p-Summable Commutators in Dimension d
By W. Arveson. 2003
80 citations recorded by the source corpus.
We show that many invariant subspaces M for d-shifts (S_1,...,S_d) of finite rank have the property that the projection P onto M almost commutes with the S_k in the sense that the commutators PS_k - S_kP belong to the Schatten-von Neumann class L^p for every p > d. In such cases the d-tuple of operators (T_1,...,T_d) obtained by compressing (S_1,...,S_d) to the orthocomplement of M generates a *-algebra whose commutator ideal is contained in L^p, p > d. It follows that the C*-algebra generated by T_1,...,T_d is commutative modulo compact operators, the associated Dirac operator is Fredholm, and the index formula for the curvature invariant is stable under compact perturbations and homotopy for this restricted class of d-contractions. We conjecture that the latter conclusions persist under much more general circumstances.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/a43d032c7b36e684942550593d70284e7915b711
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In Applied Mathematics
In Mathematical Physics
Citations in the corpus
- Quotients of standard Hilbert modules (2005, unknown): “While the proof is a straightforward variation of the proof of Proposition 4.1 of [Arv04], we present the details for completeness.”
- A new kind of index theorem (2005, background): “In [ 5 ] he established this result for submodules generated by monomials.”
- Essentially Reductive Hilbert Modules II (2006, result): “In [ 4 ] Arveson showed that submodules of H 2 m ⊗ C k , 1 ≤ k 4 ] Arveson showed in this case that all submodules generated by monomials or, equiva-”