Index L, R, operators associated with D2, 20 Rs,t,u(λ), see resolvent functions, 57 T (n) , 20 [·, ·]±, graded commutator, 56 A, a ∗-algebra, 33 B1(D, p), algebra of integrable elements , 13 Bk(D, 1 p), smooth version of B1(D, p), 21 B 2 (D, p), algebra of square integrable elements, 8 Bk(D, 2 p), smooth version of B2(D, p), 21 D, a self-adjoint operator, 7 Du = D|D|−u, the deformation from D to F = D|D|−1, 57 H, a Hilbert space, 20 K(N , τ), KN , the τ-compact operators in N , 33 Lp, ˜p, Schatten ideals, 9 Pn, seminorms on B1(D, p), 13, 14 Pn,l, seminorms on Bk(D, 1 p) and B∞(D, 1 p), 21 Φr m,t , see transgression cochain, 58 Ψr M,u , auxiliary cochain, 59 Qn, seminorms on B2(D, p), 8 α(k), combinatorial factors in index formula, 56 •, parity notation, 55 N , a semifinite von Neumann algebra, 7 δ, δ , derivations associated with D, 20 δ-ϕ-topology, 46 δ-topology, 45 ηm, constant in the definition of the resolvent cocycle, 58 λ-trick, 61 . . . m,r,s,t , expectation in the resolvent cocycle, 58 . . . m,r,s,t , expectation in the transgression cochain, 58 Z2-grading, 34 O m , functions holomorphic for (z) (1 − m)/2, 64 φr m , φr m,t , see resolvent cocycle, 58 φm, residue cocycle, 57 σ n,l , combinatorial factors in index formula, 56 σz, the one parameter group associated to D2, 23 τ, a faithful, semifinite, normal trace on a von Neumann algebra, 7 τj, residue functionals, 56 ϕs, weights defined by D, 8 ζb(z), zeta functions in index formula, 56 du = [Du, ·], 57 s-trick, 61 OPr, regular pseudodifferential operators of order r, 22 OPr, 0 tame pseudodifferential operators of order r, 22 deg, grading degree for operators, 55 dom(ϕs), domain of the weight ϕs, 8 dom(ϕs)1/2, ‘L2-domain’ of the weight ϕs, 8 Auxiliary cochain, 59 Bott projector, 41, 86 Bounded geometry, 91 Chern character, 50 Clifford bundle, 89 Conditional trace, 49 Covering space, 98 Cyclic cohomology, 48 (b, B)-cocycle, 49 cyclic cocycle, 48 normalised cocycle, 48 reduced (b, B)-cocycle, 49 Dirac bundle, 89 Dirac-type operator, 89 Discrete dimension spectrum, 56 Double construction, 39 Fatou Lemma, 10 Fredholm module, 37 Index pairing, 37 K-theoretical , 37 129

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