New York Journal of Mathematics

New York J. Math. 15 (2009) 319–351.

Homological index formulas for elliptic operators over C^{∗}-algebras

Charlotte Wahl

Abstract. We prove index formulas for elliptic operators acting between spaces of sections of C^{∗}-vector bundles on a closed manifold. The formulas involve Karoubi’s Chern character from K-theory of a C^{∗}- algebra to de Rham homology of smooth subalgebras. We show how they apply to the higher index theory for coverings and to flat foliated bundles, and prove an index theorem for C^{∗}-dynamical systems associated to actions of compact Lie groups. In an Appendix we relate the pairing of odd K-theory and KK-theory to the noncommutative spectral flow and prove the regularity of elliptic pseudodifferential operators over C^{∗}-algebras.

Contents

1. Introduction 320 2. De Rham homology and the Chern character 321

2.1. Definition 321 2.2. Chern character and tensor products 325 2.3. Pairing with cyclic cocycles 328

3. Index theorems 329 4. Applications 332

4.1. Higher index theory for coverings and flat foliated bundles332 4.2. An index theorem for C^{∗}-dynamical systems 340

Appendix A. 342 A.1. Index theory and KK-theory 342 A.2. Pseudodifferential operators over C^{∗}-algebras 348

References 350

Received May 12, 2007; revised July 19, 2009. Mathematics Subject Classification. 19D55,58G12,19K35. Key words and phrases. Index theory, cyclic homology, KK-theory, spectral flow. This research was funded by a grant of AdvanceVT.

319^{ISSN 1076-9803/09}

1. Introduction

One of the generalizations of the Atiyah–Singer index theorem is to elliptic pseudodifferential operators associated to C^{∗}-vector bundles. Mishchenko– Fomenko introduced these operators and their index, an element in the K-theory of the C^{∗}-algebra [MF80]. Furthermore they defined a Chern character for C^{∗}-vector bundles and used it to formulate and prove an analogue of the Atiyah–Singer index theorem. However, in general it is not clear how to calculate the Mishchenko–Fomenko Chern character of a C^{∗}-vector bundle: its definition is based on the map

K_{0}(C(M, A)) ⊗ ℂ → K_{0}(C(M)) ⊗ K_{0}(A) ⊗ ℂ ⊕ K_{1}(C(M)) ⊗ K_{1}(A) ⊗ ℂ

for a closed manifold M and a unital C^{∗}-algebra A, which exists by the Künneth formula.

In this paper we prove index theorems for the same situation using Karoubi’s Chern character from the K-theory of a C^{∗}-algebra to the de Rham homology of smooth subalgebras [K87]. Karoubi’s Chern character is a generalization of the Chern character in differential geometry and is closely related to the Chern character in cyclic homology. Karoubi’s de Rham homology has been used especially in noncommutative superconnection proofs beginning with [Lo92].

We also prove (in the Appendix) that the pairing

K_{1}(A) × KK_{1}(A, B) → K_{1}(B),

where A, B are unital C^{∗}-algebras, can be expressed in terms of the noncommutative spectral flow, which was introduced in the context of family index theory by Dai–Zhang [DZ98]. See [Wa07] for further references and a systematic account. The formula is well-known for B = ℂ and the ordinary spectral flow.

The main ingredient of the proof of the index theorem is a result about the compatibility of Karoubi’s Chern character with the tensor product in K-theory. This allows the comparison of Karoubi’s Chern character with Mishchenko–Fomenko’s Chern character.

Our proof generalizes the derivation of Atiyah’s L^{2}-index theorem from the Mishchenko–Fomenko index theorem in [S05]. It is also closely related to the proof of an index theorem for flat foliated bundles in [J97], which is a special case of Connes’ index theorem for foliated manifolds [C94, p. 273] and implies the C^{∗}-algebraic version of the higher index theorem of Connes–Moscovici [CM90]. As an illustration we derive the latter in detail from our formula. We also show how to apply the formula to flat foliated bundles. In this context we introduce a smooth subalgebra which is defined in more general situations than the one in [J97].

We also prove an index theorem for Toeplitz operators associated to a C^{∗}-dynamical system (A, G, α) where G is a compact Lie group. The Chern character involved here has been defined in [C80]. In [Le91] a similar index theorem was proven for G = ℝ using Breuer–Fredholm operators. We relate both theorems in the case where the ℝ-action is periodic.

In the Appendix we explain how the pairing of K-theory with KK-theory is related to index theory and collect some useful facts about pseudodifferential operators over C^{∗}-algebras beyond those proven in [MF80], in particular that elliptic pseudodifferential operators are adjointable as bounded operators between appropriate Sobolev spaces and regular as unbounded operators on a fixed Sobolev space.

Unless specified otherwise, all tensor products between graded spaces are graded tensor products. Tensor products between two Fréchet spaces are understood to be the completed projective tensor products, with the exception of tensor products between C^{∗}-algebras: These are understood to be the minimal C^{∗}-algebraic tensor products.

Acknowledgements. I would like to thank Peter Haskell for helpful comments on previous versions of this paper.

2. De Rham homology and the Chern character

2.1. Definition. In this section we recall and slightly extend the definition of Karoubi’s Chern character and collect properties that are relevant for index theory. The main reference is [K87].

Let A_{∞} be a locally m-convex Fréchet algebra.

The left A_{∞}-module of differential forms of order k of A_{∞} is defined as

Ω̂_{k}A_{∞} := A_{∞} ⊗ (A_{∞}/ℂ)^{⊗k}

and the ℤ-graded space of all differential forms is

Ω̂_{∗}A_{∞} := ∏_{k=0}^{∞} Ω̂_{k}A_{∞}.

There is a differential d on Ω̂_{∗}A_{∞} of degree one defined by

d(a_{0} ⊗ … ⊗ a_{k}) = 1 ⊗ a_{0} ⊗ … ⊗ a_{k}

and a product determined by the properties that

a_{0} ⊗ … ⊗ a_{k} = a_{0} d a_{1} … d a_{k}

and that the Leibniz rule holds, which says that for α ∈ Ω̂_{k}A_{∞}, β ∈ Ω̂_{∗}A_{∞}

d(αβ) = (d α)β + (−1)^{k}α d β.

With these structures Ω̂_{∗}A_{∞} is a graded differential locally m-convex Fréchet algebra.

For a closed manifold M

Ω̂^{p,q}(M, A_{∞}) := Ω̂^{p}(M, Ω̂_{q}A_{∞}) = Ω̂^{p}(M) ⊗ Ω̂_{q}A_{∞},

where Ω̂^{∗}(M) is the space of smooth differential forms on M.

We call an open subset U ⊂ M regular if the compactly supported de Rham cohomology H_{c}^{∗}(U) is finite-dimensional, and if there are open subsets U_{0}, U_{1}, with U_{0} ⊂ U ⊂ U ⊂ U_{1}, for which there is a smooth homotopy F : [0, 1]×U_{1} → U_{1} such that F (0, x) = x, for all x ∈ U, F^{−1}({1}×U) ⊂ {1}×U_{0} and such that F^{−1}({t} × U) ⊂ {t} × U for all t ∈ [0, 1].

For a regular open subset U in M we define Ω̂_{0}^{p,q}(U, A_{∞}) to be the closure of the subspace of Ω̂^{p,q}(M, A_{∞}) spanned by forms with support in U.

The product on Ω̂_{0}^{∗∗}(U, A_{∞}) is determined by the natural isomorphism Ω̂_{0}^{∗∗}(U, A_{∞}) ≅ Ω̂_{0}^{∗}(U) ⊗ Ω̂_{∗}A_{∞}. Here the right-hand side is understood as a graded tensor product of graded algebras. Let d_{U} be the de Rham differential on U. The differential of the total complex of the double complex (Ω̂_{0}^{∗∗}(U, A_{∞}), d_{U}, d) is denoted by d_{tot} and its homology by H_{0}^{∗}(U, A_{∞}). The definition does not depend on the embedding of U into M as a regular subset.

For a closed manifold M we usually omit the suffix and write H^{∗}(M, A_{∞}). For F as above let f_{t} = F (t, ·) : U → U. Then f_{1}^{∗} : H_{0}^{∗}(U) → H_{c}^{∗}(U) is

inverse to the map H_{c}^{∗}(U) → H_{0}^{∗}(U), since for a closed form ω ∈ Ω̂_{0}^{∗}(U) the form f_{1}^{∗}ω is a closed form supported in U and f_{1}^{∗}ω − ω = d_{U} ∫_{0}^{1} F^{∗}ω. Hence H_{c}^{∗}(U) ≅ H_{0}^{∗}(U).

The isomorphism Ω̂_{0}^{∗∗}(U, A_{∞}) ≅ Ω̂_{0}^{∗}(U) ⊗ Ω̂_{∗}A_{∞} induces isomorphisms

Ω̂_{0}^{∗∗}(U, A_{∞})/\overline{[Ω̂_{0}^{∗∗}(U, A_{∞}), Ω̂_{0}^{∗∗}(U, A_{∞})]}_{s} ≅ Ω̂_{0}^{∗}(U) ⊗ Ω̂_{∗}A_{∞}/\overline{[Ω̂_{∗}A_{∞}, Ω̂_{∗}A_{∞}]}_{s},

H_{0}^{n}(U, A_{∞}) ≅ ⊕_{p+q=n}H_{0}^{p}(U) ⊗ H_{0}^{q}(A_{∞}) ≅ ⊕_{p+q=n}H_{0}^{p}(U, H_{q}(A_{∞})).

These isomorphisms have been proven in [K87, §§4.7, 4.8] in a slighly different situation. The proof carries over. It uses completed tensor products, therefore we use Ω̂_{0}^{∗}(U) instead of compactly supported forms for the definition of cohomology. The proof uses furthermore the fact that H_{0}^{∗}(U) is finite-dimensional.

We call a smooth possibly noncompact manifold M regular if there is a covering (U_{n})_{n∈ℕ} by regular subsets with U_{n} ⊂ U_{n+1}.

Extending a form by zero induces a well-defined push forward map

H_{0}^{∗}(U_{n}, A_{∞}) → H_{0}^{∗}(U_{n+1}, A_{∞})

so that we can define

H_{c}^{∗}(M, A_{∞}) = \underrightarrow{lim}_{n→∞} H_{0}^{∗}(U_{n}, A_{∞}).

It is clear that H_{c}^{∗}(M) agrees with the compactly supported de Rham cohomology of M.

If M → B is a fiber bundle of regular oriented manifolds, then integration over the fiber yields a homomorphism

∫_{M_{b}} : H_{c}^{∗}(M, A_{∞}) → H_{c}^{∗−dim M_{b}}(B, A_{∞}).

The de Rham homology of A_{∞} is

H_{∗}(A_{∞}) := H^{∗}(∗, A_{∞}),

where ∗ is the point.

If M is a closed manifold, we usually write H^{∗}(M, A_{∞}) for H_{0}^{∗}(M, A_{∞}). Note that then the quotient map Ω̂_{n}(C^{∞}(M, A_{∞})) → ⊕_{p+q=n}Ω̂^{p,q}(M, A_{∞}) induces a homomorphism

H_{∗}(C^{∞}(M, A_{∞})) → H^{∗}(M, A_{∞}).

We proceed with the definition and the properties of the Chern character. Let A be a unital C^{∗}-algebra and let A_{∞} ⊂ A be a dense subalgebra

that is closed under involution and holomorphic functional calculus in A. Assume that A_{∞} is endowed with the topology of a locally m-convex Fréchet algebra such that the inclusion A_{∞} ↪ A and the involution are continuous. We call such a subalgebra a smooth subalgebra of A.

Let M be a regular manifold. Recall that

K_{0}(C_{0}(M, A)) = Ker(K_{0}(C_{0}(M, A)^{+}) → K_{0}(ℂ)),

where C_{0}(M, A)^{+} denotes the unitalization of C_{0}(M, A). As C_{c}^{∞}(M, A_{∞})^{+} is dense and closed under holomorphic functional calculus in C_{0}(M, A)^{+}, we have that K_{0}(C_{c}^{∞}(M, A_{∞})) ≅ K_{0}(C_{0}(M, A)).

The Chern character form of a projection P ∈ M_{n}(C_{c}^{∞}(M, A_{∞})^{+}) is defined as

ch_{A_{∞}}^{M}(P) := ∑_{k=0}^{∞} \frac{(−1)^{k}}{(2πi)^{k}k!} tr P (d_{tot}P)^{2k}.

The normalization differs from the normalization in [K87] and is chosen such that the Chern character of the Bott element B ∈ K_{0}(C_{0}((0, 1)^{2})) integrated over (0, 1)^{2} equals 1. (There is also some ambiguity about the sign of the Bott element B in the literature. Here we take B = 1 − [H] ∈ Ker(K_{0}(C(S^{2})) → K_{0}(ℂ)), where H is the Hopf bundle.)

In the following proposition we denote by P_{∞} ∈ M_{n}(ℂ) the image of P ∈ M_{n}(C_{c}^{∞}(M, A_{∞})^{+}) under “evaluation at infinity”.

Proposition 2.1. (1) ch_{A_{∞}}^{M}(P) is closed.

(2) Let P : [0, 1] → M_{n}(C_{c}^{∞}(M, A_{∞})^{+}) be a differentiable path of projections and let U ⊂ M with supp(P (t) − P_{∞}(t)) ⊂ U for all t ∈ [0, 1]. Then there is a form

α ∈ Ω̂_{0}^{∗∗}(U, A_{∞})/\overline{[Ω̂_{0}^{∗∗}(U, A_{∞}), Ω̂_{0}^{∗∗}(U, A_{∞})]}_{s}

such that d_{tot}α = ch_{A_{∞}}^{M}(P (1)) − ch_{A_{∞}}^{M}(P (0)).

(3) The Chern character form induces a well-defined homomorphism

K_{0}(C_{0}(M, A)) → H_{c}^{∗}(M, A_{∞}).

Proof. For M compact the proofs are standard. We include the proof of (2) in order to show that it works in the noncompact case as well:

From the Leibniz rule, one deduces that the terms P P′P, (1−P)P′(1−P), P (d_{tot}P)P, (1 − P)(d_{tot}P)(1 − P) all vanish.

Hence

tr(P (d_{tot}P)^{2k})′ = tr P′(d_{tot}P)^{2k} + tr P ((d_{tot}P)^{2k})′

= tr P ((d_{tot}P)^{2k})′

= ∑_{i=0}^{2k−1} tr P (d_{tot}P)^{i}(d_{tot}P)′(d_{tot}P)^{2k−i−1}.

This vanishes for k = 0.

For i even and k ≠ 0

tr P (d_{tot}P)^{i}(d_{tot}P)′(d_{tot}P)^{2k−i−1}

= tr(d_{tot}P)^{i}P (d_{tot}P′)(d_{tot}P)^{2k−i−1}

= tr(d_{tot}P)^{i}(d_{tot}(P P′))(d_{tot}P)^{2k−i−1} − tr(d_{tot}P)^{i}(dP)P′(d_{tot}P)^{2k−i−1}

= tr(d_{tot}P)^{i}(d_{tot}(P P′))(d_{tot}P)^{2k−i−1}

= d_{tot} tr P (d_{tot}P)^{i−1}(d_{tot}(P P′))(d_{tot}P)^{2k−i−1}.

Note that tr P (d_{tot}P)^{i−1}(d_{tot}(P P′))(d_{tot}P)^{2k−i−1} vanishes on U for k ≠ 0. For i odd the argument is similar. □

We define the odd Chern character via the following diagram:

(2.1.1)

K_{0}(C_{0}((0, 1) × M, A)) ⟶^{≅} K_{1}(C_{0}(M, A))
↓^{ch}_{A∞}^{(0,1)×M}  ↓^{ch}_{A∞}^{M}
H_{c}^{ev}((0, 1) × M, A_{∞}) \underrightarrow{∫{1 \atop 0}} H_{c}^{odd}(M, A_{∞}).

Note that ∫_{0}^{1} : H_{c}^{∗}((0, 1) × M, A_{∞}) → H_{c}^{∗}(M, A_{∞}) is an isomorphism by H_{c}^{∗}((0, 1) × M, A_{∞}) ≅ H_{c}^{∗}((0, 1)) ⊗ H_{c}^{∗}(M, A_{∞}) ≅ H_{c}^{∗}(M, A_{∞}).

In the following we derive a formula for the odd Chern character. It is analogous to those well-known in de Rham cohomology and cyclic homology (compare with [Gl93]).

Proposition 2.2. For u ∈ U_{n}(C_{c}^{∞}(M, A_{∞})^{+}) in H_{c}^{∗}(M, A_{∞})

ch_{A_{∞}}^{M}([u]) = ∑_{k=1}^{∞} (\frac{−1}{2πi})^{k} \frac{(k − 1)!}{(2k − 1)!}u^{∗}(d_{tot}u)((d_{tot}u^{∗})(d_{tot}u))^{k−1}.

Proof. Here we use the fact that the Chern character can be defined in terms of noncommutative connections [K87].

Let P_{n} ∈ M_{2n}(ℂ) be the projection onto the first n components. Let

W (t) ∈ C^{∞}([0, 1], U_{2n}(C_{c}^{∞}(M, A_{∞})^{+}))

with W (0) = 1 and W (1) = diag(u, u^{∗}). Then the isomorphism

K_{1}(C_{0}(M, A)) → K_{0}(C((0, 1) × M, A))

maps [u] to [W P_{n}W^{∗}] − [P_{n}].

The Chern character is independent of the choice of the connection [K87, Theorem 1.22], thus we may use the connection

W P_{n}(d_{tot} + dx ∂_{x} + xW (1)^{∗}d_{tot}(W (1)))W^{∗}

on the projective C_{c}^{∞}((0, 1) × M, A_{∞})^{+}-module

W P_{n}W^{∗}(C_{c}^{∞}((0, 1) × M, A_{∞})^{+})^{n}

for its calculation. It follows that

ch_{A_{∞}}^{M}(u) = ch_{A_{∞}}^{(0,1)×M}(W P_{n}W^{∗})

= ∑_{k=0}^{∞} ∫_{0}^{1} \frac{(−1)^{k}}{(2πi)^{k}k!}

· tr(x^{2}u^{∗}(d_{tot}u)u^{∗}(d_{tot}u) + x(d_{tot}u^{∗})(d_{tot}u) + dx u^{∗}(d_{tot}u))^{k}

= ∑_{k=0}^{∞} \frac{(−1)^{k}}{(2πi)^{k}k!} ∫_{0}^{1} tr((x − x^{2})(d_{tot}u^{∗})(d_{tot}u) + dx u^{∗}(d_{tot}u))^{k}

= ∑_{k=1}^{∞} \frac{(−1)^{k}}{(2πi)^{k}k!} ∫_{0}^{1} dx (x − x^{2})^{k−1}u^{∗}(d_{tot}u)((d_{tot}u^{∗})(d_{tot}u))^{k−1}

= ∑_{k=1}^{∞} (\frac{−1}{2πi})^{k} \frac{(k − 1)!}{(2k − 1)!}u^{∗}(d_{tot}u)((d_{tot}u^{∗})(d_{tot}u))^{k−1},

as desired. □

2.2. Chern character and tensor products. From now on assume that M is a closed manifold.

Let K_{i}(A)_{ℂ} := K_{i}(A) ⊗ ℂ.

In the following we prove the compatibility of the Chern character with the Bott periodicity map K_{1}(C_{0}((0, 1), A)) ≅ K_{0}(A) and with the Künneth formulas

K_{0}(C(M))_{ℂ} ⊗ K_{0}(A)_{ℂ} ⊕ K_{1}(C(M))_{ℂ} ⊗ K_{1}(A)_{ℂ} ≅ K_{0}(C(M, A))_{ℂ}

and

K_{0}(C(M))_{ℂ} ⊗ K_{1}(A)_{ℂ} ⊕ K_{1}(C(M))_{ℂ} ⊗ K_{0}(A)_{ℂ} ≅ K_{0}(C(M, A))_{ℂ}.

These isomorphisms are defined via the tensor product

K_{i}(C(M)) ⊗ K_{j}(A) → K_{i+j}(C(M, A)), i, j ∈ ℤ/2.

This map is injective, hence we may consider K_{i}(C(M)) ⊗ K_{j}(A) as a subspace of K_{i+j}(C(M, A)).

First recall the definition of the tensor product. For i, j = 0 the tensor product is induced by the tensor product of projections. The remaining three cases are derived from the tensor product of projections using Bott periodicity, for example

K_{1}(C(M)) ⊗ K_{1}(A) ≅ K_{0}(C_{0}((0, 1) × M)) ⊗ K_{0}(C_{0}((0, 1)) ⊗ A)
→ K_{0}(C_{0}((0, 1)^{2} × M, A))
≅ K_{0}(C(M, A))

and

K_{0}(C(M)) ⊗ K_{1}(A) ≅ K_{0}(C(M)) ⊗ K_{0}(C_{0}((0, 1), A))
→ K_{0}(C_{0}((0, 1) × M, A))
≅ K_{1}(C(M, A)).

A standard calculation (see [K87, Theorem 1.26]) shows that the tensor product for i = j = 0 is compatible with the Chern character, namely for a ∈ K_{0}(C(M)) and b ∈ K_{0}(A)

ch_{A_{∞}}^{M}(a ⊗ b) = ch^{M}(a) ch_{A_{∞}}(b).

In the following proposition

β : K_{0}(C(M, A)) → K_{0}(C_{0}((0, 1)^{2} × M, A))
a ↦ a ⊗ B

is the Bott periodicity map.

Proposition 2.3. (1) For a ∈ K_{0}(C_{0}((0, 1)^{2} × M, A))

ch_{A_{∞}}^{M} β^{−1}(a) = ∫_{(0,1)^{2}} ch_{A_{∞}}^{(0,1)^{2}×M}(a).

(2) For a ∈ K_{0}(C_{0}((0, 1) × M)) and b ∈ K_{0}(C_{0}((0, 1), A))

ch_{A_{∞}}^{M} β^{−1}(a ⊗ b) = ∫_{(0,1)^{2}} ch^{(0,1)×M}(a) ch_{A_{∞}}^{(0,1)}(b).

Proof. Consider K_{0}(C_{0}((0, 1)^{2}×M, A)) as a subgroup of K_{0}(C(T^{2}×M, A)). (1) Let b ∈ K_{0}(C(M, A)) with a = B ⊗ b. Then

∫_{(0,1)^{2}} ch_{A_{∞}}^{(0,1)^{2}×M}(B ⊗ b) = ∫_{T^{2}} ch_{A_{∞}}^{T^{2}×M}(B ⊗ b)
= ch_{A_{∞}}^{M}(b) ∫_{T^{2}} ch^{T^{2}}(B)
= ch_{A_{∞}}^{M}(b).

(2) The assertion follows from the commutative diagram

\xymatrix{K_{0}(C_{0}((0, 1) × M))⊗K_{0}(C_{0}((0, 1), A)) \ar[r] \ar[d] & K_{0}(C(S^{1} × M))⊗K_{0}(C(S^{1}, A)) \ar[d]^{⊗} \\ K_{0}(C_{0}((0, 1)^{2} × M, A)) \ar[r]^{↪} \ar[d]^{ch_{A∞}^{M} ∘β^{−1}} & K_{0}(C(T^{2} × M, A)) \ar[d]^{_{∫_{T 2} ch{T 2×M \atop A∞}}} \\ H^{∗}(M, A_{∞}) \ar[r]^{=} & H^{∗}(M, A_{∞}).}

Since the horizontal arrows are inclusions, the first vertical map on the lefthand side is determined by the first vertical map on the right-hand side. The second square commutes by (1). □

Corollary 2.4. The diagram

K_{1}(C_{0}((0, 1) × M, A) ⟶^{≅} K_{0}(C(M, A))
↓^{ch}_{A∞}^{(0,1)×M}  ↓^{ch}_{A∞}^{M}
H_{c}^{odd}((0, 1) × M, A_{∞}) \underrightarrow{∫{1 \atop 0}} H^{ev}(M, A_{∞})

commutes.

Proof. Consider the diagram

\xymatrix{K_{1}(C_{0}((0, 1) × M, A)) \ar[d]^{_{ch{(0,1)×M \atop A∞}}} & K_{0}(C_{0}((0, 1)^{2} × M, A)) \ar[d]^{_{ch{(0,1)2×M \atop A∞}}} \ar[l]_{≅} \ar[r]^{β^{−1}} & K_{0}(C(M, A)) \ar[d]^{_{ch{M \atop A∞}}} \\ H_{c}^{odd}((0, 1) × M, A_{∞}) & H_{c}^{ev}((0, 1)^{2} × M, A_{∞}) \ar[l]_{_{∫{1 \atop 0}}} \ar[r]^{_{∫(0,1)2}} & H^{ev}(M, A_{∞}).}

The first square commutes by diagram (2.1.1) applied to (0, 1) × M. The second square commutes by the first part of the previous proposition. □

We denote by

R_{jk} : K_{i}(C(M, A))_{ℂ} → K_{j}(C(M))_{ℂ} ⊗ K_{k}(A)_{ℂ} ⊂ K_{i}(C(M, A))_{ℂ}

the projections induced by the Künneth formulas.

We have a tensor product

ch^{M} ⊗ ch_{A_{∞}} : K_{j}(C(M))_{ℂ} ⊗ K_{k}(A)_{ℂ} → H^{∗}(M) ⊗ H_{∗}(A_{∞}) ≅ H^{∗}(M, A_{∞}).

Proposition 2.5. (1) On K_{0}(C(M, A))_{ℂ}

ch_{A_{∞}}^{M} = (ch^{M} ⊗ ch_{A_{∞}}) ∘ R_{00} + (ch^{M} ⊗ ch_{A_{∞}}) ∘ R_{11}.

(2) On K_{1}(C(M, A))_{ℂ}

ch_{A_{∞}}^{M} = (ch^{M} ⊗ ch_{A_{∞}}) ∘ R_{01} + (ch^{M} ⊗ ch_{A_{∞}}) ∘ R_{10}.

Proof. (1) Let a ⊗ b ∈ K_{0}(C(M, A)) with a ∈ K_{1}(C(M))) and b ∈ K_{1}(A). Let a correspond to ã ∈ K_{0}(C_{0}((0, 1) × M)) and b to b̃ ∈ K_{0}(C_{0}((0, 1), A)).

Then by definition ch^{M}(a) = ∫_{0}^{1} ch^{(0,1)×M}(ã) and ch_{A_{∞}}(b) = ∫_{0}^{1} ch_{A_{∞}}^{(0,1)}(b̃). Now by the previous proposition and its corollary

ch_{A_{∞}}^{M}(a ⊗ b) = ∫_{(0,1)^{2}} ch^{(0,1)×M}(ã) ch_{A_{∞}}^{(0,1)}(b̃)

= ∫_{0}^{1} ch^{(0,1)×M}(ã) ∫_{0}^{1} ch_{A_{∞}}^{(0,1)}(b̃)
= ch^{M}(a) ch_{A_{∞}}(b).

(2) follows applying by (1) to K_{0}(C_{0}((0, 1) × M, A)) since the Chern character interchanges the suspension isomorphisms in K-theory and de Rham homology. □

Define Ch_{A}^{M} as the map

K_{0}(C(M, A)) → K_{0}(C(M, A))_{ℂ}

≅ K_{0}(C(M))_{ℂ} ⊗ K_{0}(A)_{ℂ} ⊕ K_{1}(C(M))_{ℂ} ⊗ K_{1}(A)_{ℂ}

⟶^{ch^{M}} H^{ev}(M) ⊗ K_{0}(A) ⊕ H^{odd}(M) ⊗ K_{1}(A).

and analogously for K_{1}(C(M, A)). This is the Chern character introduced by Mishchenko–Fomenko [MF80].

The previous proposition is equivalent to the equation

(2.2.1)  ch_{A_{∞}} ∘Ch_{A}^{M} = ch_{A_{∞}}^{M} .

2.3. Pairing with cyclic cocycles. In noncommutative geometry it is more common to consider the Chern character with values in cyclic homology than the one with values in de Rham homology. De Rham homology can be paired with normalized cyclic cocycles; in this pairing both Chern characters agree up to normalization:

Let C_{n}^{λ}(A_{∞}) be the quotient of the algebraic tensor product (A_{∞}/ℂ)^{⊗n+1} by the action of ℤ/(n + 1)ℤ. Let

b : C_{n}^{λ}(A_{∞}) → C_{n−1}^{λ}(A_{∞}),

b(a_{0} ⊗ ⋯ ⊗ a_{n}) = (−1)^{n}a_{n}a_{0} ⊗ ⋯ ⊗ a_{n−1}

+ ∑_{i=0}^{∞}(−1)^{i}a_{0} ⊗ ⋯ ⊗ a_{i}a_{i+1} ⊗ ⋯ ⊗ a_{n}.

The homology of the complex (C_{∗}^{λ}(A_{∞}), b) is the reduced cyclic homology HC_{∗}(A_{∞}). Using the completed projective tensor product instead of the algebraic one we obtain the topological reduced cyclic homology HC_{∗}^{top}(A_{∞}). Furthermore we denote by HC_{∗}^{sep}(A_{∞}) the topological homology of (C_{n}^{λ}(A_{∞}), b), i.e., we use the completed projective tensor product and quotient out the closure of the range of b.

The reduced cyclic cohomology HC^{∗}(A_{∞}) is the homology of the dual complex (C_{λ}^{n}(A_{∞}), b^{t}) (in the algebraic sense). Elements of C_{λ}^{n}(A_{∞}) are called normalized cochains. The continuous reduced cyclic cohomology HC_{top}^{∗}(A_{∞}) is the homology of the topological dual complex.

The pairing HC_{top}^{∗}(A_{∞}) ⊗ HC_{∗}^{top}(A_{∞}) → ℂ descends to a pairing

HC_{top}^{∗}(A_{∞}) ⊗ HC_{∗}^{sep}(A_{∞}) → ℂ.

Furthermore the quotient map Ω̂_{n}(A_{∞}) → C_{λ}^{n}(A_{∞}) induces an homomorphism H_{n}(A_{∞}) → HC_{n}^{sep}(A_{∞}), which is an embedding for n ≥ 1 (see [K87, §§4.1 and 2.13]). In degree zero there is a pairing of H_{0}(A_{∞}) = A_{∞}/[A_{∞}, A_{∞}] with traces on A_{∞}.

The Chern character ch^{λ} : K_{0}(A_{∞}) → HC_{∗}(A_{∞}) is defined by

ch^{λ}(p) = ∑_{m=0}^{∞}(−1)^{m} tr p^{⊗2m+1}

for a projection p ∈ M_{n}(A_{∞}). Hence the composition

K_{0}(A_{∞}) ⟶^{ch^{λ}} HC_{∗}(A_{∞}) → HC_{∗}^{sep}(A_{∞})

agrees up to normalization with the map

K_{0}(A_{∞}) ⟶^{ch_{A∞}} H_{∗}(A_{∞}) ↪ HC_{∗}^{sep}(A_{∞}).

In particular if ϕ ∈ HC_{top}^{m}(A_{∞}), then

ϕ ∘ ch^{λ} = (2πi)^{m}m! ϕ ∘ ch_{A_{∞}} .

3. Index theorems

In the following we give a formulation of the Mishchenko–Fomenko index theorem, which is different from the original one and adapted to the applications. Furthermore we translate its proof into the language of KKtheory: We show the compatibility of the Chern character with the pairing K_{i}(C(M, A)) ⊗ KK_{j}(C(M), ℂ) → K_{i+j}(A) for i, j ∈ ℤ/2, where on KK_{j}((C(M), ℂ) we use the Chern character from K-homology to de Rham homology of M. We refer to Appendix A.1 for some facts about the connection of KK-theory to index theory.

Lemma 3.1. (1) For x ∈ K_{i}(C(M)) ⊗ K_{i}(A) ⊂ K_{0}(C(M, A)) and y ∈ KK_{j}(C(M), ℂ) with i ≠ j

x ⊗_{C(M)} y = 0 ∈ K_{j}(A).

(2) For x ∈ K_{i}(C(M)) ⊗ K_{j}(A) ⊂ K_{1}(C(M, A)) and y ∈ KK_{j}(C(M), ℂ)

with i ≠ j

x ⊗_{C(M)} y = 0 ∈ K_{i}(A).

It follows that for x ∈ K_{i}(C(M, A)) and y ∈ KK_{j}(C(M), ℂ)

x ⊗_{C(M)} y = R_{j,i+j}(x) ⊗_{C(M)} y ∈ K_{i+j}(A)_{ℂ}.

Proof. (1) Let B_{0} ∈ KK_{0}(C_{0}((0, 1)^{2}), ℂ) be the Bott element. By the standard isomorphism K_{i}(A) ≅ KK_{i}(ℂ, A) and the fact that the tensor product in K-theory is a special case of the Kasparov product all we have to show is that for a ∈ KK_{j}(ℂ, C_{0}((0, 1), A)) and b ∈ KK_{j}(ℂ, C_{0}((0, 1) × M))

((a ⊗ b) ⊗_{C_{0}((0,1)^{2})} B_{0}) ⊗_{C(M)} y = 0.

This follows from the associativity of the product and the fact that

(b ⊗_{C_{0}((0,1))} B_{0}) ⊗_{C(M)} y ∈ KK_{0}(C_{0}((0, 1)), ℂ) = 0.

(2) Let i = 0, j = 1. Let B_{1} ∈ KK_{1}(C_{0}((0, 1)), ℂ) be the Bott element. Let a ∈ K_{0}(C_{0}((0, 1), A)) and b ∈ K_{0}(C(M)). Then

((a ⊗ b) ⊗_{C_{0}((0,1)^{2})} B_{1}) ⊗_{C(M)} y = 0

by associativity and since

(b ⊗_{C_{0}((0,1))} B_{1}) ⊗_{C(M)} y ∈ KK_{0}(C_{0}((0, 1)), ℂ) = 0.

The proof for i = 1, j = 0 is analogous. □

Let now ch_{M} : KK_{i}(C(M), ℂ) → H_{∗}(M) be the homological Chern character where H_{∗}(M) is the de Rham homology of M with complex coefficients.

Note that the pairing ⟨ , ⟩ : H^{∗}(M) × H_{∗}(M) → ℂ induces a pairing ⟨ , ⟩ : (H^{∗}(M) ⊗ K_{∗}(A)) × H_{∗}(M) → K_{∗}(A)_{ℂ}.

Lemma 3.2. For x ∈ K_{i}(C(M, A)) and y ∈ KK_{j}(C(M), ℂ)

⟨Ch_{A}^{M}x, ch_{M}y⟩ = ⟨Ch_{A}^{M}R_{j,i+j}(x), ch_{M}y⟩ ∈ K_{i+j}(A)_{ℂ}.

Proof. Consider the case i, j = 0. We must show ⟨Ch_{A}^{M}R_{11}(x), ch_{M}y⟩ = 0 or equivalently that ⟨Ch_{A}^{M}(x), ch_{M}y⟩ = 0 for x = x_{1} ⊗ x_{2} with x_{1} ∈ K_{1}(C(M)) and x_{2} ∈ K_{1}(A).

Clearly Ch_{A}^{M}x = (ch^{M} x_{1})x_{2}.

Hence

⟨Ch_{A}^{M}(x), ch_{M}(y)⟩ = ⟨ch^{M} x_{1}, ch_{M}(y)⟩x_{2}.

Since ch^{M} x_{1} ∈ H^{odd}(M) and ch_{M}(y) ∈ H_{ev}(M), ⟨ch^{M} x_{1}, ch_{M}(y)⟩ vanishes. The remaining three cases are analogous. □

Proposition 3.3. If x ∈ K_{i}(C(M, A)) and y ∈ KK_{j}(C(M), ℂ), then

x ⊗_{C(M)} y = ⟨Ch_{A}^{M}x, ch_{M}y⟩ ∈ K_{i+j}(A)_{ℂ}.

Proof. By the first lemma x ⊗_{C(M)} y = R_{j,i+j}(x) ⊗_{C(M)} y ∈ K_{i+j}(A)_{ℂ}. By the previous lemma the right-hand side of the formula also only depends on R_{j,i+j}(x). Therefore and by linearity we may restrict to the case where x = x_{1} ⊗ x_{2} with x_{1} ∈ K_{j}(C(M)) and x_{2} ∈ K_{i+j}(A). Then in K_{i+j}(A)_{ℂ}

x ⊗_{C(M)} y = (x_{1} ⊗_{C(M)} y)x_{2}

= ⟨ch^{M} x_{1}, ch_{M}y⟩x_{2}

= ⟨Ch_{A}^{M}x, ch_{M}y⟩. □

Using formula (2.2.1) and considering the pairing

⟨ , ⟩ : H^{∗}(M, A_{∞}) × H_{∗}(M) → H_{∗}(A_{∞})

we obtain:

Corollary 3.4. If x ∈ K_{i}(C(M, A)) and y ∈ KK_{j}(C(M), ℂ), then

ch_{A_{∞}}(x ⊗_{C(M)} y) = ⟨ch_{A_{∞}}^{M} x, ch_{M}y⟩ ∈ H_{∗}(A_{∞}).

In the following we translate these results into a more classical language (see Appendix A.1):

Now let M be a closed Riemannian manifold and let E be a hermitian, possibly ℤ/2-graded, complex vector bundle on M.

Let D : C^{∞}(M, E) → C^{∞}(M, E) be an elliptic symmetric pseudodifferential operator of order 1. If E is graded, then D is assumed to be odd. In the ungraded case the symbol σ(D) defines an element in K_{1}(C_{0}(T^{∗}M)), in the graded case [σ(D^{+})] ∈ K_{0}(C_{0}(T^{∗}M)). If E is ungraded, then

[(L^{2}(M, E), D)] ∈ KK_{1}(C(M), ℂ),

else [(L^{2}(M, E), D)] ∈ KK_{0}(C(M), ℂ). In the ungraded case the values of the index ind are in K_{1}(A), in the graded case in K_{0}(A).

Define the A-vector bundle

L(U) := ([0, 1] × M × A^{n})/(0, x, v) ∼ (1, x, U(x)v)

on S^{1} ×M and let ∂/_{L(U)} be the operator \frac{1}{i} \frac{d}{dx} acting on the sections of L(U). Pull E back to S^{1} ⊗ M. Then ϕ(t)D + (1 − ϕ(t))UDU^{∗} is well-defined on L^{2}(S^{1} × M, L(U) ⊗ E).

Let π_{!} : H_{c}^{∗}(T M) → H^{∗}(M) be integration over the fiber and k = \frac{dim M(dim M+1)}{2}.

Theorem 3.5. (1) Let P ∈ M_{n}(C^{∞}(M, A)) be a projection.

(a) Assume that E is ℤ/2-graded. Then

ch_{A_{∞}} ind P (⊕^{n}D^{+})P = (−1)^{k} ∫_{M} Td(M)π_{!} ch^{T M}[σ(D^{+})] ch_{A_{∞}}^{M}[P].

(b) If E is ungraded, then

ch_{A_{∞}} ind P (⊕^{n}D)P = (−1)^{k} ∫_{M} Td(M)π_{!} ch^{T M}[σ(D)] ch_{A_{∞}}^{M}[P].

(2) Let U ∈ U_{n}(C^{∞}(M, A)) be a unitary.

(a) If E is ungraded, then

ch_{A_{∞}} sf((1 − t)D + tUDU^{∗}) = (−1)^{k} ∫_{M} Td(M)π_{!} ch^{T M}[σ(D)] ch_{A_{∞}}^{M}[U].

(b) If E is ℤ/2-graded and σ is the grading operator, then

ch_{A_{∞}} ind(−σ∂/_{L(U)} + iσ(χ(t)D + (1 − χ(t))UDU^{∗}))

= (−1)^{k} ∫_{M} Td(M)π_{!} ch^{T M}[σ(D^{+})] ch_{A_{∞}}^{M}[U].

See Appendix A.1 for more possibilities to express the left-hand side of 2(a) and 2(b).

4. Applications

4.1. Higher index theory for coverings and flat foliated bundles. In the following we deduce the higher index theorem for coverings of Connes– Moscovici [CM90] from the previous formulas. We do not recover the theorem in full generality (which calculates the pairing of an index in algebraic K-theory with group cocycles), but for extendable cocycles.

Let Γ be a discrete group.

We begin by recalling some facts about the group cohomology H^{∗}(Γ), in particular how to embed it into HC^{∗}(Γ).

Let

C^{n}(Γ) = {τ : Γ^{n+1} → ℂ, τ(gg_{0}, …, gg_{n}) = τ(g_{0}, …, g_{n}) for all g ∈ Γ}

and let

d_{Γ} : C^{n}(Γ) → C^{n+1}(Γ),

d_{Γ}τ(g_{0}, …, g_{n+1}) = ∑_{j=0}^{n+1}(−1)^{j}τ(g_{0}, …, g_{j−1}, g_{j+1}, … g_{n+1}).

We denote by C_{al}^{n}(Γ) ⊂ C^{n}(Γ) the subspace of alternating elements. The homology of (C_{al}^{∗}(Γ), d_{Γ}) is H^{∗}(Γ) (as is the homology of (C^{∗}(Γ), d_{Γ})).

Furthermore let

C_{λ}^{n}(ℂΓ)_{⟨e⟩} = {c ∈ C_{λ}^{n}(ℂΓ) | c(g_{0}, g_{1}, … g_{n}) = 0 for g_{0}g_{1} … g_{n} ≠ e}

and let

C_{λ}^{n}(ℂΓ)_{⟨g⟩≠⟨e⟩} = {c ∈ C_{λ}^{n}(ℂΓ) | c(g_{0}, g_{1}, … g_{n}) = 0 for g_{0}g_{1} … g_{n} = e}.

The complex (C_{λ}^{∗}(ℂΓ), b^{t}) decomposes into a direct sum (C_{λ}^{∗}(ℂΓ)_{⟨e⟩}, b^{t}) ⊕ (C_{λ}^{∗}(ℂΓ)_{⟨g⟩≠⟨e⟩}, b^{t}).

In the following we assume n ≥ 1.

For c ∈ C_{λ}^{n}(ℂΓ)_{⟨e⟩} define τ_{c} ∈ C_{al}^{n}(Γ) by

τ_{c}(e, g_{1}, …, g_{n}) := c(g_{n}^{−1}, g_{1}, g_{1}^{−1}g_{2}, g_{2}^{−1}g_{3}, … g_{n−1}^{−1}g_{n})

and for τ ∈ C_{al}^{n}(Γ) define c_{τ} ∈ C_{λ}^{n}(ℂΓ)_{⟨e⟩} by

c_{τ} (g_{0}, g_{1}, …, g_{n}) = τ(e, g_{1}, g_{1}g_{2}, …, g_{1} … g_{n}) if g_{0}g_{1} … g_{n} = e

and

c_{τ} (g_{0}, g_{1}, …, g_{n}) = 0 if g_{0}g_{1} … g_{n} ≠ e.

The maps

C_{λ}^{n}(ℂΓ)_{⟨e⟩} → C_{al}^{n}(Γ), c ↦ τ_{c}

and

C_{al}^{n}(Γ) → C_{λ}^{n}(ℂΓ)_{⟨e⟩}, τ ↦ c_{τ}

are inverse to each other and compatible with the differentials. The isomorphism of complexes

(C_{al}^{∗}(Γ), d_{Γ}) ≅ (C_{λ}^{∗}(ℂΓ)_{⟨e⟩}, b^{t})

induces an injection H^{n}(Γ) → HC^{n}(ℂΓ), n ≥ 1.

The case n = 0 is different but easy, therefore we leave it to the reader. Let M be a closed Riemannian manifold with fundamental group Γ and

universal covering π : M̃ → M. Let C_{r}^{∗}Γ be the reduced C^{∗}-algebra of Γ.

We recall the definition of the higher index in K_{0}(C_{r}^{∗}Γ) of an elliptic differential operator on M.

Let E be ℤ/2-graded hermitian vector bundle on M and let Ẽ := π^{∗}E. Then Ẽ is endowed with a right Γ-action. There is an induced left Γaction R_{g}^{∗} : C^{∞}(M̃, Ẽ) → C^{∞}(M̃, Ẽ), (R_{g}^{∗}s)(x) = R_{g^{−1}}(s(xg)). Let D : C^{∞}(M, E) → C^{∞}(M, E) be an odd symmetric elliptic differential operator. It lifts to a Γ-invariant odd elliptic operator D̃ on C_{c}^{∞}(M̃, Ẽ). On the right ℂΓ-module C_{c}^{∞}(M̃, Ẽ) (which is defined using the right Γ-action R_{g^{−1}}^{∗}) we have a C_{r}^{∗}Γ-valued scalar product

⟨x, y⟩ := ∑_{g∈Γ} g ∫M̃⟨x, R_{g}^{∗}y⟩ẼdvolM̃.

The completion of C_{c}^{∞}(M̃, Ẽ) with respect to the corresponding norm is a Hilbert C_{r}^{∗}Γ-module denoted by H. The higher index of D is defined as the index of the closure of D̃^{+} : C_{c}^{∞}(M̃, Ẽ^{+}) → H^{−}. In the following we show that D̃ is unitarily equivalent to an elliptic differential operator on M. In particular its closure is indeed regular and Fredholm (see Appendix §A.2).

If V is a vector space with a left Γ-action, there is a left Γ-action on (M̃ × V) defined by (x, v) ↦ (xg^{−1}, gv). The Mishchenko–Fomenko bundle on M is

P := M̃ ×_{Γ} C_{r}^{∗}Γ = (M̃ × C_{r}^{∗}Γ)/Γ.

It inherits a C_{r}^{∗}Γ-valued scalar product from the standard C_{r}^{∗}Γ-valued scalar product on C_{r}^{∗}Γ.

Let P_{alg} = M̃ ×_{Γ} ℂΓ ⊂ P.

Lemma 4.1. There is an isometric isomorphism between the Hilbert C_{r}^{∗}Γmodules H and L^{2}(M, E ⊗ P) inducing an isomorphism between C_{c}^{∞}(M̃, Ẽ) and C^{∞}(M, E ⊗ P_{alg}).

Proof. A left Γ-action on C^{∞}(M̃, Ẽ ⊗ ℂΓ) is defined by

L_{h}(sg) = (R_{h}^{∗}s)hg

for s ∈ C^{∞}(M̃, Ẽ) and g ∈ Γ. Let C^{∞}(M̃, Ẽ ⊗ ℂΓ)^{Γ} be the subspace of Γ-invariant sections. There is an C_{r}^{∗}Γ-valued scalar product on C^{∞}(M̃, Ẽ ⊗ ℂΓ)^{Γ} given by

⟨x, y⟩ = ∫_{F} ⟨x, y⟩_{Ẽ⊗C_{r}^{∗}Γ}dvolM̃

where F ⊂ M̃ is a fundamental domain. We denote the completion of C^{∞}(M̃, Ẽ ⊗ ℂΓ)^{Γ} with respect to the induced norm by H_{1}.

Any s ∈ C^{∞}(M, E ⊗ P_{alg}) lifts uniquely to an element ∑_{g∈Γ} s_{g}g ∈ C^{∞}(M̃, Ẽ ⊗ ℂΓ)^{Γ} with s_{g} ∈ C_{c}^{∞}(M̃, Ẽ). The induced map is an isometric isomorphism

C^{∞}(M, E ⊗ P_{alg}) ≅ C^{∞}(M̃, Ẽ ⊗ ℂΓ)^{Γ}

hence we get an isometric isomorphism L^{2}(M, E ⊗ P) ≅ H_{1}.

It can easily be checked that the isomorphism

C_{c}^{∞}(M̃, Ẽ) → C^{∞}(M̃, Ẽ ⊗ ℂΓ)^{Γ}, s ↦ ∑_{g∈Γ}(R_{g}^{∗}s)g

is an isometry as well, inducing an isometry H ≅ H_{1}. □

By the lemma and its proof D̃ : C_{c}^{∞}(M̃, Ẽ) → H is unitarily equivalent to an elliptic differential operator D : C^{∞}(M, E ⊗ P_{alg}) → L^{2}(M, E ⊗ P).

In order to apply Theorem 3.5 we embed P into a trivial bundle as follows: Let {U_{i}}_{i∈I} be a finite open covering of M such that π^{−1}U_{i} is diffeomor-

phic to U_{i} × Γ. By refining the covering we may assume that U_{i} ∩ U_{j} is connected for each i, j ∈ I. Let {χ_{i}^{2}}_{i∈I} be a subordinate partition of unity. For each i ∈ I fix an open set U′_{i} ⊂ π^{−1}U_{i}, such that π : U′_{i} → U_{i} is a diffeomorphism. The projection p : M̃ × C_{r}^{∗}Γ → P induces isometric isomorphisms p_{i} : U′_{i}×C_{r}^{∗}Γ → P|_{U_{i}}. Hence we get an isometry P → M ×C_{r}^{∗}Γ^{|I|} by mapping s_{x} ∈ P_{x} to (x, χ_{i}(x)v_{i})_{i∈I}, where v_{i} is defined by the equation p_{i}^{−1}s_{x} = (x′, v_{i}) for x ∈ U_{i}.

Let g_{ij} ∈ Γ be the deck transformation inducing a diffeomorphism

U′_{i} ∩ π^{−1}U_{j} → U′_{j} ∩ π^{−1}U_{i}.

One verifies easily that

P = (χ_{i}χ_{j}g_{ij})_{ij} ∈ C^{∞}(M, M_{|I|}(C_{r}^{∗}Γ))

is the projection onto the image of the embedding P → M × C_{r}^{∗}Γ^{|I|}.

D and P (⊕^{|I|}D)P are elliptic pseudodifferential operators on L^{2}(M, E ⊗ P) with the same symbol, so we get

ind D̃^{+} = ind D^{+} = ind((P (⊕^{|I|}D)P)^{+}) ∈ K_{0}(C_{r}^{∗}Γ).

Let B_{∞} ⊂ C_{r}^{∗}Γ be a smooth subalgebra containing ℂΓ. Let τ ∈ C_{al}^{n}(Γ) be a cocycle such that c_{τ} extends to a continuous cyclic cocycle on B_{∞}. Such a cocycle τ is called extendable.

Note that P ∈ C^{∞}(M) ⊗ M_{|I|}(ℂΓ), hence its Chern character form is in Ω̂^{∗}(M) ⊗ Ω_{∗}ℂΓ/[Ω_{∗}ℂΓ, Ω_{∗}ℂΓ], which we indicate by writing ch_{ℂΓ}^{M}(P). Here Ω_{∗}ℂΓ is defined using the algebraic tensor product.

By Theorem 3.5

c_{τ} ch_{B_{∞}} ind((P (⊕^{|I|}D)P)^{+}) = (−1)^{k} ∫_{M} Td(M)π_{!} ch^{T M}[σ(D^{+})]c_{τ} ch_{ℂΓ}^{M}(P)

with k = \frac{dim M(dim M+1)}{2}.

It remains to identify c_{τ} ch_{ℂΓ}^{M}(P).

Let ν : M → BΓ be the classifying map of the covering M̃ → M.

Lift the functions χ_{i} to functions χ′_{i} : U′_{i} → ℝ and define h = ∑_{i} χ_{i}^{′2}. In [Lo92] it was shown that for τ ∈ C_{al}^{n}(Γ) with d_{Γ}τ = 0 the differential

form

ω̃_{τ} = ∑_{g_{1},...,g_{n}}(R_{g_{1}}^{∗}d_{M}h) ∧ (R_{g_{2}}^{∗}d_{M}h) ∧ … (R_{g_{n}}^{∗}d_{M}h)τ(e, g_{1}, …, g_{n})

on M̃ is closed and Γ-invariant and that the form ω_{τ} ∈ Ω^{n}(M) defined by π^{∗}ω_{τ} = ω̃_{τ} fulfills

ν^{∗}[τ] = [ω_{τ}] ∈ H^{n}(M)

under the identification H^{∗}(Γ) ≅ H^{∗}(BΓ).

Proposition 4.2. Let τ ∈ C_{al}^{n}(Γ) n ≥ 1 with d_{Γ}τ = 0. Then on the level of differential forms

c_{τ} (ch_{ℂΓ}^{M}(P)) = (−1)^{\frac{(n−1)n}{2}} ((2πi)^{n}n!)^{−1} ω_{τ} .

Hence in H^{n}(M)

[c_{τ} (ch_{ℂΓ}^{M}(P))] = (−1)^{\frac{(n−1)n}{2}} ((2πi)^{n}n!)^{−1} ν^{∗}[τ].

Proof. The following calculation is a modification of an argument in [Lo92]. See also [Wu97, §3].

First we show that the connection P d P behaves like a flat connection in the sense that in the expansion of c_{τ} (ch_{ℂΓ}^{M}(P)) with respect to the decomposition d_{tot}P = d_{M}P + d P all terms containing the factor P d P d P vanish.

By the cyclicity of the trace we only need to consider terms of the form c_{τ} (tr P (d_{tot}P)^{m}P (d P)(d P)). Using the Leibniz rule for d we deduce that tr P (d_{tot}P)^{m}P (d P)(d P) can be written as

∑_{i_{0},i_{1},...,i_{n}∈I} χ_{i_{0}}χ_{i_{1}}f_{i_{1}...i_{n−1}}χ_{i_{n−1}}χ_{i_{n}}^{2}χ_{i_{0}}g_{i_{0}i_{1}} d g_{i_{1}i_{2}} … d g_{i_{n−1}i_{n}} d g_{i_{n}i_{0}}

in Ω^{∗}M ⊗ Ω̂_{∗}ℂΓ/[Ω̂_{∗}ℂΓ, Ω̂_{∗}ℂΓ] with f_{i_{1}...i_{n−1}} ∈ Ω^{∗}M. Since

χ_{i_{0}}χ_{i_{1}}f_{i_{1}...i_{n−1}}χ_{i_{n−1}}χ_{i_{n}}^{2}χ_{i_{0}}c_{τ} (g_{i_{0}i_{1}} d g_{i_{1}i_{2}}, …, d g_{i_{n−1}i_{n}} d g_{i_{n}i_{0}})

= χ_{i_{1}}f_{i_{1}...i_{n−1}}χ_{i_{n−1}}χ_{i_{n}}^{2}χ_{i_{0}}^{2}τ(e, g_{i_{1}i_{2}}, g_{i_{1}i_{2}}g_{i_{2}i_{3}}, …, g_{i_{1}i_{2}} … g_{i_{n−1}i_{n}}g_{i_{n}i_{0}})

= χ_{i_{1}}f_{i_{1}...i_{n−1}}χ_{i_{n−1}}χ_{i_{n}}^{2}χ_{i_{0}}^{2}τ(e, g_{i_{1}i_{2}}, g_{i_{1}i_{3}}, …, g_{i_{1}i_{n}}, g_{i_{1}i_{0}})

and since τ is antisymmetric, the summands are antisymmetric with respect to the pair (i_{0}, i_{n}), hence c_{τ} (tr P (d_{tot}P)^{m}(d P)(d P)) = 0.

Thus

c_{τ} (ch_{ℂΓ}^{M}(P)) = \frac{(−1)^{n}}{(2πi)^{n}n!}c_{τ} tr (P (d_{M}P)(d P) + P (d P)(d_{M}P))^{n} .

We have the equation

(P (d_{M}P)(d P)P)_{im}

= ∑_{j,k,l∈I} χ_{i}χ_{j}g_{ij}d_{M}(χ_{j}χ_{k})g_{jk}χ_{k}χ_{l}(d g_{kl})χ_{l}χ_{m}g_{lm}

= ∑_{j,k,l∈I} χ_{i}χ_{j}d_{M}(χ_{j}χ_{k})χ_{k}χ_{l}^{2}χ_{m}(g_{ik} d g_{km} − g_{il} d g_{lm})

= \frac{1}{2} ∑_{j,k,l∈I}(χ_{i}(d_{M}χ_{j}^{2})χ_{k}^{2}χ_{l}^{2}χ_{m} + χ_{i}χ_{j}^{2}(d_{M}χ_{k}^{2})χ_{l}^{2}χ_{m})(g_{ik} d g_{km} − g_{il} d g_{lm})

= \frac{1}{2} ∑_{k∈I} χ_{i}(d_{M}χ_{k}^{2})χ_{m}g_{ik} d g_{km}.

Here we used ∑_{i∈I} χ_{i}^{2} = 1.

A similar calculation for P (d P)(d_{M}P)P shows that

P (d_{M}P)(d P)P = P (d P)(d_{M}P)P.

It follows that

((P (d_{M}P)(d P)P + P (d P)(d_{M}P)P)^{n})_{im}

= (−1)^{\frac{(n−1)n}{2}} ∑_{i_{1},...,i_{n}} χ_{i}d_{M}(χ_{i_{1}}^{2})d_{M}(χ_{i_{2}}^{2}) … d_{M}(χ_{i_{n}}^{2})

· χ_{m}g_{ii_{1}} d g_{i_{1}i_{2}} d g_{i_{2}i_{3}} … d g_{i_{n}m}.

Hence c_{τ} (ch_{ℂΓ}^{M}(P)) equals, up to the factor (−1)^{\frac{(n+1)n}{2}} ((2πi)^{n}n!)^{−1},

∑_{i_{0},i_{1},...,i_{n}} χ_{i_{0}}^{2}d_{M}(χ_{i_{1}}^{2})d_{M}(χ_{i_{2}}^{2}) … d_{M}(χ_{i_{n}}^{2})c_{τ} (g_{i_{0}i_{1}} d g_{i_{1}i_{2}} d g_{i_{2}i_{3}} … d g_{i_{n}i_{0}})

= ∑_{i_{0},i_{1},...,i_{n}} χ_{i_{0}}^{2}d_{M}(χ_{i_{1}}^{2})d_{M}(χ_{i_{2}}^{2}) … d_{M}(χ_{i_{n}}^{2})τ(g_{i_{0}i_{1}}, g_{i_{0}i_{2}}, g_{i_{0}i_{3}}, …, g_{i_{0}i_{n}}, e).

The Γ-invariant lift of the sum to M̃ equals

∑_{g_{0},g_{1},...,g_{n}∈Γ} R_{g_{0}}^{∗}hR_{g_{1}}^{∗}d_{M}hR_{g_{2}}^{∗}d_{M}h … R_{g_{n}}^{∗}d_{M}hτ(g_{1}, g_{1}g_{2}, …, g_{1} … g_{n}, e)

= (−1)^{n} ∑_{g_{0},g_{1},...,g_{n}} R_{g_{0}}^{∗}(h(R_{g_{1}}^{∗}d_{M}h)(R_{g_{1}g_{2}}^{∗}d_{M}h) …

… (R_{g_{1}...g_{n}}^{∗}d_{M}h))τ(e, g_{1}, g_{1}g_{2}, …, g_{1} … g_{n})

= (−1)^{n} ∑_{g_{0},g_{1},...,g_{n}} R_{g_{0}}^{∗}(h(R_{g_{1}}^{∗}d_{M}h)(R_{g_{2}}^{∗}d_{M}h) …

… (R_{g_{n}}^{∗}d_{M}h))τ(e, g_{1}, g_{2}, …, g_{n})

= (−1)^{n} ∑_{g_{0}} R_{g_{0}}^{∗}(hω̃_{τ}) = (−1)^{n}ω̃_{τ} ,

as desired. □

Corollary 4.3. For any cocycle τ ∈ C_{al}^{n}(Γ), n ≥ 1 such that c_{τ} extends to a continuous cocycle on C_{n}^{λ}(B_{∞})

c_{τ} ch_{B_{∞}} ind D̃^{+} = (−1)^{k}((2πi)^{n}n!)^{−1} ∫_{M} Td(M)π_{!} ch^{T M}[σ(D^{+})]ν^{∗}[τ]

with k = \frac{dim M(dim M+1)}{2} + \frac{n(n−1)}{2}.

The formula for the higher index in K_{1}(C_{r}^{∗}Γ) of an ungraded operator is analogous.

If D is the signature operator on M, then the right-hand side equals the higher signature associated to τ, up to normalization. The Novikov conjecture for extendable τ follows then from the homotopy invariance of ind D̃^{+}.

In the following we give an example of a smooth subalgebra B_{∞} with ℂΓ ⊂ B_{∞} ⊂ C_{r}^{∗}Γ. Its construction is typical for the construction of smooth subalgebras from unbounded derivations. Up to minor details the construction is due to Connes–Moscovici [CM90] who showed that for Γ Gromovhyperbolic every class in H^{∗}(Γ) has a representative that is extendable with respect to this particular B_{∞}, which implies the Novikov conjecture for Gromov-hyperbolic groups.

Let l be a word length function on Γ. Define an unbounded operator D_{l} on l^{2}(Γ) with domain ℂΓ by D_{l}g := l(g)g. Let A be the smallest subalgebra of B(l^{2}(Γ)) containing ℂΓ and multiplication by elements of l^{∞}(Γ) . We define an unbounded derivation

δ : A → B(l^{2}(Γ)), δ(T) := [D_{l}, T].

Note that in general δ(g) is not Γ-invariant, hence δ(ℂΓ) is not a subset of C_{r}^{∗}(Γ).

Lemma 4.4. (1) δ is closable.

(2) δ(A) ⊂ A.

Proof. (1) Let (T_{n})_{n∈ℕ} ⊂ dom δ converge to zero in B(l^{2}(Γ)) with

lim_{n→∞}[D_{l}, T_{n}] = L ∈ B(l^{2}(Γ)).

For f ∈ l^{2}(Γ) we have that [D_{l}, T_{n}]f = D_{l}(T_{n}f) − T_{n}D_{l}(f). The second term on the right-hand side converges to zero, hence the first converges to Lf. Since T_{n}f converges to 0 and D_{l} is closable, it follows that Lf = 0.

(2) It is clear that δ annihilates multiplication operators. From

[D_{l}, g]h = (l(gh) − l(h))gh

and |l(gh) − l(h)| ≤ l(g) it follows that δ(g) ∈ A. □

Let δ_{1} be the closure of δ. For i ∈ ℕ define inductively the Banach algebra A_{i} = dom δ_{i} with norm ∥a∥_{i} = ∥a∥_{i−1} + ∥δ_{i}(a)∥_{i−1} and the operator δ_{i+1} as the closure of δ on A_{i}. Let A_{∞} be the projective limit of the Banach algebras A_{i}. Then A_{∞} is a locally m-convex Fréchet algebra. Let B_{i} be the Banach algebra A_{i} ∩ C_{r}^{∗}Γ and B_{∞} = A_{∞} ∩ C_{r}^{∗}Γ. Then A ∩ C_{r}^{∗}Γ ⊂ ℂΓ ⊂ B_{∞}, hence B_{∞} is dense in C_{r}^{∗}Γ and ℂΓ is dense in B_{∞}.

Lemma 4.5. The algebras B_{i}, i ∈ ℕ, and B_{∞} are closed under holomorphic functional calculus in C_{r}^{∗}Γ.

Proof. We set B_{0} = C_{r}^{∗}Γ and show that B_{i+1} is closed under holomorphic functional calculus in B_{i} for each i ∈ ℕ_{0} .

If x ∈ B_{i+1} with ∥1 − x∥_{i} < 1, then x^{−1} = ∑_{n=0}^{∞}(1 − x)^{n} ∈ B_{i}. Since δ_{i+1}((1 − x)^{n}) ∈ A_{i} and

∥δ_{i+1}((1 − x)^{n})∥_{i} ≤ n∥(1 − x)∥_{i}^{n−1}∥δ_{i+1}(1 − x)∥_{i},

we have that x^{−1} ∈ dom δ_{i+1} ∩ C_{r}^{∗}Γ = B_{i+1}. Since A is dense in A_{i}, for general x ∈ B_{i+1} such that x^{−1} ∈ B_{i} exists there is y ∈ A ∩ B_{i} such that ∥1 − xy∥_{i} ≤ \frac{1}{2}. Hence x^{−1} = y(xy)^{−1} ∈ B_{i+1}. □

We can also obtain an index theorem for the operator D twisted by the Mishchenko–Fomenko bundle P(m) := M̃×_{Γ}C^{∗}Γ, where C^{∗}Γ is the maximal group algebra of Γ. This is a refined version of the theorem for C_{r}^{∗}Γ since there is a surjective homomorphism p : C^{∗}Γ → C_{r}^{∗}Γ. We set B(m) = C^{∗}Γ and B(m)_{i} = p^{−1}B_{i}. The norm on B(m)_{i} is given by ∥a∥_{i} = ∥a∥ + ∥p(a)∥_{i}, where ∥a∥ is the norm of a in C^{∗}Γ. It is straight-forward to check that B(m)_{i} is closed under holomorphic functional calculus in B(m).

In the following we briefly discuss how these results modify in the situation of flat foliated bundles. Our approach is motivated by the approach taken in [J97].

If not specified, the notation is as before. Let X be a closed manifold. We assume that Γ acts on X from the right by diffeomorphisms. By pullback one gets an induced left action on the algebra Ω^{∗}(X) of continuous differential forms on X.

Thus we get reduced crossed products B := C(X) ×_{r} Γ and Ω^{∗}(X) ×_{r} Γ. We denote elements of Ω^{∗}(X) by greek letters in the following and assume

they are homogeneous, where necessary. Let g, h ∈ Γ.

Multiplication on Ω^{∗}(X) ×_{r} Γ is given by the formula

(αh)(βg) = (α ∧ h^{∗}β)hg.

Ω^{∗}(X) ×_{r} Γ acts faithfully on the Hilbert Ω^{∗}(X)-module Ω^{∗}(X, l^{2}(Γ)) by (ωg)(αv) = (ω ∧ g^{∗}α)gv, where v ∈ l^{2}(Γ).

A smooth subalgebra of Ω^{∗}(X) ×_{r} Γ was constructed in [J97] if the group acts isometrically with respect to some Riemannian metric on X. The following construction, which is a generalization of the construction above, works in general.

Let A ⊂ B(Ω^{∗}(X, l^{2}(Γ))) be the algebra generated by multiplication operators associated to elements in l^{∞}(Γ) and by the algebraic crossed product C^{∞}(X, ΛT^{∗}X) ×_{alg} Γ ⊂ Ω^{∗}(X) ×_{r} Γ. One checks that δ(T) = [D_{l}, T] is a derivation on A. We denote by A the closure of A in B(Ω^{∗}(X, l^{2}(Γ))). The de Rham operator d_{X} on X defines a derivation on C^{∞}(X, ΛT^{∗}X) ×_{alg} Γ by d_{X}(ωg) := (d_{X}ω)g. The Leibniz rule follows from

d_{X}(αhβg) = d_{X}(α ∧ h^{∗}β)hg
= (d_{X}α ∧ h^{∗}β + (−1)^{|α|}α ∧ h^{∗}d_{X}β)hg
= (d_{X}α)hβg + (−1)^{|α|}αh(d_{X}β)g.

We extend d_{X} to a derivation on A by letting it commute with multiplication operators coming from elements in l^{∞}(Γ).

Lemma 4.6. The derivations d_{X} and δ on A are closable.

Proof. Let (a_{n})_{n∈ℕ} ⊂ A be a sequence converging to zero in A and with lim_{n→∞} d_{X}(a_{n}) = a ∈ A. For f ∈ C^{∞}(X, ΛT^{∗}X) ⊗ ℂΓ ⊂ Ω^{∗}(X, l^{2}(Γ))

af = lim_{n→∞} d_{X}(a_{n}f).

Choose a Riemannian metric on X and let Ω_{(2)}^{∗}(X) be the L^{2}-completion of Ω^{∗}(X). The unbounded operator d_{X} is densely defined and closable on the Hilbert space Ω_{(2)}^{∗}(X) ⊗ l^{2}(Γ). Since there is a continuous injection Ω^{∗}(X, l^{2}(Γ)) → Ω_{(2)}^{∗}(X) ⊗ l^{2}(Γ), the operator d_{X} is closable on Ω^{∗}(X, l^{2}(Γ)) (considered here as a Banach space) as well. Hence af = 0.

The proof of the closability of δ is analogous, using the closability of D_{l} on Ω_{(2)}^{∗}(X) ⊗ l^{2}(Γ). □

Let now δ_{1}, d_{1}^{X} be the closure of δ, d_{X} respectively. We define, generalizing the above construction, for i ∈ ℕ inductively the Banach algebra A_{i} = dom δ_{i} ∩ dom d_{i}^{X} with norm

∥a∥_{i} = ∥a∥_{i−1} + ∥δ_{i}(a)∥_{i−1} + ∥d_{i}^{X}(a)∥_{i−1}

and the operator δ_{i+1}, d_{i+1}^{X} as the closure of δ, d_{X}, respectively, on A_{i}. Then A_{i} is closed under holomorphic functional calculus in A. Again, let B_{i} be the Banach algebra A_{i} ∩ B. We have that A ∩ B = C^{∞}(X) ×_{alg} Γ, which is dense in B_{i} for any i.

If Γ is trivial, then there is a continuous embedding from B_{i} to C^{1}(X). Examples of cyclic cocycles on C^{1}(X) are the traces f ↦ ∫_{X} α ∧ df for α a closed k form, k = dim X − 1.

We define a B-vector bundle P^{X} := M̃ ×_{Γ} B. An isometric embedding P^{X} → M × B^{|I|} can constructed as above. (Indeed, it holds that P^{X} = P ⊗_{C_{r}^{∗}Γ} B.) Now let F be a hermitian vector bundle on M̃ ×_{Γ} X and F → (M̃×_{Γ}X)×ℂ^{p} an isometric embedding. We let P_{F} : (M̃×_{Γ}X)×ℂ^{p} → F be the orthonormal projection and denote by P̃_{F} the Γ-invariant lift to (M̃ × X) × ℂ^{p}.

Lemma 4.7. Let T be a C^{∞}(X)-linear Γ-equivariant differential operator on C_{c}^{∞}(M̃ × X). Then T descends to a differential operator T on C^{∞}(M, P^{X}). The map T ↦ T is compatible with taking sums and products of differential operators.

Proof. The operator T induces a Γ-equivariant operator on C_{c}^{∞}(M̃× X) ⊗ ℂΓ and extends to a Γ-equivariant operator on C^{∞}(M̃, C(X) ×_{r} Γ), thus is well-defined on C^{∞}(M̃, C(X) ×_{r} Γ)^{Γ} ≅ C^{∞}(M, P^{X}). □

Thus the projection P̃_{F} defines a projection P_{F} on C(M, P^{X})^{p} and therefore a C(X) ×_{r} Γ-vector bundle F on M. We denote the composition of the projection M × B^{|I|p} → (P^{X})^{p} with P_{F} by P_{F} again. We have that P_{F} ∈ C^{∞}(M, M_{|I|p}(B_{∞})).

Now one can apply Theorem 3.5 to the pairing of [D] ∈ KK_{0}(C(M), ℂ) with [P_{F}] ∈ K_{0}(C(M, C(X) ×_{r} Γ)), where D is as before.

For cyclic cocycles concentrated at the conjugacy class of the identity (see [GL03] for the terminology) on B_{∞} one can evaluate the pairing with ch_{B_{∞}}^{M}(P_{F}) further, in a similar but more complicated way as in Prop. 4.2, see [J97][GL03] for related calculations. We refrain from giving details since the formula would be a special case of [GL03]. (To be precise, in [GL03] it was assumed that the holonomy groupoid is Hausdorff and remarked that the results might hold in general. Here we do not make this assumption.)

As above we also get index formulas if we take the maximal crossed product.

4.2. An index theorem for C^{∗}-dynamical systems. Let G be an ndimensional oriented compact Lie group with n odd. We assume that G is endowed with an invariant Riemannian metric with unit volume. Let D be an invariant Dirac operator associated to an invariant Dirac bundle E on G.

Let (A, G, α) be C^{∗}-dynamical system associated to G. Hence A is a C^{∗}-algebra, which we assume to be unital, and α : G → Aut A is a group homomorphism such that the map A → C(G, A), a ↦ α(a) is a well-defined homomorphism of C^{∗}-algebras.

The operator P = 1_{≥0}(D) acts on the Hilbert A-module H := L^{2}(G, E ⊗ A). We define the following Toeplitz type extension: Let T (D, α) be the C^{∗}-algebra generated by the compact operators K(P H) on P H and the Toeplitz operators P α(a)P, a ∈ A.

Let Ψ_{A}(E) be the closure in B(L^{2}(G, E ⊗ A)) of the algebra of classical pseudodifferential operators of order smaller than or equal to zero. We obtain a commutative diagram with exact rows

0 ⟶ K(P H) ⟶ T (D, α) ⟶  A  ⟶ 0
↓  ↓  ↓
0 ⟶ K(H) ⟶ Ψ_{A}(E) ⟶^{σ} C(SG, End E) ⊗ A ⟶ 0.

See Appendix A.2 for the exactness of the second row. Here the second vertical map is defined by P α(a)P ↦ (P α(a)P + (1 − P)) and the last vertical map is defined by a ↦ σ(1 − P) + (α(a) ∘ p)σ(P) where p : SG → G is the projection. Since this map is injective, the last map in the first row, defined as P α(a)P ↦ a, is well-defined.

The connecting map K_{1}(A) → K_{0}(K(P H)) ≅ K_{0}(A) maps [u] ∈ K_{1}(A) with u ∈ U_{k}(A) to ind((⊕^{k}P)α(u)(⊕^{k}P)) ∈ K_{0}(A), where (⊕^{k}P)α(u)(⊕^{k}P) is understood as a Fredholm operator on (P H)^{k} and α(u) is defined by applying α componentwise. For notational simplicity we assume that k = 1 in the following. If μ : C(G) → B(L^{2}(G, E)) is the multiplication operator, then (L^{2}(G, E), μ, D) is an unbounded Kasparov (C(G), ℂ)-module, which can be paired with [α(u)] ∈ K_{1}(C(G, A)). By Proposition A.6

ind(P α(u)P) = [α(u)] ⊗_{C(G)} [(L^{2}(G, E), μ, D)].

In order to apply Theorem 3.5 define the algebra

A_{∞} = {x ∈ A | (g ↦ α_{g}(x)) ∈ C^{∞}(G, A)}.

Endowed with the subspace topology of C^{∞}(G, A) this is a smooth subalgebra of A. Let 𝔤 = T_{e}G and define

d_{e} : A_{∞} → 𝔤^{∗} ⊗ A_{∞}, a ↦ d_{G}(α(a))(e).

We denote by [𝔤] ∈ Λ^{n}𝔤 the dual of the volume form of G at e. Let τ be an invariant trace on A_{∞}. Then for u ∈ U(A_{∞}) we have that α(u) ∈ U(C^{∞}(G, A_{∞})). We obtain the following index formula:

τ(ind(P α(u)P)) = τ(ch_{A_{∞}}[α(u)][D])

= (−1)^{\frac{n(n+1)}{2}} ∫_{G} Â(G) ch(E/S)τ(ch_{A_{∞}}^{G} α(u))

= (−1)^{\frac{n(n+1)}{2}} (\frac{−1}{2πi})^{n} \frac{(n − 1)!}{(2n − 1)!}⟨τ(u^{∗}d_{e}u((d_{e}u^{∗})(d_{e}u))^{k−1}), [𝔤]⟩.

The second equality follows from Theorem 3.5.

As an example consider G = S^{1}, E = S^{1} × ℂ and D = \frac{1}{i} \frac{d}{dx}. Let

δ(u) := \frac{d}{dt}α_{t}(u)|_{t=0} = d_{e}(u)[T_{e}S^{1}].

Then we get

(4.2.1)  τ(ind(P α(u)P)) = −\frac{1}{2πi}τ(u^{∗}δ(u)) = \frac{1}{2πi}τ(uδ(u^{∗})).

If furthermore A = C(S^{1}) with the S^{1}-action α given by translation and τ(1) = 1, then for u = e^{−2πit} the formula gives τ(ind(P α(u)P)) = 1. Hence in this case the above connecting map K_{1}(C(S^{1})) → K_{0}(C(S^{1})) is given by the Bott periodicity isomorphism.

We relate formula (4.2.1) to an index theorem proven in [Le91]: Assume that the trace is faithful, normal and τ(1) = 1. Let A_{τ} be the Hilbert space completion of A with respect to the scalar product ⟨a, b⟩ := τ(a^{∗}b). Let D̃ be the closure of \frac{1}{i} \frac{d}{dx} acting on L^{2}(ℝ, A_{τ}) and let P̃ = 1_{≥0}(D̃). Let α̃ be the lift of α to an action of ℝ on A, furthermore π the left regular representation of A on A_{τ} and λ the representation of ℝ on L^{2}(ℝ) given by (λ(y)f)(x) = f(x − y). Then (π, λ) is a covariant representation of (A, ℝ, α̃) and induces a representation π × λ of the cross product A ×α̃ ℝ on L^{2}(ℝ, A_{τ}). Let N be the von Neumann algebra generated by the image of π × λ in B(L^{2}(ℝ, A_{τ})). By [Le91] the operator P̃α̃(u)P̃ is Breuer–Fredholm in P̃N P̃ and

ind_{τ} (P̃α̃(u)P̃) = \frac{1}{2πi}τ(uδ(u^{∗})),

where ind_{τ} denotes the index with respect to the trace τ. We conclude:

Proposition 4.8.

ind_{τ} (P̃α̃(u)P̃) = τ(ind(P α(u)P)).

Appendix A.

A.1. Index theory and KK-theory. Let A, B be unital C^{∗}-algebras. We recall the notion of a truly unbounded Kasparov (A, B)-module from [Wa07]. Such modules define elements in KK_{∗}(A, B). We then express the pairing of K_{∗}(A) with a truly unbounded Kasparov (A, B)-module in terms of index theory. We refer to [B98] for more about KK-theory.

Let H be a countably generated Hilbert B-module.

Recall that a densely defined selfadjoint operator D on H is regular if (1 + D^{2}) has a bounded inverse. For a selfadjoint regular operator D on H we denote by H(D) the Hilbert B-module whose underlying B-module is dom D and whose B-valued scalar product is given by

⟨x, y⟩_{D} := ⟨x, y⟩ + ⟨Dx, Dy⟩,

where ⟨ , ⟩ is the B-valued scalar product on H. We say that a regular selfadjoint operator D is Fredholm if F := D(1 + D^{2})^{− \frac{1}{2}} is invertible in the Calkin algebra B(H)/K(H). This is equivalent to D : H(D) → H being Fredholm and also to the existence of an odd nondecreasing smooth function χ whose limit at ±∞ is ±1 such that χ(D)^{2} − 1 ∈ K(H). Such a function is called a normalizing function for D.

For the definition of the index note that by the Stabilization Theorem

H ⊕ H_{B} ≅ H_{B}.

Hence there is an inclusion K(H) → K(H_{B}). The induced map

K_{i}(K(H)) → K_{i}(K(H_{B}))

does not depend on the choice of the isomorphism. Thus we get a map

K_{i}(B(H)/K(H)) → K_{i+1}(K(H)) → K_{i+1}(K(H_{B})) ≅ K_{i+1}(B).

If H is ℤ/2-graded with H^{+} ≅ H^{−} and D is odd, then we identify H^{+} with H^{−} and define ind D^{+} as the image of [D^{+}(1+D^{2})^{− \frac{1}{2}} ]∈ K_{1}(B(H^{+})/K(H^{+})) in K_{0}(B). If H^{+} is not isomorphic to H^{−}, we define the index of D as the index of the direct sum of D with an invertible odd operator on H_{B}^{+} ⊕ H_{B}^{−}. This works by the Stabilization Theorem.

In the case where H is ungraded, the index ind(D) ∈ K_{1}(B) is defined as the image of [2χ(D) − 1] ∈ K_{0}(B(H)/K(H)) in K_{1}(B).

Definition A.1. Let H be a Hilbert C^{∗}-module and ρ : A → B(H) a unital C^{∗}-homomorphism.

Let D be a selfadjoint regular Fredholm operator and assume that there is a dense subset A_{∞} ⊂ A such that for all a ∈ A_{∞} the operator [D, ρ(a)] is defined on a core for D and extends to a compact operator from H(D) to H and that there is x ∈ [0, \frac{1}{2}) such that [D, ρ(a)](1 + D^{2})^{−x} is bounded.

Then (H, ρ, D) is called a truly unbounded odd Kasparov (A, B)-module. If in addition H is ℤ/2-graded, ρ is even and D is odd, then (H, ρ, D) is

called a truly unbounded even Kasparov (A, B)-module.

A truly unbounded Kasparov (A, B)-module (H, ρ, D) is called an unbounded Kasparov (A, B)-module if (D^{2} + 1)^{−1} ∈ K(H) and if [ρ(a), D] is bounded for a ∈ A_{∞}.

A truly unbounded Kasparov (A, B)-module (H, ρ, F) is a called a bounded Kasparov (A, B)-module if F is bounded and F^{2} − 1 ∈ K(H).

If (H, ρ, D) is a truly unbounded odd (resp. even) Kasparov (A, B)-module and χ is a normalizing function for D, then (H, ρ, χ(D)) is a bounded odd (resp. even) Kasparov (A, B)-module. The class [(H, ρ, D)] in KK_{1}(A, B) (resp. KK_{0}(A, B)) is defined as the class [(H, ρ, χ(D))].

For example let D be an elliptic scalar selfadjoint pseudodifferential operator on a closed manifold M and let A = C(M), A_{∞} = C^{∞}(M) and for a ∈ A let ρ(a) ∈ B(L^{2}(M)) be the multiplication operator. Then (L^{2}(M), ρ, D) is an odd truly unbounded Kasparov (C(M), ℂ)-module.

If A = ℂ and ρ : ℂ → B(H) is the unique unital homomorphism, then we suppress ρ in the notation in the following. We identify KK_{0}(ℂ, B) with K_{0}(B) via the natural isomorphism [(H, D)] ↦ ind(D^{+}) and KK_{1}(ℂ, B) with K_{1}(B) via [(H, D)] ↦ ind(D).

For the following lemma note that (H, ρ, D) is an even (resp. odd) truly unbounded Kasparov (A, B)-module, then (H^{n}, M_{n}(ρ), ⊕^{n}D) is an even (resp. odd) Kasparov (M_{n}(A), B)-module and that a projection P ∈ M_{n}(A) defines a class [P]_{1} ∈ K_{0}(A) as well as [P]_{n} ∈ K_{0}(M_{n}(A)).

In the following we write a for ρ(a).

Lemma A.2. Let P ∈ M_{n}(A) be a projection.

Let (H, ρ, D) be an even (resp. odd) truly unbounded (A, B)-Kasparov module.

Then in K_{0}(B) (resp. K_{1}(B))

[P]_{n} ⊗_{M_{n}(A)} [(H^{n}, M_{n}(ρ), ⊕^{n}D)] = [P]_{1} ⊗_{A} [(H, ρ, D)].

Proof. By Morita-equivalence the map i : A → M_{n}(A), a ↦ aE_{11} induces an isomorphism in KK-theory. We have that

[(H, ρ, D)] = i^{∗}[(H^{n}, M_{n}(ρ), ⊕^{n}D)],

hence

[P]_{1} ⊗_{A} [(H, ρ, D)] = [P]_{1} ⊗_{A} i^{∗}[(H^{n}, M_{n}(ρ), ⊕^{n}D)]
= i_{∗}[P]_{n} ⊗_{M_{n}(A)} [(H^{n}, M_{n}(ρ), ⊕^{n}D)].

Since i_{∗}[P]_{1} = [P]_{n},

[P]_{1} ⊗_{A} [(H, ρ, D)] = [P]_{n} ⊗_{M_{n}(A)} [(H^{n}, M_{n}(ρ), ⊕^{n}D)].  □

Lemma A.3. Let P ∈ A be a projection.

Let (H, ρ, F) be an even (resp. odd) bounded (A, B)-Kasparov module. If F = P F P + (1 − P)F (1 − P), then in KK_{0}(ℂ, B) (resp. KK_{1}(ℂ, B))

[P] ⊗_{A} [(H, ρ, F)] = [P H, P F P].

Proof. Define the C^{∗}-algebra A_{P} = P AP ⊂ A and let i : A_{P} → A be the injection. Let p : ℂ → A_{P} the unique unital homomorphism. Then [(A_{P} , p, 0)] ∈ KK_{0}(ℂ, A_{P}) and i_{∗}[(A_{P} , p, 0)] = [P]. Hence

\begin{aligned}[P] ⊗_{A} [(H, ρ, F)] & = [p] ⊗_{A_{P}} i^{∗}[(H, ρ, F)] \\ & = p^{∗}i^{∗}[P H, ρ, P F P] \\ & = [P H, ρ ∘ i ∘ p, P F P] \\ & = [P H, P F P]. \end{aligned}  □

Lemma A.4. Let D be a regular selfadjoint operator and let A be an unbounded symmetric operator on H with dom D contained in dom A. Assume that A : H(D) → H is compact. Then D+A is regular. If furthermore there is x < \frac{1}{2} such that A(1 + D^{2})^{−x} is bounded, then

f(D + A) − f(D) ∈ K(H_{A})

for any function f ∈ C(ℝ) such that lim_{x→∞} f(x) and lim_{x→−∞} f(x) exist.

Proof. Let (ϕ_{n})_{n∈ℕ} ⊂ C_{c}(ℝ) be a uniformly bounded sequence converging uniformly to 1 on each compact subset of ℝ. Then ϕ_{n}(D)Aϕ_{n}(D) converges to A in B(H(D), H). Furthermore ϕ_{n}(D)Aϕ_{n}(D) ∈ B(H). Since D is regular, the operators D ± i are invertible. Hence there is n such that D + A − ϕ_{n}(D)Aϕ_{n}(D) ± i are invertible, thus D + A − ϕ_{n}(D)Aϕ_{n}(D) is regular. A bounded perturbation of a regular operator is regular, thus D+A is regular.

For D_{0} = D + A and D_{1} = D let F_{i} = D_{i}(1 + D_{i}^{2})^{−1/2}. As in the proof of [Wa07, Proposition 3.7.] it follows that F_{0} − F_{1} ∈ K(H_{A}) . Then π(F_{0}) = π(F_{1}), where π : B(H) → B(H)/K(H) is the projection, hence π(f(F_{0})) = π(f(F_{1})) for any function f ∈ C([−1, 1]). □

Proposition A.5. Let P ∈ M_{n}(A_{∞}) be a projection.

Let (H, ρ, D) be an even (resp. odd) truly unbounded Kasparov (A, B)- module. Then in KK_{0}(ℂ, B) (resp. KK_{1}(ℂ, B))

[P] ⊗_{A} [(H, ρ, D)] = [P H^{n}, P (⊕^{n}D)P].

Hence for D even

[P] ⊗_{A} [(H, ρ, D)] = ind(P (⊕^{n}D^{+})P) ∈ K_{0}(B),

and for D odd

[P] ⊗_{A} [(H, ρ, D)] = ind(P (⊕^{n}D)P) ∈ K_{1}(B).

Proof. By the first lemma it is enough to consider the case P ∈ A. Let D_{P} = P DP + (1 − P)D(1 − P). Since D_{P} = D − 2P [D, P] and 2P [D, P] : H(D) → H is compact, the

operator D_{P} is Fredholm. Furthermore by assumption there is x < \frac{1}{2} such that 2P [D, P](1 − D^{2})^{−x} is bounded. Let χ be a normalizing function of D. Then by the previous lemma χ(D) − χ(D_{P}) ∈ K(H), hence in KK_{0}(A, B) (resp. KK_{1}(A, B)) we have that

[(H, ρ, D)] = [(H, ρ, χ(D))] = [(H, ρ, χ(D_{P})].

Hence

[P] ⊗_{A} [(H, ρ, χ(D_{P})] = [P H, P χ(D_{P})P]
= [P H, χ(P D_{P} P)]
= [P H, P DP],

where the first equation follows from the second lemma. □

In the following we use the definition of and results on the relative index of projections and the noncommutative spectral flow from [Wa07]. We also refer to [Wa07] for history and references concerning the noncommutative spectral flow, which generalizes the family spectral flow introduced by Dai– Zhang [DZ98]. We denote the even and the odd spectral flow both by sf. The relative index of a pair of projections and the relative index of pair of Lagrangian projections are denoted by ind.

Let D be a regular selfadjoint Fredholm operator on H. Recall that a selfadjoint operator A ∈ K(H) is called a trivializing operator of D if D +A is invertible. If H is ℤ/2-graded and D is odd, we assume furthermore that A is odd. We point out that in the following proposition we deal with unbounded Kasparov (A, B)-modules.

Proposition A.6. Let (H, ρ, D) be an unbounded even (resp. odd) Kasparov (A, B)-module.

Let U ∈ M_{n}(A_{∞}) be a unitary such that U^{∗} ∈ M_{n}(A_{∞}) as well and let [U] be its class in K_{1}(A).

If there is a trivializing operator A of ⊕^{n}D, then with P = 1_{≥0}(⊕^{n}D +A)

ind(P, UP U^{∗}) = sf((1 − t)(⊕^{n}D) + tU(⊕^{n}D)U^{∗}, A, UAU^{∗})
= [U] ⊗_{A} [(H, ρ, D)].

Without the assumption on the existence of trivializing operators formula (A.1.1) below holds in the ungraded case and formula (A.1.2) in the graded case.

Proof. The equality

ind(P, UP U^{∗}) = sf((1 − t)(⊕^{n}D) + tU(⊕^{n}D)U^{∗}, A, UAU^{∗})

was proven in [Wa07, Example after Proposition 3.15] in the ungraded case. The proof of this formula in the graded case is analogous.

For the second equality it is enough to consider the case n = 1 by the first lemma.

Assume that H is ungraded.

Let χ ∈ C^{∞}(ℝ) be a monotonous function with χ(x) = 0 for x < \frac{1}{3} and χ(x) = 1 for x > \frac{2}{3}.

Let ∂/_{S^{1}} = \frac{1}{i} \frac{d}{dx} on L^{2}(S^{1}) and let [∂/_{S^{1}}] ∈ KK_{1}(C(S^{1}), ℂ) be the corresponding class, where the C(S^{1})-action on L^{2}(S^{1}) is given by multiplication.

Define the A-vector bundle L(U) := ([0, 1] × A)/(0, v) ∼ (1, Uv) on S^{1} and let ∂/_{L(U)} be the Dirac operator ∂/_{S^{1}} twisted by the bundle L(U) with the trivial connection. Then i∂/_{L(U)} +χ(t)D +(1−χ(t))UDU^{∗} is densely defined and and its closure is Fredholm on the Hilbert B-module L^{2}(S^{1}, L(U))⊗_{ρ} H. We claim that in K_{0}(A)

(A.1.1)  [U][D] = ind(i∂/_{L(U)} + χ(t)D + (1 − χ(t))UDU^{∗}).

If D admits a trivializing operator A ∈ B(H), then

sf((1 − t)D + tUDU^{∗}, A, UAU^{∗})
= sf(χ(t)D + (1 − χ(t))UDU^{∗})
= ind(i∂/_{L(U)} + χ(t)D + (1 − χ(t))UDU^{∗})),

where the last equation follows from [Wa07, Proposition 3.15] and the relative K-theoretic index theorem.

Let χ_{0} (resp. χ_{2}) be a smooth positive function equal to 1 on [0, \frac{1}{3}] (resp. on [\frac{2}{3}, 1]) and equal to 0 on [\frac{1}{2}, 1] (resp. on [0, \frac{1}{2}]). Let χ_{1} = \sqrt{1 − \sqrt{χ_{0} + χ_{2}}}; hence χ_{1}^{2} + (χ_{0} + χ_{2})^{2} = 1. It is easy to check that the map M from the bundle L(U) to the range of the projection

P (U) = ( \begin{matrix}χ_{1}^{2} & χ_{1}(χ_{0} + χ_{2}U) \\ χ_{1}(χ_{0} + χ_{2}U^{∗}) & (χ_{0} + χ_{2})^{2} \end{matrix} )

on S^{1} × A^{2} defined by

(x, v) ↦ (x, χ_{1}(x)v ⊕ (χ_{0}(x)v + χ_{2}(x)Uv))

is an isometric isomorphism. It induces an isometric isomorphism between P (U)(L^{2}(S^{1}, A^{2}) ⊗_{ρ} H) and L^{2}(S^{1}, L(U)) ⊗_{ρ} H, denoted by M as well. On L^{2}(S^{1}, L(U)) ⊗_{ρ} H the maps

∂/_{L(U)} − M^{−1}P (U)(⊕^{2}∂/_{S^{1}})P (U)M ∈ M_{2}(C(S^{1}, A))

and χ(t)D+(1−χ(t))UDU^{∗}−M^{−1}P (U)(⊕^{2}D)P (U)M are bounded. Hence in K_{0}(B)

ind(−i∂/_{L(U)}+χ(t)D+(1−χ(t))UDU^{∗}) = ind(P (U)(−i⊕^{2}∂/_{S^{1}} +⊕^{2}D)P (U)).

We have that [U] ⊗_{A} [(H, ρ, D)] = [P (U)] ⊗_{C(S^{1},A)} ([∂/_{S^{1}}] ⊗ [(H, ρ, D)]). Since the Kasparov product [∂/_{S^{1}}] ⊗ [(H, ρ, D)] ∈ KK_{0}(C(S^{1}, A), B) is

represented by the odd selfadjoint operator

( \begin{matrix}0 & −i∂/_{S^{1}} + D \\ i∂/_{S^{1}} + D & 0 \end{matrix} )

on L^{2}(S^{1}, H^{+} ⊕ H^{−}), the previous proposition implies that

([P (U)]⊗_{C(S^{1},A)}[∂/_{S^{1}}])⊗[(H, ρ, D)] = ind(P (U)((−i⊕^{2}∂/_{S^{1}})+(⊕^{2}D))P (U)).

If H is graded, let σ be the grading operator. One proves analogously that in K_{1}(B)

(A.1.2)  [U][D] = ind(−σ∂/_{L(U)} + iσ(χ(t)D + (1 − χ(t))UDU^{∗})).

This, [Wa07, Remark after Proposition 8.4] and the relative index theorem imply the assertion. □

By the following argument, which was pointed out to the author by Ryszard Nest, the pairing of U with (H, ρ, D) as in the proposition can always be expressed in terms of a spectral flow (in the ungraded case): let d be an invertible unbounded operator on H_{B} with compact resolvents and such that the ranges of 1_{≥0}(d) and 1_{≤0}(d) contain a copy of H_{B}. Then the pairing of [U] with [(H, ρ, D)] coincides with the pairing of [U] with the odd Kasparov (A, B)-module (H ⊕ H ⊕ H_{A}, ρ ⊕ 0 ⊕ 0, D ⊕ (−D) ⊕ d). The index of D ⊕ (−D) ⊕ d in K_{1}(B) vanishes, and there are trivializing operators (see [Wa07, §3.1] and references therein). Now apply the proposition.

A.2. Pseudodifferential operators over C^{∗}-algebras. See [MF80] for the definition and general facts about pseudodifferential operators over C^{∗}- algebras.

Let A be a C^{∗}-algebra with unit. Let M be a closed Riemannian manifold and let E be an A-vector bundle over M endowed with an A-valued metric. Endow A^{n} with the standard A-valued scalar product and let E → M × A^{n} be a smooth isometry (such an isometry always exists). Let F be the complement of E in M × A^{n}.

Let Δ be the scalar Laplacian on M. For any p ∈ ℝ we define the Sobolev space H^{p}(M, A^{n}) as the completion of C^{∞}(M, A^{n}) with respect to the norm induced by the A-valued scalar product

⟨f, g⟩_{H^{p}} := ⟨(1 + Δ)^{p/2}f, (1 + Δ)^{p/2}g⟩_{L^{2}} .

We first assume that E = A^{n}.

Let P : C^{∞}(M, E) → C^{∞}(M, E) be a symmetric pseudodifferential operator of order s ∈ ℝ. The nonsymmetric case can be reduced to the symmetric

by considering ( \begin{matrix}0 & P^{∗} \\ P & 0 \end{matrix} ).

In [MF80] it was shown that P : H^{p+s}(M, E) → H^{p}(M, E) is continuous.

Lemma A.7. The operator P : H^{p+s}(M, E) → H^{p}(M, E) is adjointable.

Proof. It is straightforward to check that

P^{T} = (1 + Δ)^{−2s−p}P (1 + Δ)^{p} : H^{p}(M, E) → H^{p+s}(M, E)

is the adjoint of P. □

The proof of the following lemma is analogous to the classical case and is given here for completeness.

Lemma A.8. Let s > 0. If P is elliptic, then P as an unbounded operator on L^{2}(M, E) with domain H^{s}(M, E) is selfadjoint.

Proof. Let Q be the parametrix of P and P^{∗} the adjoint of P. The closure of QP^{∗} equals the closure of QP on L^{2}(M, E). Thus Kh = (1 − QP^{∗})h ∈ C^{∞}(M, E) for h ∈ dom P^{∗}. Clearly QP^{∗}h ∈ dom P . Hence

h = QP^{∗}h − Kh ∈ dom P.  □

Proposition A.9. Let s > 0. If P is elliptic, then P is regular as an unbounded operator on L^{2}(M, E) with domain H^{s}(M, E).

Proof. From

∥(P^{2} + 1)f∥_{L^{2}} ≥ ∥f∥_{L^{2}}, f ∈ C^{∞}(M, E),

it follows that the operator P^{2} + 1 : H^{2s}(M, E) → L^{2}(M, E) is injective and its range is closed. Furthermore it is adjointable by Lemma A.7.

It follows that the range of (1 + P^{2}) : H^{2s}(M, E) → L^{2}(M, E) is complemented. By the previous lemma (1 + P^{2}) is selfadjoint, hence

Coker(1 + P^{2}) = Ker(1 + P^{2}) = {0}.

Therefore (1 + P^{2}) is surjective and thus P is regular. □

Corollary A.10. Let s > 0 and assume that P is elliptic. The identity induces an adjointable isomorphism between the Hilbert A-modules H(P) and H^{s}(M, E).

Proof. The identity H(P) → H^{s}(M, E) equals the composition of

(1 + P^{2})^{\frac{1}{2}} : H(P) → L^{2}(M, E)

with (1 + P^{2})^{− \frac{1}{2}} : L^{2}(M, E) → H^{s}(M, E). The first map is an isometry and the second is adjointable since (1 + P^{2})^{− \frac{1}{2}} is a pseudodifferential operator of order −s. □

Since (1 + P^{2})^{q} is a pseudodifferential operator for any q ∈ ℝ we also get adjointable isomorphisms of Hilbert A-modules

(1 + P^{2})^{q} : H^{p+2qs}(M, E) → H^{p}(M, E).

We conclude that if E is ℤ/2-graded and P is odd, then the index of P^{+} : H^{p+s}(M, E^{+}) → H^{s}(M, E^{−}) is independent of p and equals the index of P^{+}(1 + P^{2})^{− \frac{1}{2}} : H^{p}(M, E^{+}) → H^{p}(M, E^{−}).

For general E let e ∈ C^{∞}(M, M_{n}(A)) be the orthogonal projection onto E. Define Δ_{e} := eΔe+(1−e)Δ(1−e). Since Δ_{e} : H^{2}(M, A^{n}) → L^{2}(M, A^{n}) is regular, the restriction of Δ_{e} to L^{2}(M, E) is regular as well. We define H^{p}(M, E) as the completion of C^{∞}(M, E) with respect to the norm induced by the A-valued scalar product

⟨f, g⟩_{H^{p}} := ⟨(1 + Δ_{e})^{p/2}f, (1 + Δ_{e})^{p/2}g⟩_{L^{2}}.

After replacing Δ by Δ_{e}, the statements of this sections hold for general E.

350 Charlotte Wahl

Furthermore from the previous corollary one can deduce that the injection H^{p}(M, E) → H^{p}(M, A^{n}) is adjointable.

In the end we note the exactness of the sequence associated to the symbol map for classical pseudodifferential operators over C^{∗}-algebras.

We assume that E is the trivial vector bundle with fiber A. The general case can be derived from this. Let Ψ_{A} be the closure of the algebra of classical pseudodifferential operators of order smaller than or equal to zero in B(L^{2}(M, A)). Let SM be the sphere bundle of T M. Since C(SM) and K(L^{2}(M)) are nuclear, the algebra Ψ_{ℂ} is nuclear as an extension of C(SM) by K(L^{2}(M)) [B98, Theorem 15.8.2]. Hence Ψ_{ℂ} ⊗ A ≅ Ψ_{A} where ⊗ is any tensor product of C^{∗}-algebras, and there is the commutative diagram

0 ⟶ K(L^{2}(M, A)) ⟶ Ψ_{ℂ} ⊗ A ⟶ C(SM, A) ⟶ 0
↓  ↓  ↓
0 ⟶ K(L^{2}(M, A)) ⟶ Ψ_{A}  ⟶^{σ} C(SM, A) ⟶ 0.

In particular the second row is exact.

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Leibniz-Forschungsstelle der Göttinger Akademie der Wissenschaften, Waterloostr. 8, 30169 Hannover, Germany wahlcharlotte@googlemail.com

This paper is available via http://nyjm.albany.edu/j/2009/15-18.html.