A Canonical Trace Associated with Certain Spectral Triples

In the abstract pseudodifferential setup of Connes and Moscovici, we prove a general formula for the discrepancies of zeta-regularised traces associated with certain spectral triples, and we introduce a canonical trace on operators, whose order lies outside (minus) the dimension spectrum of the spectral triple.


Introduction
Connes and Moscovici's setup for abstract pseudodifferential calculus [4] (see also [6]) associated with a certain type of spectral triple (A, H, D) provides a framework, in which the ordinary trace on trace-class operators can be extended to a linear form on all abstract pseudodifferential operators using zeta regualarisation type methods. Here, A is an involutive algebra represented in a complex Hilbert space H and D a self-adjoint operator in H. This linear extension, which depends on the operator D used as a regulator in the zeta regularisation procedure, does not vanish on commutators as it can be seen from results of [4].
These constructions mimick the classical pseudodifferential calculus setup on a closed manifold, where similar linear extensions of the ordinary trace are built using the same zeta regularisation type procedure. On non-integer order operators, these linear extensions are independent of the regulator used in the zeta regularisation procedure. They define the canonical trace popularised by Kontsevich and Vishik [9], which vanishes on non-integer order commutators of classical pseudodifferential operators. The canonical trace is unique, in so far as any linear form on non-integer order classical pseudodifferential operators which vanishes on non-integer order commutators, is proportional to the canonical trace [12].
In the abstract pseudodifferential calculus framework, and under a mild reinforcement (see assumption (H) in Section 4 and assumptions (Dk) and (T) in Section 5) of the usual assumptions on a regular spectral triple with discrete dimension spectrum Sd (used to define zeta regularised type linear extensions of the trace), we show that a similar canonical linear form on operators whose order lies outside the discrete set −Sd can be defined, and that this canonical linear form vanishes on commutators of operators, whose order lies outside the discrete set −Sd.
The assumptions (H), (Dk) and (T) we put on the spectral triple generalise the usual assumptions one puts on spectral triples to ensure the existence and a reasonable pole structure of meromorphic extensions of traces of holomorphic families of operators of the type a|D| −z with a in A. Our strengthened assumptions ensure the existence and a reasonable pole structure of meromorphic extensions of traces of more general holomorphic families a(z)|D| −z where a(z) is a holomorphic family in A. 1 Broadly speaking, strengthening the assumptions on the spectral triple amounts to embedding the spectral triple (A, H, D) into a holomorphic family (A(z), H, D) of spectral triples such that A(0) = A. Under these strengthened assumptions, not only can one tackle traces of holomorphic families of the type A|D| −z where A is an abstract pseudodifferential operator, but one can also deal with more general holomorphic families (see Definition 5 in Section 4) where b j (z) is a holomorphic family in the algebra B generated by elements of the type δ n (a(z)) or the type δ n ([D, a(z)]) with n varying in Z ≥0 and a(z) any holomorphic family in A.
We derive a general formula (see Theorem 1) 2 which expresses the complex residues at zero of the meromorphic extensions Tr(A(z))| mer of the trace Tr of a holomorphic family A(z) in terms of residue type linear forms (see formula (5.1)) introduced by Connes and Moscovici in [4]. Here Tr (A |D| −z ) | mer stands for the meromorphic extension of the holomorphic function Tr (A |D| −z ) defined on an adequately chosen half-plane. Specialising to appropriate holomorphic families yields explicit formulae for the discrepancies of the linear extensions of the ordinary trace. With the help of a holomorphic family (|D|+ P ) −z defined in (6.1) where P is a zero order perturbation, we can change the weight |D| to |D| + P in the linear forms τ |D| j in order to define the linear form τ |D|+P j (see formula (6.1)). The subsequent identity (see (6.2 measures the sensitivity of τ |D| j to zero order perturbations of the weight |D| and the following identity (see (6.3 which was derived in Proposition II.1 in [4] by other means, measures the obstruction to its vanishing on commutators.
1 A family {f (z)}z∈Ω in a topological vector space E, parametrised by a complex domain Ω, is holomorphic at z0 ∈ Ω if the corresponding function f : Ω → E is uniformly complex-differentiable on compact subsets in a neighborhood of z0. When f takes its values in a Banach space E, the existence of a complex derivative implies (via a Cauchy formula) that a holomorphic function is actually infinitely differentiable and admits a Taylor expansion which converges uniformly on a compact disk centered at z0 (see e.g. [6,Theorem 8.1.7] and [5, Section XV.5.1]). Holomorphy of a Banach space valued function is therefore equivalent to its analyticity. 2 We have set [[a, b]] = [a, b] ∩ Z for two integers a < b, Res j 0 stands for the j-th order residue at zero and Res 0 0 stands for the finite part at z = 0. 3 We have set L( On the grounds of these identities, under assumptions (H), (Dk) and (T) we show that the linear form τ |D| −1 is invariant under zero order perturbations of |D| (see Proposition 6) and that it vanishes on commutators for operators whose order lies outside minus the dimension spectrum (see Proposition 7). These remarkable features allow us to consider this linear form as a substitute in the abstract pseudodifferential operator set up for Kontsevish and Vishik's canonical trace.
To conclude, at the cost of embedding a spectral triple in a holomorphic family of spectral triples, we have built a linear form on operators whose order lies outside a given discrete set, which vanishes on commutators, whose orders lie outside that discrete set and which is insensitive to zero order perturbations of the weight |D| used to build this linear form.

Spectral triples and abstract dif ferential operators
We briefly recall the setup of abstract differential calculus as explicited in [6]. It encompasses essential features of ordinary pseudodifferential calculus on manifolds, as illustrated on a typical example throughout this paragraph.
Starting from a Hilbert space H with scalar product ·, · and associated norm · together with a self-adjoint operator D on H, we build a non-negative self-adjoint operator ∆ = D 2 . For simplicity, we assume the operator ∆ is invertible and hence positive; if this is not the case we replace it with 1 + ∆.
The self-adjoint operator ∆ and its powers ∆ k are defined on dense domains Dom(∆ k ) = Dom(D 2k ) in H and we consider the set . Example 2. With the notation of Example 1, the operators ∆ k = D 2k acting on the space of smooth sections of E are essentially self-adjoint and have unique self-adjoint extensions (denoted by the same symbol ∆ k ) to the H k -Sobolev spaces Dom(∆ k ) = H 2k (M, E), obtained as closures of the space C ∞ (M, E) of smooth sections of the bundle E for the Sobolev norm (where · stands for the norm on H), and whose intersection H ∞ coincides with C ∞ (M, E).
We now introduce spectral triples [2], which are the building blocks for abstract pseudodifferential operators [4,6].   For any complex number s, let Op s be the set of operators defined by: Let us also introduce the algebra 6 We henceforth assume that the spectral triple is regular and that H ∞ is stable under left multiplication by A.
Following Higson [6] (see Definition 4.28 for the integer order case), for a complex number a we consider the set E a (A, D) of operators A : H ∞ → H ∞ , called basic pseudodif ferential operators of order a, that have the following expansion: meaning by this, that for any non-negative integer N Any finite linear combination of basic pseudodifferential operators of order a is called an abstract pseudodif ferential operator of order a. By Proposition 4.31 in [6] and Appendix B in [4], abstract pseudodifferential operators of integer order form an algebra filtered by the order, which is stable under the adjoint operator δ = [|D|, ·].
Example 4. With the notation of Example 1, the triple consisting of A = C ∞ (M ), H = L 2 (M, E) and D a generalised Dirac operator acting on C ∞ (M, E) form a regular spectral triple. Abstract pseudodifferential operators of integer order correspond to a subalgebra of the algebra Cℓ Z (M, E) of classical pseudodifferential operators of integer order [7,18,19,20], acting on smooth sections of the vector bundle E over M . Since D is a priori not an abstract pseudodifferential operator, the subalgebra corresponding to all integer order abstract pseudodifferential operators is a priori smaller than Cℓ Z (M, E).
By Lemma 4.30 in [6] (see also (11) in Part II of [4]), for any abstract pseudodifferential operator A and any complex number α we have: for any positive integer k. Note that if A has order a, then δ k (A) |D| α−k has order a + α − k, whose real value decreases as k increases. c α,k δ k (A)|D| α−k holds in the Fréchet topology of classical pseudodifferential operators with constant order (see [9]), here a + α where a is the order of A. Remark 1. In particular, for any element a in A, the operator a|D| −z is trace-class on the half-plane Re(z) > n. Holomorphicity of the map z → Tr (a|D| −z ) on the half-plane Re(z) > n then follows. Indeed, on any half-plane Re(z) > n + ǫ for some positive ǫ, the operator a|D| −z can be written as the product of a fixed trace-class operator A := a|D| −n−ǫ and a holomorphic family of bounded operators B(z) := |D| −z+n+ǫ on that half-plane. Since holomorhicity implies analyticity for Banach spaces valued functions (see footnote 1), there are bounded operators B n , n ∈ Z ≥0 such that B(z) = ∞ n=0 B n z n converges uniformly on any compact disk centered at a point of the half-plane Re(z) > n + ǫ. Since A is trace-class, so are the operators AB n and it follows Tr(AB n )z n is holomorphic on every half-plane Re(z) > n+ǫ for positive ǫ and hence on the half-plane Re(z) > n.
More generally, given a finitely summable regular spectral triple (A, H, D) with degree of summability n, for any A in E a (A, D), the map z → Tr(A |D| −z )| mer is holomorphic on some half-plane Re(z − a) > n in C depending on the order a of A.
Definition 4. A regular and finitely summable spectral triple has discrete dimension spectrum, if there is a discrete subset S ⊂ C such that for any operator A ∈ E a (A, D) with order a, the map z → Tr(A|D| −z ) extends to a meromorphic map z → Tr(A|D| −z )| mer on C with poles in the set S − a. Let Sd denote the smallest such set S called the dimension spectrum.
The dimension spectrum is simple if all the poles are simple. It has finite multiplicity k ∈ N if the poles are at most of order k. Example 6. With the notation of Example 1, let A be a classical pseudodifferential operator of real order a, acting on C ∞ (M, E). The operator z → Tr(A|D| −z ) is holomorphic on the half-plane Re(z) > a + n, where n is the dimension of the underlying manifold M , and has a meromorphic extension to the whole complex plane with simple poles in ]−∞, a + n] ∩ Z (see [17]). This meromorphic extension can be written in terms of the canonical trace popularised by Kontsevich and Vishik [9] (see also [10]) The regular spectral triple (A, H, D) arising from Example 1 therefore has a discrete simple dimension spectrum given by Sd =]−∞, n] ∩ Z, which is stable under translations by negative integers. Note that the meromorphic extension is holomorphic at zero if the order a is noninteger.

Logarithms
Given a regular spectral triple 7 (A, H, D), the logarithm being a continuous function on the spectrum of the operator |D| = √ ∆, one can define the unbounded self-adjoint operator log |D| by Borel functional calculus. It can also be viewed as the derivative at zero of the complex power |D| z . For any positive ǫ, the map z → |D| z−ǫ ∈ Op z−ǫ defines a holomorphic function on the half plane Re(z) < ǫ with values in B (H) and we have log |D| = |D| ǫ (∂ z (|D| z−ǫ )) | z=0 . For any positive ǫ, the operator log |D||D| −ǫ = |D| −ǫ log |D| lies in Op 0 , so that log |D| lies in Op ǫ for any positive ǫ.
For any complex number a and any positive integer L, we now introduce the set E a,L (A, D) of operators A : H ∞ → H ∞ , which have the following expansion: We call a linear operator A : H ∞ → H ∞ in E a,L (A, D), a basic pseudodif ferential operator of order a and logarithmic type L. An abstract pseudodifferential operator of order a and logarithmic type L is a finite linear combination of basic abstract pseudodifferential operators of order a and logarithmic type L.
Setting a = z in (2.1) and differentiating with respect to z at zero yields Recall that ∆ is assumed to be invertible; otherwise we replace ∆ with ∆ + 1.
for any abstract pseudodifferential operator A, with c ′ 0,k := ∂ z c z,k | z=0 = k−1 j=1 (−1) j j for positive integers k. In particular, this implies that the logarithmic type does not increase by the adjoint action with log |D|, so that for any complex number a and any integer k, we have Remark 3. This is reminiscent of the fact that in the classical setup, the bracket of the logarithms of an elliptic operator (with appropriate conditions on the spectrum for the logarithm to be defined) with a classical pseudodifferential operator, is classical in spite of the fact that the logarithm is not classical.
The following proposition compares the logarithms of two operators of the same order. Proof . By (2.1) applied to α = −1, the operator |D| −1 P is an abstract pseudodifferential operator of order −1 and hence a bounded operator on H. The operator log(1 + |D| −1 P ) defined by Borel functional calculus reads: and hence lies in E 0 (A, D) by (2.1). The Campbell-Hausdorff formula (see [13] in the classical case) then yields: where C (k) (·, ·) are Lie monomials given by: with the inner sum running over j− tuples of pairs (α i , β i ) such that α i +β i > 0 and and with the following notational convention: Here ad X := [X, ·] denotes the adjoint action by an operator X. Property (3.1) applied to A = log(1 + |D| −1 P ), yields that lies in E 0 (A, D). By induction on k, one shows that C (k) log |D|, log(1 + |D| −1 P ) lies in E 0 (A, D) for any positive integer k, so that the difference log(|D| + P ) − log |D| also lies in E 0 (A, D). Note that by (2.1) the adjoint operation ad log(1+|D| −1 B) decreases the order by 1 unit and that by (3.1) the same property holds for the adjoint operation ad log |D| . Remark 4. Proposition 1 is reminiscent of the fact that in the classical setup, the difference of the logarithms of two elliptic operators (with appropriate conditions on the spectrum for their logarithms to be defined) is classical in spite of the fact that each logarithm is not classical.
Let (A, H, D) be a regular spectral triple with discrete finite dimension spectrum Sd and degree of summability n. Under the regularity assumption, one can equip A with a locally convex topology by means of semi-norms a → δ k (a) and a → δ k ([D, a]) . The completion of A for this topology is a Fréchet space [21] (and even a Fréchet C * -algebra), which we denote by the same symbol A, so that we shall henceforth assume that A is a Fréchet space. Equivalence between holomorphicity and analyticity for Banach space valued functions (see footnote 1) actually extends to the case of Fréchet valued functions; the proof in the Banach case, which uses the Cauchy formula indeed extends to a Fréchet space in replacing the norm on the Banach space by each of the semi-norms that define the topology on the Fréchet space.
We call a B-valued function b(z) holomorphic, if it is a linear combination of products of operators δ n i (a i (z)) and δ n j ([|D|, a j (z)]) for some non-negative integers n i and n j , where a i (z) and a j (z) are holomorphic families in A.

• Assumption (H). For any holomorphic family
is holomorphic on the half plane Re(z) > n and extends to a meromorphic map with poles in the same set Sd. This is actually a particular instance (with A(z) = f (z)(1 + D 2 ) −z/2 ) of the following more general result, which can be found in [9] (see also [14] for a review). For any holomorphic family of classical pseudodifferential operators A(z) with non-constant affine order α(z) acting on C ∞ (M, E), the map z → TR (A(z)) is meromorphic with simple poles in the discrete set α −1 (Z ∩ [−n, +∞[). It is therefore natural to introduce the following definition inspired from the notion of holomorphic family of operators used by Kontsevich and Vishik in [9].

Definition 5. We call a family
of linear operators acting on H ∞ parametrised by z ∈ C, holomorphic, if 1) α(z) is a complex valued holomorphic function, 2) b j (z) is a B-valued holomorphic function for any non-negative integer j, The following lemma extends this to families A(z) = |D| −z A. (A, D), the product |D| −z A is a holomorphic family of order a − z acting on H ∞ , and the commutator [|D| −z , A] is a holomorphic family of order a − z − 1 acting on H ∞ .

Lemma 1. For any complex number a and any operator
Proof . Applying (2.1), with α replaced with −z, to the operators b j |D| a−j for any non-negative integer j, yields We shall also need the following technical result.

Proposition 2. Given a holomorphic family
Proof . By assumption, the family A(z)|D| −α(z) is holomorphic in the Banach space B(H), with derivatives that lie in E α(z),l (A, D), the Leibniz rule yields the following identity of (unbounded) operators:

Traces of holomorphic families
Let (A, H, D) be a regular spectral triple which satisfies assumption (H) of the previous section. We make two further assumptions • Assumption (Dk). The poles of the meromorphic map z → Tr (b(z)|D| −z ) | mer introduced in (4.1) have multiplicity ≤ k + 1.
• Assumption (T). The dimension spectrum Sd is invariant under translation by negative integers:

Remark 5.
Under assumption (Dk) we have where for a meromorphic function f with a pole of order N at zero we write in a neighborhood of zero: Res j 0 f z j + O(z).
Example 10. Results of Kontsevich and Vishik [9] tell us that assumptions (Dk) and (T ) are satisfied in the classical geometric setup corresponding to Example 1. Indeed, assumption (Dk) is satisfied for k = 0 since TR (b(z)|D| −z ) is a meromorphic map with simple poles. These lie in the set ]−∞, −n] ∩ Z, which is stable under translation by negative integers so that assumption (T) is satisfied.
We henceforth assume that (H), (Dk) and (T) are satisfied.
b j (z)|D| α(z)−j be a holomorphic family of operators acting on H ∞ with non-constant affine order α(z). The map z → Tr (A(z)) is well-defined and holomorphic on the half plane Re(α(z)) < −n, and further extends to a meromorphic function Tr (A(z)) | mer on the complex plane with poles in α −1 (−Sd) with multiplicity ≤ k + 1.
Proof . Let us set α(z) := −qz + a for some positive real number q (a similar proof holds for negative q). Under assumption (H), for any non-negative integer j the map is holomorphic on the half-plane Re(z) > Re(a) + n − j q ⇐⇒ Re(α(z)) < −n + j.
Under Assumption (Dk), it extends to a meromorphic map Tr b j (z)|D| −q z− a−j q | mer with poles of multiplicity ≤ k + 1 in the set as a consequence of Assumption (T). On the other hand, given a real number r and an integer N > Re(a) + n − rq, under Assumption (H), the map z → Tr(K N (z)) is holomorphic on the half-plane Re(z) > r, where is the remainder operator. Combining these observations yields a meromorphic map on the half-plane Re(z) > r given by Tr b j (z)|D| α(z)−j | mer + Tr(K N (z)), with poles in Sd+a q = α −1 (−Sd) with multiplicity ≤ k + 1. Since r can be chosen arbitrarily, this provides a meromorphic extension to the whole complex plane with poles in α −1 (−Sd) with multiplicity ≤ k + 1. The result then follows.
2. In particular, where as before, fp z=0 stands for the finite part at z = 0.
By Proposition 2, the higher derivatives (∂ n z (A(z)|D| qz )) | z=0 are operators of order a. We have since for any abstract pseudodifferential operator A, we have τ |D| k+l (A) = 0 ∀ l > 0. 2. The finite part at z = 0 is obtained from setting j = −1 in the previous formula.
Remark 7. These formulae are similar to formulae for the higher residues of traces of holomorphic families of log-polyhomogeneous pseudodifferential operators of logarithmic type k, acting on smooth sections of a vector bundle, which were derived in [10].

Discrepancies
As before we assume that (H), (Dk) and (T) are satisfied.
We want to measure the obstructions preventing the linear forms τ |D| j from having the expected properties of a trace, which we interpret as discrepancies of the linear forms τ |D| j . The |D|-dependence of the |D|-weighted residue trace τ |D| j (A) of an abstract pseudodifferential operator A is one of these defects. We show how a zero order perturbation |D| → |D| + P of the weight |D| by some P ∈ E 0 (A, D) affects |D|-weighted residue traces. We express this variation in terms of expressions of the type τ |D| j+n (P n (A, |D|)), where P n (A, |D|) are abstract pseudodifferential operators indexed by positive integers n.
On the grounds of formulae which hold in the classical setup (see formulae (14) and (15) in [15] with φ(x) = x −z ), similar to formulae derived in [4] and [6] (see e.g., the proof of Proposition 4.14), for an operator P ≃ ∞ j=0 b j |D| −j ∈ E 0 (A, D) and a complex number z we define (|D| + P ) −z by where we have set k! = k 1 ! · · · k n ! and |k| = k 1 + · · · + k n . Note that the real part of the order −z − |k| − n of the operators δ k 1 (P ) · · · δ kn (P )|D| −z−|k|−n decreases with n and |k|. Using (2.1), holds for any operator A in E a (A, D) of any complex order a. Moreover, is independent of the perturbation P and when k = 0 the above formula reads Proof . We apply Theorem 1 to the holomorphic family By Proposition 2, the higher derivatives at zero have order zero, and by Theorem 1, we have since the n = 0 term vanishes. If j = k, we have τ Remark 8. When k = 0, the linear form τ 0 generalises Wodzicki's noncommutative residue [22,23] on classical pseudodifferential operators acting on smooth sections of a vector bundle E over a closed manifold M (see [8] for a review).
The linear form τ |D| j does not vanish on commutators as could be expected of a trace, leading to another discrepancy. The following result yields back Proposition II.1 in [4]. We can apply Theorem 1 to the holomorphic family Indeed, where we have set σ(z)(A) = |D| −z A|D| z . Since it follows that where as before c α,k = α(α−1)···(α−k+1) k! for any positive integer k. By Proposition 2, the higher derivatives at zero ∂ n z (A(z)|D| z ) | z=0 have order a + b, where a is the order of A and b that of B. Since These discrepancy formulae are similar to the anomaly formulae for weighted traces of classical pseudodifferential operators, see [11,3,16] and [14] for an extension to log-polyhomogeneous operators.

7
An analog of Kontsevich and Vishik's canonical trace In [9] Kontsevich and Vishik popularised what is known as the canonical trace on non-integer order classical pseudodifferential operators acting on smooth sections of a vector bunlde E over a closed manifold M . This linear form which vanishes on commutators of non-integer order classical pseudodifferential operators, is the unique (up to a multiplicative constant) linear extension of the ordinary trace to the set of non-integer order operators with that property. The non integrality assumption on the order can actually be weakened to the order not lying in the set [−n, +∞[∩Z which corresponds to −Sd, with Sd the dimension spectrum of the spectral triple C ∞ (M ), L 2 (M, E), D . For a spectral triple which fulfills assumptions (H), (Dk) and (T), we replace this assumption on the order with the requirement that the order lies outside the set −Sd.
Lemma 2. For an abstract pseudodifferential operator A whose order a lies outside the set −Sd and for any holomorphic family A(z) such that A(0) = A, the map z → Tr (A(z)) | mer is holomorphic at z = 0.