Tau-compact operator - Imported Knowlpedia Content

Definition

Let $\mathcal M$ be a semifinite von Neumann algebra with a faithful normal semifinite trace $\tau$ . The tau-compact ideal is

\mathcal K(\mathcal M,\tau) =\overline{\operatorname{span}}^{\|\cdot\|} \{xey:x,y\in\mathcal M,\ e=e^*=e^2,\ \tau(e)<\infty\}.

An element of $\mathcal K(\mathcal M,\tau)$ is a tau-compact operator. Thus tau-compactness is norm-closed two-sided ideal membership inside $\mathcal M$ , with finite-trace projections playing the role of finite-rank projections. It depends on both the represented algebra and the chosen trace, not only on the underlying Hilbert-space operator.

Equivalent characterizations

For $x\in\mathcal M$ , tau-compactness is equivalent to decay of its generalized singular-value function:

\mu_t(x)\longrightarrow0\qquad(t\to\infty).

Equivalently, $x$ can be approximated in operator norm by elements supported on finite-trace projections. This is the relative compact ideal used in Breuer's Fredholm theory; see Breuer, §§1–2.

Examples and boundary cases

For $\mathcal M=B(H)$ with the usual trace, $\mathcal K(\mathcal M,\tau)$ is the ordinary compact-operator ideal. If $\tau(1)<\infty$ , then the identity itself has finite trace and $\mathcal K(\mathcal M,\tau)=\mathcal M$ . Hence tau-compactness may be much weaker than Hilbert-space compactness.

Role in semifinite geometry

An unbounded affiliated operator $D$ is said to have tau-compact resolvent when a bounded transform such as $(1+D^2)^{-1/2}$ lies in $\mathcal K(\mathcal M,\tau)$ . This replaces ordinary compact resolvent in semifinite spectral triples and underlies the associated Breuer–Fredholm theory.

References

M. Breuer, “Fredholm theories in von Neumann algebras. I,” Mathematische Annalen 178 (1968), 243–254. DOI record. Relevant: the relative compact ideal and Fredholm theory.
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, semifinite compactness and spectral triples.