On the index of equivariant Toeplitz operators

Ulrich Bunke^{∗}

March 10, 2000

Contents

1 Introduction 0

2 The index of Toeplitz operators 0

3 Γ-equivariant Toeplitz operators 3

4 Examples 7

1 Introduction

The goal of the present paper is to understand aspects of the recent preprint [6]. While this paper studies traces of commutators the present paper concentrates on the index theoretic aspects. This allows for studying the index Toeplitz operators under quite general assumptions. The basic observation (compare [5]) is that the index of the Toeplitz operator is equal to the index of an associated Callias type operator, i.e. a Dirac operator with potential. Callias type operators were thoroughly studied in [3]. In particular, their index is very accessible to computation. In the present note we show how to extend all that to the equivariant case.

2 The index of Toeplitz operators

In this Section we review the non-equivariant situation. The main ideas can be traced back to [5].

∗Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, GERMANY, Email:bunke@uni-math.gwdg.de

Let (M, g) be a complete Riemannian manifold and E → M be a Dirac bundle which may be Z_{2}-graded or ungraded. The associated Dirac operator is an unbounded essentially selfadjoint operator on the Hilbert space H := L^{2}(M, E) with domain C_{c}^{∞}(M, E).

Assumption 2.1 We assume that 0 is an isolated point of the spectrum of D.

Let H denote the kernel of D, and let P be the orthogonal projection onto H. In the graded case we have an orthogonal splitting H = H^{+} ⊕ H^{−}, P = P^{+} + P^{−}.

Let C_{g}^{∞}(M) denote the space of all bounded smooth functions such that df vanishes at infinity of M. The commutative C^{∗}-algebra C_{g}(M) is defined as the closure of C_{g}^{∞}(M) inside C(M). Furthermore, let C_{0}(M) denote the closure of C_{c}^{∞}(M) in C(M).

If f ∈ C(M), then M_{f} denotes the multiplication operator on H induced by f.

Definition 2.2 We define the following operators on H, resp. H^{±}:

T_{f} := P M_{f}P, T_{f}^{±} := P^{±}M_{f}P^{±} .

We extend this definition to matrix valued functions f ∈ Mat(n, C(M)) such that M_{f} ∈ B(H ⊗ C^{n}), T_{f} ∈ B(H ⊗ C^{n}), etc.

Lemma 2.3 If f ∈ C_{0}(M), then T_{f} is compact.

Proof. If f ∈ C_{c}^{∞}(M), then M_{f}P is compact by Rellich’s Lemma since H ⊂ H^{1}, where the latter space is the Sobolev space defined as the domain of the closure of the elliptic operator D. The map f ↦ M_{f} is continuous. Since the space of compact operators is norm-closed we conclude that M_{f}P is compact for all f ∈ C_{0}(M). □

Lemma 2.4 If f ∈ C_{g}(M), then [P, M_{f}] is compact.

Proof. It is here where we use the assumption that 0 is an isolated point of σ(D). Let B ⊂ C be a small ball arround zero such that σ(D) ∩ B = {0}. For λ ∉ σ(D) let R_{D}(λ) := (λ − D)^{−1} denote the resolvent. By holomorphic function calculus we can write

P = \frac{1}{2πı} ∫_{∂B} R_{D}(λ)dλ .

Let f ∈ C_{g}^{∞}(M). Then we have

[R_{D}(λ), M_{f}] = R_{D}(λ)[D, M_{f}]R_{D}(λ)
= R_{D}(λ)c(gradf)R_{D}(λ) ,

where c denotes Clifford multiplication. Since gradf vanishes at infinity we conclude that c(gradf)R_{D}(λ) is compact. In fact, we can approximate gradf uniformly by vector fields X of compact support, and XR_{D}(λ) is again compact by Rellich’s Lemma since R_{D}(λ) maps H continuously to H^{1}. Thus [R_{D}(λ), M_{f}] is compact, too. Again refering to the norm-closedness of the space of compact operators on H we see that

[P, M_{f}] = \frac{1}{2πı} ∫_{∂B}[R_{D}(λ), M_{f}] dλ

is compact for all f ∈ C_{g}^{∞}(M), and hence for all f ∈ C_{g}(M). □

Note that Lemma 2.3 and 2.4 extend to matrix valued functions.

The underlying topological space of M can be considered as the spectrum of the commutative C^{∗}-algebra C_{0}(M). The exact sequence of C^{∗}-algebras

0 → C_{0}(M) → C_{g}(M) → C(∂_{g}M) → 0

defines the compactification of M by the Higson corona ∂_{g}M, where ∂_{g}M is the spectrum of the commutative C^{∗}-algebra C(∂_{g}M) := C_{g}(M)/C_{0}(M).

We now consider a continuous unitary matrix valued function F : ∂_{g}M → U(n), or equivalently, a unitary F ∈ U(n, C(∂_{g}M)). In the ungraded case we assume in addition that F is selfadjoint. Let f, g ∈ Mat(n, C_{g}(M)) be extensions of F and F^{−1} to M.

Lemma 2.5 The operator T_{f} is a Fredholm operator. Its index only depends on F, where we define the index in the graded case as index(T_{f}) := index(T_{f}^{+}) − index(T_{f}^{−}).

Proof. A parametrix of T_{f} is given by T_{g}. In fact, if ”∼” denotes equality modulo compact operators, then we have by Lemma 2.4 that T_{f}T_{g} ∼ T_{fg} = 1 + T_{fg−1}. Since fg − 1 ∈ Mat(n, C_{0}(M)) we have by Lemma 2.3 that T_{fg−1} ∼ 0. Similarly we show that T_{g}T_{f} ∼ 1. If f, f′ are two extensions of F, then f − f′ ∈ Mat(n, C_{0}(M)). By Lemma 2.3 we conclude that T_{f} ∼ T_{f′} and hence equality of the indices. □

The goal of the present section is to relate the index of T_{f} with the index of the Calliastype operator constructed in [3] from D and f, g. The Theorems 2.9 and 2.16 of [3] reduce the computation of the index of the Callias type operator and thus of T_{f} to an application of the Atiyah-Singer Index theorem for elliptic differential operators on closed manifolds. In the ungraded case [3], Prop. 2.8 implies that index(T_{f}) = 0, since we assume that σ(D) has a gap.

In the ungraded case we define the Callias-type operator C := D + iM_{f} on E ⊗ C^{n} and put ϵ := 1. In the graded case we define ϵ := −1 and C := D+diag(−M_{g}, M_{f}) on E^{+}⊗C^{n}⊕E^{−}⊗C^{n}.

As shown in Sec. 2 of [3] these Callias-type operators have a well-defined index

index(C) := dim ker_{L^{2}}(C) − dim ker_{L^{2}}(C^{∗}) .

Our main result is

Proposition 2.6

index(T_{f}) = ϵ index(C)

Proof. We first consider the graded case. Let H^{1} be the domain of the closure of (the extension of) D (from E to E ⊗ C^{n}) and put H := L^{2}(M, E ⊗ C^{n}). Then C : H^{1} ↦ H is Fredholm in the usual sense. Set Φ := diag(−M_{g}, M_{f}) and Q := 1 − P. P and Q act on H as well as on H^{1}. It follows from Lemma 2.5 that QΦP and P ΦQ are compact operators from H^{1} to H. Therefore we can replace C by P CP ⊕ QCQ without changing the index. Using the homotopy QC_{t}Q, C_{t} := D + tΦ we can deform QCQ to the invertible operator QC_{0}Q through Fredholm operators. Here we use the fact that C_{t} is Fredholm for all t > 0. We conclude that

index(C) = index(P CP) = index(P ΦP)
= index(T_{g}^{+}) + index(T_{f}^{−}) = −(index(T_{f}^{+}) − index(T_{f}^{−}))
= −index(T_{f}) .

We now come to the ungraded case. We again have

index(C) = index(P CP) + index(QCQ) = index(P CP) = index(T_{f}) ,

since QCQ can be deformed to the invertible operator QC_{0}Q, where C_{t} := D + itM_{f}. □

As explained above we conclude with [3], Prop. 2.8 that

Corollary 2.7 In the ungraded case we have index(T_{f}) = 0.

3 Γ-equivariant Toeplitz operators

Let M̃ → M be a Galois cover with group of deck transformations Γ. We reserve the symbol ”˜ ” to denote lifts of various objects to Γ-coverings.

Assumption 3.1 We can choose a cut-off function χ^{Γ} ∈ C_{g}^{∞}(M̃) such that ∑_{γ∈Γ} γ^{∗}χ^{Γ} = 1, ♯{γ ∈ Γ | γ^{∗}χ^{Γ}χ^{Γ} ≠ 0} < ∞.

In our motivating example M̃ is a symmetric space of rank one such that Γ is a convex-cocompact group of isometries. In this case χ^{Γ} exists by [4], Lemma 6.4.

Assumption 3.2 We assume that 0 is an isolated point of the spectrum of D̃.

If D̃ is a homogeneous Dirac operator on a symmetric space M̃ of rank one and M is a quotient by a convex cocompact subgroup, then by the result of [4] we have σ_{ess}(D) = σ_{ess}(D̃). In particular, if D̃ is one of the Dirac operators constructed by [1] in order to realize the representations of the discrete series, then the assumptions 2.1 and 3.2 are satisfied.

We consider the Hilbert space H̃ := L^{2}(M̃, Ẽ) which carries an unitary representation of Γ. Let B_{Γ} := ^{Γ}B(H̃) denote the Γ-equivariant bounded operators on H̃. Using a fundamental domain F ⊂ M̃ we can write H̃ = L^{2}(Γ) ⊗ H in a Γ-equivariant way. Let N (Γ) ⊂ B(L^{2}(Γ)) be the group von Neumann algebra of all operators commuting with left translations. Then B_{Γ} = N (Γ) ⊗ B(H). Let K_{Γ} ⊂ B_{Γ} be the ideal of Γ-compact operators corresponding to N (Γ) ⊗ K(H). An operator A ∈ B_{Γ} is called Γ-Fredholm if it is invertible modulo K_{Γ}. If A is Γ-Fredholm, then its index is an element index(A) ∈ K_{0}(K_{Γ}) = K_{0}(N (Γ)). The normalized trace τ (which is given by τ(B) := ⟨δ_{e}, Bδ_{e}⟩, B ∈ N (Γ), δ_{e} ∈ L^{2}(Γ), δ_{e}(γ) := 0 or 1 if γ ≠ e or γ = e) on N (Γ) induces a homomorphism τ : K_{0}(N (Γ)) → R. We define the Γ-index of A by index_{Γ}(A) := τ(index(A)).

Lemma 3.3 If A ∈ B_{Γ} and M_{χ^{Γ}}A is compact, then A is Γ-compact.

Proof. We make the isomorphism I : B_{Γ} →^{∼} N (Γ) ⊗ B(H) explicit. First we identify H =

L^{2}(F, Ẽ_{|F}). Then i : H̃ →^{∼} L^{2}(Γ) ⊗ L^{2}(F, Ẽ_{|F}) is given by i(ϕ) := ∑_{γ∈Γ} δ_{γ} ⊗ χ_{F} γ^{−1}ϕ, where

χ_{F} denotes the characteristic function of F. The inverse of this identification is given by i^{−1}(∑_{γ∈Γ} δ_{γ} ⊗ ϕ_{γ}) = ∑_{γ∈Γ} γϕ_{γ}. We now compute I(A) = i ∘ A ∘ i^{−1}

I(A)(∑_{γ∈Γ} δ_{γ} ⊗ ϕ_{γ}) = ∑_{γ,γ′∈Γ} δ_{γ′} ⊗ χ_{F} (γ′)^{−1}γAϕ_{γ}
= ∑_{γ,γ′∈Γ} δ_{γγ′} ⊗ χ_{F} (γ′)^{−1}Aϕ_{γ}
= (∑_{γ′∈Γ} R(γ′) ⊗ A_{γ′})(∑_{γ∈Γ} δ_{γ} ⊗ ϕ_{γ}) ,

thus I(A) = ∑_{γ∈Γ} R(γ) ⊗ A_{γ}, where A_{γ} := χ_{F} γ^{−1}Aχ_{F} and R(γ) ∈ N (γ) is the right translation by γ. If M_{χ^{Γ}}A is compact, then so is M_{γ^{∗}χ^{Γ}}A for all γ ∈ Γ. For all γ ∈ Γ we see that χ_{F} γ^{−1}Aχ_{F} = ∑_{γ′∈Γ} χ_{F} γ^{−1}M_{(γ′)^{∗}χ^{Γ}}Aχ_{F} is compact since χ_{F} γ^{−1}M_{(γ′)^{∗}χ^{Γ}} ≠ 0 for at most finitely many γ′ ∈ Γ. Thus A is Γ-compact. □

If f ∈ C_{g}(M), then we have the multiplication operator Mf̃ ∈ ^{Γ}B_{Γ}. Let H̃ denote the kernel of D̃ and P̃ ∈ ^{Γ}B(H̃) the orthogonal projection onto H̃.

Definition 3.4 We define T̃_{f} := P̃Mf̃P̃ ∈ B_{Γ}.

Lemma 3.5 If f ∈ C_{0}(M), then T̃_{f} is Γ-compact.

Proof. We have

M_{χ^{Γ}} T̃_{f} = M_{χ^{Γ}} P̃Mf̃P̃ ∼^{Lemma2.4} P̃M_{χ^{Γ}f̃}P̃ ∼^{Lemma2.3} 0

since χ^{Γ}f̃ ∈ C_{0}(M̃). The assertion now follows from Lemma 3.3. □

Lemma 3.6 If f ∈ C_{g}(M), then [P̃, Mf̃] is Γ-compact.

Proof. We employ the same method as in the proof of Lemma 2.4 using

M_{χ^{Γ}} R̃(λ)c(gradf̃)R̃(λ) ∼ R̃(λ)M_{χ^{Γ}}c(gradf̃)R̃(λ) ∼ 0

since M_{χ^{Γ}}c(gradf̃) vanishes at infinity of M̃. The assertion now follows from Lemma 3.3. □

Consider F ∈ U(n, C(∂_{g}M)) which is selfadjoint in the ungraded case. Let f, g ∈ Mat(n, C_{g}(M)) be lifts of F, F^{−1}.

Lemma 3.7 The operator T̃_{f} is Γ-Fredholm. Its index only depends on F.

Proof. The proof is analogous to the proof of Lemma 2.5. One has to replace ”compact” by ”Γ-compact and applies Lemma 3.5 and 3.6 instead of Lemma 2.3 and 2.4. □

The following theorem is the main result of the present paper.

Theorem 3.8

index_{Γ}(T̃_{f}) = index(T_{f}) .

Proof. The lifts f̃, g̃ give rise to a Γ-equivariant Callias type operator C̃. In the first step we show that C̃ is Γ-Fredholm, and that its index coincides with the index of T̃_{f} (up to the sign ϵ). In [3], Sec.2 the computation of the index of C was reduced to the computation of the index of an elliptic differential operator R on a closed manifold S^{1} × N using a relative index theorem and a cut-and-past procedure. Doing this cut-and-paste procedure equivariantly in the second step we reduce the computation of the index of C̃ to the computation of the index of the lift R̃ of R to a certain cover S^{1} × Ñ. In the third and final step we apply the Atiyah’s index theorem for coverings in order to conclude that the Γ-index of R̃ coincides with the index of R.

We form the flat bundle of von Neumann algebras V := M̃ ×_{Γ} N (Γ) and let D_{V} be the N (Γ)- equivariant twisted Dirac operator on E ⊗ N (Γ). We now form the N (Γ)-equivariant Callias type operators Ĉ := D_{V} +iM_{f} on E ⊗V ⊗C^{n} in the ungraded and Ĉ := D_{V} +diag(−M_{g}, M_{f}) on E^{+} ⊗ V ⊗ C^{n} ⊕ E^{−} ⊗ V ⊗ C^{n} in the graded case. Combining [3], Lemma 2.6, 2.14, and the proof of Lemma 1.18 we show that the operator Ĉ is invertible at infinity (see [3], Ass. 1). It follows that Ĉ induces a Fredholm operator between the Hilbert-N (Γ) modules H^{1}(M, E ⊗ V ⊗ C^{n}) and L^{2}(M, E ⊗ V ⊗ C^{n}). The tensor products over N (Γ) of these modules with L^{2}(Γ) identify with H^{1}(M̃, Ẽ ⊗ C^{n}) and L^{2}(M̃, Ẽ ⊗ C^{n}), respectively. The operator Ĉ gives rise to the Γequivariant Callias type operator C̃ which is just the lift of C. In particular, we see that C̃ is Γ-Fredholm and index(Ĉ) = index(C̃). We can now apply exactly the same argument as in the proof of Proposition 2.6 in order to show that index(C̃) = ϵ index(T̃_{f}), replacing compactness and Fredholm by the corresponding Γ-equivariant notions. This ends the first step of the proof.

We now come to the second step. In the ungraded case we can repeat the argument of the proof of [3], Prop. 2.8 in order to see that index(Ĉ) = 0 since by assumption there is a gap in the spectrum of D_{V} (note that the spectrum of D_{V} coincides with that of D̃). Thus 0 = index_{Γ}(T̃_{f}). Since index(T_{f}) = 0 by Corollary 2.7 we obtain the assertion on the theorem in the ungraded case. It remains to consider the graded case. Doing the construction [3], 2.4.2 with D and D_{V} at the same time we arrive at a Dirac operator R and its twist R_{W} over a compact manifold S^{1} × N such that index(C) = index(R) and index(Ĉ) = index(R_{W}). Here N is a certain closed hypersurface of M, R is associated to a Dirac bundle L ↦ S^{1} × N, and W is the pull-back to S^{1} × N of the restriction of V to N. Let Ñ be the restriction of the cover M̃ → M to N. The tensor products over N (Γ) of H^{1}(S^{1} × N, L ⊗ W), L^{2}(S^{1} × N, L ⊗ W) with L^{2}(Γ) identify with H^{1}(S^{1} × Ñ, L̃) and L^{2}(S^{1} × Ñ, L̃), respectively. The operator R_{W} induces the Γ-equivariant Dirac operator R̃ on L̃ which is just the lift of R. We have index(R̃) = index(R_{W}). This accomplishes the second step.

In the last step we apply Atiyah’s index theorem for coverings [2] in order to conclude that index_{Γ}(R̃) = index(R). This implies index_{Γ}(T̃_{f}) = index(T_{f}) by Proposition 2.6 and the first two steps. □

4 Examples

We first consider the two-dimensional example. Let M̃ be the hyperbolic plane and Γ be a convex cocompact subgroup of the group of isometries of M̃. The geodesic boundary ∂M̃ can be decomposed into a limit set Λ and its complement Ω. The group Γ acts freely and properly on M̃ ∪Ω, and the compact manifold with boundary M̄ := Γ\M̃ ∪Ω is the geodesic compactification of M := Γ\M̃. The boundary B := ∂M̄ is a finite union of circles Γ\Ω. There is a natural projection of the Higson corona ∂_{g}M to B. Thus any U(1)-valued function F on B can be lifted to ∂_{g}M, and we will denote this lift by the same symbol.

Note that M is a complex manifold. Let K be the canonical bundle of M. We fix k ∈ Z and consider the graded Dirac operator D = ¯∂ + (¯∂)^{∗} on E = K^{k} ⊕ K^{k−1}, where ¯∂: C^{∞}(M, K^{k}) → C^{∞}(M, K^{k−1}) is the Dolbeault operator.

The complex structure fixes an orientation of M which induces an orientation of B. We now use the notation of [3], 2.16. The Dirac operator D_{N} is just i\frac{∂}{∂t} on any component of B, where t is the coordinate of S^{1} compatible with the orientation. The index index(T_{f}) is minus the spectral flow of the family connecting D_{N} and F^{∗}D_{N}F, and this is equal to the total winding number n(F) of F, i.e.

index(T_{f}) = −\frac{1}{2πı} ∫_{B} F^{−1}dF .

Let P_{k} denote the projection onto the space of holomorphic and antiholomorphic square integrable sections of K̃^{k}. Note that P_{0} = 0. For k > 0 (k < 0) the range of P_{k} is the holomorphic (antiholomorphic) discrete series representation of P SL(2, R), the orientation-preserving isometry group of M̃. Let T̃_{f}^{k} := P_{k}f̃P_{k} be the Toeplitz compressions. Then by the computation above and Theorem 3.8 for k ≠ 0 we have index_{Γ}(T̃_{f}^{k}) = −sign(k)n(F).

This has the following higher-dimensional generalization. Let M̃ := Spin(1, 2n)/Spin(2n) be the real hyperbolic space of dimension 2n and Γ ⊂ Spin(1, 2n) be a convex cocompact subgroup. Let S̃ be the spinor bundle of M̃ and Ṽ be any further Spin(1, 2n)-homogeneous bundle. We put Ẽ := S̃ ⊗ Ṽ and let D̃ be the associated Dirac operator. We assume that Ṽ is such that 0 is an isolated point of the spectrum of D̃. In this case ker D̃ decomposes into a finite sum of discrete series representations of Spin(1, 2n).

We again have a decomposition of the geodesic boundary of M̃ into a limit set and a domain of discontinuity Ω. The locally symmetric space M := Γ\M̃ can be compactified by adjoining the boundary B := Γ\Ω. The topology of B can be quite complicated. Since Spin(1, 2n) acts on the sphere ∂M̃ by orientation-preserving conformal transformations B admits a locally conformally flat structure. In particular, all Pontrjagin classes of T B and all associated bundles vanish.

Again we have a natural map from the Higson corona of M to B. Let F : B → U(m) be a continuous function and f, g ∈ Mat(m, C_{g}(M)) extensions of F, F^{−1}. The function F represents an element [F] in K^{1}(B). Let ch([F]) ∈ H^{odd}(B, R) be the Chern character of [F]. We define the degree of F by deg(F) := ch_{2n−1}([F])([B]). Here [B] is the orientation of B as the boundary of M, where the orientation of M is determined by the Z_{2}-grading of S.

Proposition 4.1 index_{Γ}(T̃_{f}) = index(T_{f}) = − dim(V) deg(F).

Proof. We have to compute the index of the Callias type operator C on M which is associated to f, g. By [3], Thm. 2.16 it is equal to the index of the Dirac operator D_{L} on S^{1} × B twisted with a bundle L = L_{F} ⊗ L_{V}. Here L_{V} is the restriction of V to B. The bundle L_{F} is obtained from the trivial bundle [0, 1] × B × C^{m} by glueing (1, b, v) with (0, b, F (b)v), b ∈ B, v ∈ C^{m}. The bundle L_{V} is associated to the tangent bundle of B and thus has vanishing Chern classes. Further note that Â(T (S^{1} × N)) = 1. The index theorem for twisted Dirac operators thus gives

index(D_{L}) = dim(V)ch(L_{F})_{[2n]}([S^{1} × N]) = dim(V)ch_{2n−1}([F])([B]) .

This finishes the proof of the proposition. □

In the situation above we know that ker D is infinite-dimensional by [4]. The proposition above would give an alternative index-theoretic proof of this fact.

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