Breuer–Fredholm operator - Imported Knowlpedia Content

Definition

Let $\mathcal M\subseteq B(H)$ be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace, and let $\mathcal K(\mathcal M,\tau)$ be its tau-compact ideal. A bounded operator $T\in\mathcal M$ is Breuer–Fredholm if its image in the quotient $C^*$ -algebra

\mathcal M/\mathcal K(\mathcal M,\tau)

is invertible. Equivalently, there is $S\in\mathcal M$ such that $1-ST$ and $1-TS$ are tau-compact. This is the semifinite analogue of a Fredholm operator, with tau-compactness replacing ordinary compactness.

Kernel criterion and index

A bounded $T\in\mathcal M$ is Breuer–Fredholm exactly when $\tau(P_{\ker T})<\infty$ and there is a projection $p\in\mathcal M$ such that $\tau(1-p)<\infty$ and $pH\subseteq\operatorname{Ran}(T)$ . In that case the cokernel projection $P_{\ker T^*}$ also has finite trace, and the Breuer index is

\operatorname{Ind}_\tau(T) =\tau(P_{\ker T})-\tau(P_{\ker T^*}).

Unlike the classical Fredholm index, this value may be any real number because the trace of a projection in a semifinite algebra need not be integral. The criterion and index originate in Breuer, §§1–3.

Stability and examples

The Breuer–Fredholm operators form an open subset of $\mathcal M$ , and their index is locally constant. Adding an element of $\mathcal K(\mathcal M,\tau)$ preserves Breuer–Fredholmness and the index. For $\mathcal M=B(H)$ with the ordinary trace, tau-compact operators are the usual compact operators, so the definition and index reduce to classical Fredholm theory Breuer, §§1–3.

If $\tau(1)<\infty$ , then $\mathcal K(\mathcal M,\tau)=\mathcal M$ , the quotient is zero, and the notion becomes degenerate: every operator is Breuer–Fredholm under the standard zero-quotient convention. Applications generally concern traces for which the relative compact ideal is proper.

Unbounded version

A closed densely defined self-adjoint operator affiliated with $\mathcal M$ is called Breuer–Fredholm when an appropriate bounded transform, such as

T(1+T^2)^{-1/2},

is Breuer–Fredholm. Equivalent formulations use invertibility modulo tau-compact operators for the resolvent or spectral projections near zero. This unbounded form is the one used in semifinite spectral triples and spectral flow Carey–Phillips–Rennie–Sukochev, §2.

References

M. Breuer, “Fredholm theories in von Neumann algebras. I,” Mathematische Annalen 178 (1968), 243–254. DOI record. Relevant: §§1–3 on relative compactness, Fredholm operators, and the trace index.
A. L. Carey, J. Phillips, A. Rennie, and F. A. Sukochev, “The local index formula in semifinite von Neumann algebras I: Spectral flow,” Advances in Mathematics 202 (2006), 451–516. Preprint record. Relevant: §2 on semifinite Fredholm theory and spectral flow.