Classification of Traces and Associated Determinants on Odd-Class Operators in Odd Dimensions

To supplement the already known classification of traces on classical pseudodifferential operators, we present a classification of traces on the algebras of odd-class pseudodifferential operators of non-positive order acting on smooth functions on a closed odd-dimensional manifold. By means of the one to one correspondence between continuous traces on Lie algebras and determinants on the associated regular Lie groups, we give a classification of determinants on the group associated to the algebra of odd-class pseudodifferential operators with fixed non-positive order. At the end we discuss two possible ways to extend the definition of a determinant outside a neighborhood of the identity on the Lie group associated to the algebra of odd-class pseudodifferential operators of order zero.


Introduction
From the connection between the trace of a matrix with scalar coefficients and its eigenvalues, one can derive a relation between the trace and the determinant of a matrix, namely det(A) = exp(tr(log A)).
At the level of Lie groups, a trace on a Lie algebra is the derivative of the determinant at the identity on the associated Lie group. Using the exponential mapping between a Lie algebra and its Lie group, one recovers in this setting the relation (1). This exponential mapping always exists in the case of finite-dimensional Lie groups, and in the infinite-dimensional case its existence is ensured by requiring regularity of the Lie group. On trace-class operators over a separable Hilbert space one can promote the trace on matrices to an operator trace. Further generalizing to classical pseudodifferential operators one can consider traces on such operators. In the case of a closed manifold of dimension greater than one, M. Wodzicki proved that there is a unique trace (up to a constant factor) on the whole algebra of classical pseudodifferential operators acting on smooth functions on the manifold, namely the noncommutative residue [31]. As S. Paycha and S. Rosenberg pointed out [23], this fact does not rule out the existence of other traces when restricting to subalgebras of such operators. In fact, other traces such as the leading symbol trace, the operator trace and the canonical trace appear naturally on appropriate subalgebras. The classification of the traces on algebras of classical pseudodifferential operators of non-positive order has been carried out in [17] (see also [12]).
After the construction of the canonical trace on non-integer order pseudodifferential operators, M. Kontsevich and S. Vishik [8] introduced the set of odd-class operators, which is an algebra that contains the differential operators. They also defined a trace on this algebra when the dimension of the manifold is odd, and it has been proven that this is the unique trace in this context [14,22,26]. In [21], M.F. Ouedraogo gave another proof of this fact based on the expression of a symbol of such an operator as a sum of derivatives of symbols corresponding to appropriate operators on the same algebra.
Odd-class operators are one of the rare types of operators in odd dimensions on which the canonical trace is defined, this fact serving as a motivation to investigate the classification of traces on the algebra of odd-class operators of order zero. The noncommutative residue is not of interest here since it vanishes on odd-class operators (Lemma 2). This contrasts with the algebra of ordinary zero-order operators where the only traces are linear combinations of the leading symbol trace and the noncommutative residue (see [13]). We supplement that classification of traces, showing that when restricting to odd-class zero-order operators, the only traces turn out to be linear combinations of the leading symbol trace and the canonical trace (this is the particular case a = 0 in Theorem 3).
In this article we present the classification of traces on algebras of odd-class pseudodifferential operators acting on smooth functions on a closed odd-dimensional manifold. The methods we implement combine various approaches used in the literature on the classification of traces. However, a detailed analysis is required here because of the specificity of odd-class operators (see Proposition 3). We recall the one to one correspondence between continuous traces and determinants of class C 1 on regular Lie groups 1 , and as in [13] we use this correspondence to give the classification of determinants on the Lie group associated to the algebra of odd-class operators of a fixed non-positive order. At the end we discuss two ways to extend the definition of determinants outside a neighborhood of the identity.
In Section 2 we recall some of the basic notions of classical pseudodifferential operators, including that of symbols on an open subset of the Euclidean space and odd-class operators on a closed manifold. Inspired by [14], we use the representation of an odd-class symbol as a sum of derivatives up to a smoothing symbol (Proposition 1), to express an odd-class operator in terms of commutators of odd-class operators (Proposition 3), a fact that helps considerably in the classification of traces.
In Section 3 we give a classification of traces on odd-class operators of non-positive order. For that we recall the known traces on classical pseudodifferential operators. The noncommutative residue vanishes on the algebra of odd-class operators in odd dimensions, whereas the canonical trace is the unique linear form on this set which vanishes on commutators of elements in the algebra (see [14]). Then using the fact that any odd-class operator can be expressed in terms of commutators of odd-class operators (Proposition 3), we prove that any trace on an algebra of odd-class operators of fixed non-positive order can be expressed as a linear combination of a generalized leading symbol trace and the canonical trace (Theorem 3).
In Section 4 we classify determinants on the Lie groups associated to the algebras of oddclass operators of non-positive order. We follow some of the work done in [13], concerning the one to one correspondence between continuous traces on Lie algebras and C 1 -determinants on the associated regular Lie groups (this is also discussed in [2] in specific situations). Then, we combine this correspondence with the classification of traces given in Section 3.3, to provide the classification of determinants on Lie groups associated to algebras of odd-class operators of fixed non-positive order (Theorem 4). This classification is carried out for operators in a small neighborhood of the identity, where the exponential mapping is a diffeomorphism.
At the end of this section, we give two possible ways to extend the definition of a determinant outside a neighborhood of the identity on the Lie group associated to the algebra of odd-class pseudodifferential operators of order zero; the first one (see (22)) using a spectral cut to define the logarithm of an admissible operator; in this case, for some traces this definition of determinant depends on the spectral cut; the other one (see (24)) via the definition of the determinant of an element on the pathwise connected component of the identity, using a path that connects the element with the identity. Both the first and the second extension of the definition of a determinant, provide maps which do not depend on the spectral cut and which satisfy the multiplicativity property, under the condition that the image of the fundamental group of invertible odd-class pseudodifferential operators of order zero is trivial.

Preliminaries on pseudodif ferential operators
Here we recall the basic notions of classical pseudodifferential operators following [29].

Symbols
Let U be an open subset of R n . Given a ∈ C, a symbol of order a on U is a complex valued function σ(x, ξ) in C ∞ (U ×R n ) such that for any compact subset K of U and any two multiindices α = (α 1 , . . . , α n ), β = (β 1 , . . . , β n ) ∈ N n there exists a constant C K,α,β satisfying for all (x, ξ) ∈ K × R n , where ∂ α x = ∂ α 1 x 1 · · · ∂ αn xn , |β| = β 1 + · · · + β n , and (a) stands for the real part of a. Let S a (U ) denote the set of such symbols. Notice the space of smoothing symbols on U . Given σ ∈ S m 0 (U ), The product on symbols is defined as follows: if σ 1 ∈ S a 1 (U ) and σ 2 ∈ S a 2 (U ), In particular, σ 1 σ 2 ∈ S a 1 +a 2 (U ).

Classical symbols
A symbol σ ∈ S a (U ) is called classical, and we write σ ∈ CS a (U ), if there is an asymptotic expansion Here σ a−j (x, ξ) is a positively homogeneous function in ξ of degree a − j: for all t ∈ R + , |ξ| = 0, and ψ ∈ C ∞ (R n ) is any cut-off function which vanishes for |ξ| ≤ 1 2 and such that ψ(ξ) = 1 for |ξ| ≥ 1.
We denote by the algebra generated by all classical symbols on U for the product .

Odd-class symbols
The homogeneous components in the asymptotic expansion of a classical symbol may satisfy some other symmetry relations additional to the positive homogeneity on the second variable. Now we recall the definition of odd-class symbols introduced first in [8] (see also [5]).
Definition 1 (see [8]). A classical symbol σ ∈ CS a (U ) with integer order a is odd-class if for each j ≥ 0, the term σ a−j in the asymptotic expansion (3) satisfies In other words, odd-class symbols have the parity one would expect from differential operators.
Let us denote by CS a odd (U ) the set of odd-class symbols of order a ∈ Z on U . We set Lemma 1 (see [3,8]). The odd-class symbols satisfy the following: 1. The product of two odd-class symbols is an odd-class symbol, therefore CS odd (U ) is an algebra.
2. If an odd-class symbol is invertible with respect to the product , then its inverse is an odd-class symbol.

The noncommutative residue on symbols
As before, we consider U an open subset of R n .
Definition 2 (see [6,31]). The noncommutative residue of a classical symbol σ ∼ where µ is the surface measure on the unit sphere S * x U over x in the cotangent bundle T * U .
The noncommutative residue clearly vanishes on symbols of order strictly less than −n and also on symbols of non-integer order.
Lemma 2. In odd dimensions, the noncommutative residue of any odd-class symbol vanishes.
The function b −n is smooth on U × R n and is homogeneous of degree −n + 2 in ξ for |ξ| ≥ 1. As σ −n (x, ξ) vanishes for x outside a compact set, so does b −n (x, ξ). In particular, b −n is a symbol of order −n + 2 on U . Let us define τ i,−n+1 := −∂ ξ i b −n . Since h is odd so is b −n and therefore, Moreover, we have for r = |ξ| ≥ 1

Pseudodif ferential operators
Let U ⊂ R n be an open subset, and denote by C ∞ c (U ) the space of smooth compactly supported functions on U . To the symbol σ ∈ S(U ), we associate the linear integral operator Op(σ) : where u(ξ) = U e −iy·ξ u(y) dy is the Fourier transform of u and dξ := (2π) −n dξ. In this expression k(x, y) = e i(x−y)·ξ σ(x, ξ)dξ is seen as a distribution on U × U that is smooth outside the diagonal. We say that Op(σ) is a pseudodifferential operator (ψDO) with Schwartz kernel given by k(x, y). An operator is smoothing if its Schwartz kernel is a smooth function on U × U . If ψσ a−j is a classical symbol of order a, then A = Op(σ) is called a classical pseudodifferential operator of order a. The homogeneous component σ a of σ is called the leading symbol of A, and will be denoted by σ L A . A ψDO A on U is called properly supported if for any compact K ⊂ U , the set {(x, y) ∈ supp(k A ) : x ∈ K or y ∈ K} is compact, where supp(k A ) denotes the support of the Schwartz kernel of A. Any ψDO A can be written in the form (see [29]) where P is properly supported and R is a smoothing operator. A properly supported ψDO maps C ∞ c (U ) into itself. The product on symbols defined in (2) induces a composition on properly supported ψDOs on U . The composition AB of two properly supported ψDOs A and B is a properly supported ψDO with symbol σ(AB) = σ(A) σ(B).
The notion of a ψDO can be extended to operators acting on manifolds (see Section 4.3 in [29]). Let M be a smooth closed manifold of dimension n. A linear operator A : C ∞ (M ) → C ∞ (M ) is a pseudodifferential operator of order a on M if in any atlas, A is locally a pseudodifferential operator. This means that given a local coordinate chart U of M , with diffeomorphism ϕ : U → V , from U to an open set V ⊆ R n , the operator ϕ # A defined by the following diagram is a pseudodifferential operator of order a on V : is the natural embedding, and r U : Let C a (M ) denote the set of classical ψDOs of order a on M , i.e. operators whose symbol is classical of order a in any local chart of M . We will also denote by the space generated by classical ψDOs on M whose order is non-integer or less than −n. A classical operator A ∈ C a (M ) of integer order a is odd-class if in any local chart its local symbol σ(A) is odd-class. We denote by C a odd (M ) the set of odd-class operators of order a ∈ Z and we define As in Lemma 1, the following lemma implies that C odd (M ) is an algebra: The algebra C odd (M ) contains the differential operators and their parametrices.

Remark 1.
Even though the definition of odd-class pseudodifferential operators makes sense on any closed manifold, in this paper we restrict ourselves to odd-dimensional closed manifolds. The reason is that the canonical trace (which will be explained below in Section 3.1.2) is well defined only in that case. So, from now on, the notation C odd (M ) will be used only when the dimension n of the manifold M is odd.

Fréchet topology on pseudodif ferential operators
For any complex number a, we equip the vector space C a (M ) with a Fréchet topology as follows. Let us consider a covering of M by open neighborhoods {U i } i∈I , a finite subordinated partition of unity (χ i ) i∈I and smooth functions ( χ i ) i∈I on M such that supp( χ i ) ⊂ U i and χ i = 1 near the support of χ i . By (5)

any ψDO A can be written in the form
a−j (A), and R i is a smoothing operator with smooth kernel k i which has compact support in U i × U i .
We equip C a (M ) with the following countable set of semi-norms: for any compact subset K ⊂ U i for any j ≥ 0, N ≥ 1 and for any multiindices α, β

The logarithm of a classical pseudodif ferential operator
An operator A ∈ C (M ) with positive order has principal angle θ if for every (x, ξ) ∈ T * M \ {0}, the leading symbol σ L A (x, ξ) has no eigenvalues on the ray L θ = {re iθ , r ≥ 0}; in that case A is elliptic.
Definition 3 (see e.g. [18]). An operator A ∈ C (M ) is admissible with spectral cut θ if A has principal angle θ and the spectrum of A does not meet L θ = {re iθ , r ≥ 0}. In particular such an operator is invertible and elliptic. The angle θ is called an Agmon angle of A.
Let A ∈ C (M ) be admissible with spectral cut θ and positive order a. For (z) < 0, the complex power A z θ of A is defined by the Cauchy integral (see [28]) where λ z θ = |λ| z e iz(argλ) with θ ≤ arg λ < θ + 2π. Here Γ r,θ is a contour along the ray L θ around the (non-zero) spectrum of A, and r is any small positive real number such that Γ r,θ does not meet the spectrum of A. The operator A z θ is an elliptic classical ψDO of order az; in particular, for z = 0, we have A 0 θ = I. The definition of complex powers can be extended to the whole complex plane by setting for k ∈ N and (z) < k; this definition is independent of the choice of k in N and preserves the usual properties, i.e.
Complex powers of operators depend on the choice of spectral cut. Indeed, if L θ and L φ are two spectral cuts for A outside an angle which contains the spectrum of σ L (A)(x, ξ) then where the operator is a projection (see [25,32]). Here Γ θ,φ is a contour separating the part of the spectrum of A contained in the open sector θ < arg λ < φ from the rest of the spectrum. The logarithm of an admissible operator A with spectral cut θ is defined in terms of the derivative at z = 0 of its complex power: Logarithms of classical ψDOs of positive order are not classical anymore since their symbols involve a logarithmic term log |ξ| as the following elementary result shows.
be an admissible operator with positive order and spectral cut θ. In a local trivialization, the symbol of log θ (A) reads: where σ A 0 is a classical symbol of order zero. Remark 2. If A is a classical ψDO of order zero then A is bounded, and if it admits a spectral cut, then complex powers and the logarithm of A are directly defined using a Cauchy integral formula, and they are classical ψDOs (see [18] and Remark 2.1.7 in [21]).
Just as complex powers, the logarithm depends on the choice of spectral cut. Indeed, given two spectral cuts θ, φ of the operator A such that 0 ≤ θ < φ < 2π, differentiation of (6) with respect to z and evaluation at z = 0 yields

Pseudodif ferential operators in terms of commutators
In this subsection we use the ψDO analysis techniques similar to the ones implemented in [14]; we assume that M is an n-dimensional closed manifold and n is odd. Given a function f ∈ C ∞ (M ), we also denote by f the zero-order classical ψDO given by multiplication by f .
odd (M ) and a smoothing operator R A such that Proof . Let us consider a covering of M by open neighborhoods {U j } N j=1 and a finite subordinated partition of unity {ϕ j } N j=1 , such that for every pair (j, k), both ϕ j and ϕ k have support in one coordinate neighborhood. We write (see Subsection 2.2.1) Each operator ϕ j Aϕ k may be considered as an odd-class ψDO on R n with symbol σ in CS a odd (R n ). By Proposition 1, there exist odd-class symbols τ l of order a + 1 such that For any symbol τ we have, As in (5) we write ϕ j Aψ j = Op(σ j ) + R j for some σ j ∈ CS a odd (R n ) and some smoothing opera- where α k is a smooth function on M (and represents the operator in C 0 odd (M ) of multiplication by α k ), B k lies in C a+1 odd (M ), and R A is a smoothing operator.
, and a smoothing operator R such that

Classif ication of traces on odd-class operators
The classification made in this section is essentially based on Proposition 3, namely the decomposition of an odd-class operator in terms of commutators of odd-class operators. Let us first recall the definition of a trace. Let M be a closed connected manifold of dimension n > 1, and linear in the sense that for all a, b ∈ C, whenever A, B and aA + bB belong to A we have τ (aA + bB) = aτ (A) + bτ (B), and such that for any A, B ∈ A, whenever AB, BA ∈ A it satisfies

Examples of traces on pseudodif ferential operators
Interestingly by Lemma 2, the noncommutative residue, which is the only trace on the whole space C (M ) (see [4,10,31]), vanishes on odd-class operators when the dimension of the manifold is odd. In this section we review the traces which are non-trivial on this class of operators.

The L 2 -trace
A ψDO A whose order has real part less than −n is a trace-class operator. The L 2 -trace (also called operator trace) is the functional Tr : where k A is the Schwartz kernel of the operator A. This trace is continuous for the Fréchet topology on the space of ψDOs of constant order less than −n. This is the unique trace on the algebra of smoothing operators C −∞ (M ), since we have the exact sequence (see [7]) More precisely we have Theorem 1 (Theorem A.1 in [7]). If R is a smoothing operator then, for any pseudodifferential idempotent J, of rank 1, there exist smoothing operators S 1 , . . . , S N , T 1 , . . . , T N , such that Therefore, any smoothing operator with vanishing L 2 -trace is a sum of commutators in the space Proposition 4 (see e.g. Introduction in [10] and Proposition 4.4 in [11]). The trace Tr does not extend to a trace functional neither on the whole algebra C (M ), nor does it on the algebra C 0 (M ).

The canonical trace
We start with the definition of the cut-off integral of a symbol, as in [22]. Let U be an open subset of R n .
Proof . We write Using the fact that σ a−j is positively homogeneous of degree a − j we have It follows that the integral B * x (0,R) σ(x, ξ) dξ admits the asymptotic expansion where α x (σ) is given by which may depend on the variable x.
For N sufficiently large, the integral B * x (0,R) σ N (x, ξ)dξ is well defined, so the term α x (σ) given by (11) converges when R → ∞, and taking this limit we define the cut-off integral of σ by where S * x U stands for the unit sphere in the cotangent space T * x U . This definition is independent of N > a + n − 1. Moreover, if a < −n, then − R n σ(x, ξ)dξ = R n σ(x, ξ)dξ. According to Proposition 4, there is no non-trivial trace on C (M ) which extends the L 2 -trace. However, the L 2 -trace does extend to non-integer order operators and to odd-class operators. Indeed, M. Kontsevich and S. Vishik [8] constructed such an extension, the canonical trace where the right hand side is defined using a finite covering of M , a partition of unity subordinated to it and the local representation of the symbol, but this definition is independent of such choices.
As we already stated in Remark 1, the canonical trace is well defined on C odd (M ) only when the dimension n of the manifold is odd (see [5,10]), which is always our case.
If A ∈ C a (M ), B ∈ C b (M ) and if a, b / ∈ Z, then ord(AB) = a + b may be an integer, so the linear space C / ∈Z (M ) is not an algebra; in spite of this, the canonical trace has the following properties (see [8], Section 5 in [10], and [20,22,24]

Generalized leading symbol traces
In [23], S. Paycha and S. Rosenberg introduced the leading symbol traces defined on an algebra of operators C a (M ) for a ≤ 0; in this section we follow [17] and consider a trace which actually coincides with a leading symbol trace when a = 0. Let a be a non-positive integer and consider the projection map π a from C a (M ) to the quotient space C a (M )/C 2a−1 (M ): One can see in  Remark 3. For a < 0, a leading symbol trace is the particular case of a generalized leading symbol trace when ρ a−i ≡ 0 for all i = 1, . . . , |a|.
Generalized leading symbol traces are continuous for the Fréchet topology on the space of constant order ψDOs, since they are defined in terms of a finite number of homogeneous components of the symbols of the operators.

Trace on C odd (M )
The canonical trace is the unique trace (up to a constant) on its domain. This result was proved in [14] (which goes back to 2007 but it was published in 2008) using the fact that the operator corresponding to the derivative of a symbol is, up to a smoothing operator, proportional to the commutator of appropriate operators. The latter idea was considered in [22] to show the equivalence between Stokes' property for linear forms on symbols and the vanishing of linear forms on commutators of operators. In [26] the author uses the Schwartz kernel representation of an operator to express any non-integer order operator and any operator of regular parity class as a sum of commutators up to a smoothing operator, and then to give another proof of the uniqueness of the canonical trace. The following proof, that we find in Proposition 3.2.4 of [21], is done in the spirit of [14]. Proof . By (9) any operator A in C odd (M ) can be written in the form where α k are smooth functions on M that can be seen as elements of C 0 odd (M ), B k belong to C a+1 odd (M ), and R A is a smoothing operator. By Theorem 1, we can express R A as where J is a pseudodifferential projection of rank 1 and S j , T j are smoothing operators. Summing up, the expression for A becomes Applying the canonical trace TR to both sides of this expression, since TR vanishes on commutators of operators in C odd (M ), we infer that TR(A) = Tr(R A ).

Thus (13) reads
If τ is a trace on C odd (M ), applying τ to both sides of (14) we reach the conclusion of the theorem.

Traces on C a odd (M ) for a ≤ 0
In this section we assume as before, that the dimension n is odd, and we prove that any trace on the algebra of odd-class operators of non-positive order is a linear combination of a generalized leading symbol trace and the canonical trace. We can adapt Lemma 4.5 in [12] (see also Lemma 5.1.1 in [17]) in the case of odd-class operators: Lemma 5. If a ∈ Z is non-positive, then there exists an inclusion map Proof . By Lemma 3, integer powers of an invertible differential operator are odd-class operators. Hence we proceed as in the proof of Lemma 4.5 in [12] as follows: Let A ∈ C 0 odd (M ), B ∈ C 2a odd (M ). Consider a first-order positive definite elliptic differential operator Λ. For any a ∈ R, Λ a and Λ −a are operators of order a and −a, respectively, and therefore AΛ a , Λ a A, Λ a , BΛ −a , Λ −a B, ABΛ −a , Λ −a BA are operators in C a odd (M ). Moreover, Adding up the expressions in (15) As in (12), for a non-positive integer a we also denote by π a the projection with corresponding splitting θ a : C a odd (M )/C 2a−1 odd (M ) → C a odd (M ), so that for any A ∈ C a odd (M ), A − θ a (π a (A)) ∈ C 2a−1 odd (M ). The following result adds to the known classification of traces on pseudodifferential operators, the classification of all traces on odd-class operators with fixed non-positive order in odd-dimensions.
We fix a non-positive integer a, and describe any trace on C a odd (M ) (see Section 5.1.4 in [17]).
Theorem 3. If a ∈ Z is non-positive, any trace on C a odd (M ) can be written as a linear combination of a generalized leading symbol trace and the canonical trace.
So we conclude that τ is a linear combination of a generalized leading symbol trace and the canonical trace.

Determinants and traces
We use the classification of traces on algebras of odd-class operators given in Theorem 3 to classify the associated determinants on the corresponding Fréchet-Lie group. Well-known general results in the finite-dimensional context concerning determinants associated with traces generalize to the context of Banach spaces (see [2]) and further to Fréchet spaces (see [13]).
Definition 4 (Definition 36.8 in [9]). A (possibly infinite-dimensional) Lie group G with Lie algebra Lie(G) admits an exponential mapping if there exists a smooth mapping Exp : Lie(G) → G such that t → Exp(tX) is a one-parameter subgroup, i.e. a Lie group homomorphism (R, +) → G with tangent vector X at 0.
The existence of a smooth exponential mapping for a Lie group is ensured by a notion of regularity [9,16] on the group. Following [9], for J. Milnor [16], a Lie group G modelled on a locally convex space is a regular Lie group if for each smooth curve u : [0, 1] → Lie(G), there exists a smooth curve γ u : [0, 1] → G (which is unique: Lemma 38.3 in [9]) which solves the initial value problemγ u = γ u u with γ u (0) = 1 G , where 1 G is the identity of G, with smooth evolution map For example, Banach Lie groups (in particular finite-dimensional Lie groups) are regular. If E is a Banach space, then the Banach Lie group of all bounded automorphisms of E is equipped with an exponential mapping given by the series In [19] a wider concept of infinite-dimensional Lie groups called regular Fréchet-Lie groups is introduced. In this paper we will consider the regular Fréchet-Lie groups of classical ψDOs of non-positive order (see [33]).
If G admits an exponential mapping Exp and if a suitable inverse function theorem is applicable, then Exp yields a diffeomorphism from a neighborhood of 0 in Lie(G) onto a neighborhood of 1 G in G, whose inverse is denoted by Log.
For our purpose in this section, we assume that the Lie group G is regular.
Definition 5 (see Definition 2 in [13]). Let G be a Lie group and letG be its subgroup of elements pathwise connected to the identity of G. A determinant on G is a group morphism i.e. for any g, h ∈G, We also say that Λ is multiplicative. A trace on the Lie algebra Lie(G) is a linear map λ : Lie(G) → C, such that for all u, v ∈ G, In our examples below [u, v] = uv − vu.
The following lemma gives the construction of a locally defined determinant on G from a trace on Lie(G).
Lemma 6 (see Proposition 2 and Theorem 3 in [13] which is based on [2]). A continuous trace λ : Lie(G) → C gives rise to a determinant map Λ : Exp(Lie(G)) ⊂G → C * def ined on the range of the exponential mapping by Λ(g) := exp(λ(Log(g))), where locally Log = Exp −1 , making the following diagram commutative, for any small enough neighborhood U 0 of zero in Lie(G):
Conversely, following [13] we give a construction of a trace from a determinant. Our proof here is different from the one in loc. cit.

Lemma 7 (See Proposition 2 and Theorem 3 in [13]).
A determinant Λ : Exp(Lie(G)) → C * , which is of class C 1 on G, yields a continuous trace λ : Lie(G) → C in the following way: for all u ∈ Lie(G) which makes the following diagram commutative:

Exp
Proof . Let u 1 , u 2 ∈ Lie(G). Then Here we use the fact that Λ is multiplicative, which implies that Λ(g −1 ) = Λ(g) −1 . The linearity of λ can be proved using the commutativity of the diagram. In the following we assume that M is an n-dimensional closed manifold and n is odd. We are going to classify determinants on a neighborhood of the identity in the space of invertible pseudodifferential operators (I + C a odd (M )) * , for a non-positive integer a. For that, let us first single out the Fréchet-Lie groups and Fréchet-Lie algebras we use.
The following proposition can be found in Proposition 3 in [13] for the case (C 0 (M )) * . See Proposition 6.1.4 in [21] for the case of odd-class operators. We give here an exhaustive proof of this result since we did not find it in the literature. On the other hand (see [29]), the inclusion C 0 (M ) → L(L 2 (M )) is continuous so that the inclusion i : C 0 odd (M ) → C 0 (M ) → L(L 2 (M )) is also continuous and the inverse image i −1 (V ) yields an open neighborhood of A in C 0 odd (M ) * . It follows that C 0 odd (M ) * is canonically equipped with a manifold structure which makes it a Lie group. Let us now construct an exponential mapping on C 0 odd (M ). Given any operator B in C 0 odd (M ), the differential equation has a unique solution in C 0 (M ) * given by where Γ is a contour around the spectrum of B. Note that A t is bounded since B has zero order. Let us check that A t belongs to C 0 odd (M ) * . The homogeneous components of the  u which solves the initial value problem γ −1 uγ u = u, γ u (0) = I.
Inspired by Corollary 5.12 in [19], and [33] we have the following Proposition 7. If a < 0, the space of invertible odd-class ψDOs G := (I + C a odd (M )) * = ({I + A : A ∈ C a odd (M )}) * is a Fréchet-Lie group with Fréchet-Lie algebra C a odd (M ), which admits an exponential mapping from C a odd (M ) to (I + C a odd (M )) * . Explicitly this exponential mapping is given by This map restricts to a diffeomorphism from some neighborhood of the identity in C a odd (M ) to a neighborhood of the identity in G. The inverse is given by 4.1 Classif ication of determinants on (I + C a odd (M )) * for a ≤ 0 As before we consider a non-positive integer number a. From the classification of traces on C a odd (M ) derived in Theorem 3, we infer a description of the determinants defined on the range of the exponential mapping in (I + C a odd (M )) * . The following theorem supplements the known classification of determinants on classical pseudodifferential operators. Det c 1 ,c 2 (·) = exp c 1 ρ • π a (Log(·)) + c 2 TR(Log(·)) .
Proof . By Theorem 3, any trace τ on C a odd (M ) is a linear combination of the canonical trace and a generalized leading symbol trace: for some c 1 , c 2 ∈ C, and for some linear map ρ : Cl a odd (M )/Cl 2a−1 odd (M ) → C. Moreover, τ is continuous for the Fréchet topology of C a odd (M ). Then, applying Lemma 6 to G = (I + C a odd (M )) * and Lie(G) = C a odd (M ), it follows that a determinant map on G is of the form exp c 1 ρ • π a (Log(·)) + c 2 TR(Log(·)) .
The determinants given in Theorem 4 differ from the ones sometimes used by physicists for operators of the type I+Schatten-class operator [15,30] which in contrast to these are not multiplicative but do extend the ordinary determinant (Fredholm determinant) for operators of the type I+trace-class operator.

Extension of determinants
Now we consider determinants constructed from a trace as above, by defining both sides of Equation (21), not only on a neighborhood of the identity in the range of the exponential mapping, but also on a set of admissible operators and on the pathwise connected component of the identity.
Remark 5. In this section, the word "extension" is in the sense that the determinants can be defined on operators which not necessarily lie in the range of the exponential mapping. See Remark 6 below for the other sense of this word.

First extension
The first way to define these determinants is carried out by considering the right hand side of (21). In the case of G = C 0 odd (M ) * the logarithm can be defined provided one chooses a spectral cut θ thereby to fix a determination log θ of the logarithm. We set Recall that if the operator A lies in the odd-class and has even order, then the logarithm log θ A lies also in the odd-class. If φ is another spectral cut of A such that 0 ≤ θ < φ < 2π, by formula (8) we have where P θ,φ (A) is an odd-class projection as in (7).
Proposition 8 (Proposition 6.2.3 in [21]). Let λ be any continuous trace on C 0 odd (M ). Suppose that λ takes integer values on the image of P θ,φ (A) for all θ, φ and A, where A is an admissible operator with spectral cuts θ and φ as in (8). Then λ gives rise to the map Det λ θ in (22), on admissible operators, independent of the choice of the spectral cut θ.
Let us consider this construction for the traces given in Section 3: Let A be an admissible operator in C 0 odd (M ) with spectral cut θ. With the notation of Subsection 3.1.3, the determinant associated to the leading symbol trace λ = ρ • π 0 is defined by Det λ θ (A) := exp (ρ • π 0 (log θ A)) .
In Example 2 of [13], it is shown that if P is a zero-order pseudodifferential idempotent, then its leading symbol p is also an idempotent so that the fibrewise trace tr x (p(x, ·)) is the rank rk(p(x, ·)). Hence It follows that Det λ θ (A) is independent of the choice of the spectral cut θ for any linear map ρ : Cl 0 odd (M )/Cl −1 odd (M ) → Z. Observe that if A is a zero-order odd-class operator so is log θ A. Hence, the canonical trace extends to logarithms of admissible odd-class operators of order zero with its property of cyclicity in odd dimensions. The determinant associated to the canonical trace is defined by Det TR θ (A) := exp(TR(log θ A)).
In contrast to the leading symbol trace, the canonical trace does not satisfy the requirement of Proposition 8 so that the associated determinant depends on the choice of spectral cut.

Second extension
An alternative way to define these determinants is by considering the left hand side of (21) and the expression for a determinant given in the proof of Lemma 6: where γ : [0, 1] → G is a C 1 -path with γ(0) = 1 G and γ(1) = g.
Such an approach was adopted in [2] by P. de la Harpe and G. Skandalis in the case of a Banach Lie group. In her thesis [3], C. Ducourtioux adopted this point of view to construct a determinant associated to a weighted trace with associated Lie algebras C 0 (M ) and C (M ). In [13], J.-M. Lescure and S. Paycha showed that such a construction extends to Fréchet-Lie groups with exponential mapping.
As in Definition 5, let G denote the pathwise connected component of the identity 1 G of G and P(G) the set of C 1 -paths γ : [0, 1] → G starting at 1 G (γ(0) = 1 G ) in G. On P(G) we introduce the map: Det λ : P(G) → C * defined by where ω = g −1 dg is the Maurer-Cartan form on G.
Since λ satisfies the tracial property, Det λ has the multiplicative property: Lemma 8. Let γ 1 , γ 2 be two C 1 -paths in P(G). Then Proof . The same proof applies as in Lemma 6.
In general the Maurer-Cartan form ω = g −1 dg is not exact on G so that for a C 1 -path c : [0, 1] → G with c(0) = c(1), the integral c ω = independently of the choice of path γ.
3. If g lies in the range of the exponential mapping Exp then Det λ (g) = exp (λ(Log(g))) , where Log = Exp −1 is the inverse of the exponential mapping.
Remark 6. Item 3 of this proposition shows that Det λ is an extension of the determinant map defined in Lemma 6.
Proof . 1. We want to show that two homotopic loops c 1 and c 2 have common primitive. Let us first recall the following general construction of a primitive: for ω a differential form on G, let γ : [0, 1] → G be a C 1 -path and F : [0, 1] → G be such that for any t ∈ [0, 1], F (t) = ω(γ(t))γ (t).
If ω is an exact form i.e. w = df for some f ∈ G, then F (t) = f (γ(t)) is a primitive of F . If the form ω is closed, then ω is locally exact. In this case, let 0 = t 0 < t 1 < · · · < t k = 1 be a subdivision of the interval [0, 1] such that γ([t i−1 , t i ]) is a subset of G, and there exists f i defined on [t i−1 , t i ] such that df i = ω. We can construct a function F (t) on [0, 1] in the following way: F (t) = f 0 (γ(t)) on [t 0 , t 1 ], F (t) = f 1 (γ(t)) − h 1 on [t 1 , t 2 ] where h 1 = f 1 (γ(t 1 )) − f 0 (γ(t 1 )) and for i = 3, . . . , k, F (t) = f i (γ(t)) − h i on [t i−1 , t i ] where h i = f i (γ(t i )) − f i−1 (γ(t i )) + h i−1 . Now let F (t) and G(t) be primitives of c 1 and c 2 respectively. Since c 1 and c 2 are homotopic, there exists a family of C 1 -paths (α i ) 0≤i≤k defined in a neighborhood of 1 G such that c 1 = c 2 k i=0 α i . Each path α i is closed so that c ω vanishes on α i . It follows that F (t) = G(t), and therefore the map Φ is well-defined on Π 1 (G).
From Proposition 8 we can see that in the first extension of the definition of a determinant, the non-dependency on the spectral cut is controlled by the image of the projection P θ,φ by the trace λ. From Proposition 9, the map Det λ is well defined if the image of the fundamental group of C 0 odd (M ) * is trivial. The following theorem shows that in both definitions (Subsec- Theorem 5 (see Section 4.5 in [8] and Lemma A.5 in [3]). The fundamental group of C 0 odd (M ) * is generated by the homotopy classes of loops {Exp(2iπtP )} 0≤t≤1 , where P is a projector in C 0 odd (M ).
Remark 7. Let us give some comments for the case C odd (M ). As seen in Theorem 2, any trace in C odd (M ) is proportional to the canonical trace. For the subalgebra C 0 odd (M ) in (23) we defined the determinant associated to the canonical trace of an operator A with spectral cut θ. Unfortunately, this definition cannot be extended a priori to positive order elements of C odd (M ). Indeed, as we said before (see Subsection 4.2.1), if the order of A is even, for any spectral cut θ of A, the logarithm log θ A is also odd-class, so the canonical trace extends to logarithms of odd-class operators with even order and one can extend the determinant by: Det TR θ (A) := exp(TR(log θ A)). This is no longer true if the order of A is odd. M. Braverman in [1] introduced the notion of symmetrized trace to define a symmetrized determinant on oddclass operators. It was shown in [20] that this symmetrized trace is in fact the canonical trace. The idea is to compute the average of the usual terms given by two spectral cuts of A, namely θ and θ − aπ, where a is the order of A; one obtains the following definition of symmetrized logarithm: log sym θ A := 1 2 (log θ A + log θ−aπ A).
This symmetrized logarithm is also odd-class and, once again, one can use the canonical trace to define a determinant by setting: DET sym θ (A) := exp TR log sym θ A .
Notice that if a = 0, DET sym θ (A) = Det TR θ (A). However this symmetrized determinant also depends on the spectral cut θ and under suitable assumptions it is multiplicative up to a sign (see [1]).