only if the Fredholm Index of T T is 0. In this case, the perturbation may be made as small in norm as we wish. Using the generalized Fredholm index for a von Neumann algebra developed by C. L. Olsen [3], the following generalization is obtained: let A \mathfrak {A} be a von Neumann algebra with norm closed ideal F \mathfrak {F} and canonical quotient map π \pi . Let T ∈ A T \in \mathfrak {A} be such that π ( T ) \pi (T) is invertible. Then there exists K ∈ F K \in \mathfrak {F} such that T + K T + K is invertible if and only if ind ⁡ ( T ) = 0 \operatorname {ind} (T) = 0 .