  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.
