$C^{\ast}$-algebraic approach to the principal symbol. III

We treat the notion of principal symbol mapping on a compact smooth manifold as a $\ast$-homomorphism of $C^{\ast}$-algebras. Principal symbol mapping is built from the ground, without referring to the pseudodifferential calculus on the manifold. Our concrete approach allows us to extend Connes Trace Theorem for compact Riemannian manifolds.


Introduction
This paper is motivated by the theory of pseudodifferential operators.A central notion of that theory is that of a principal symbol, which is roughly a homomorphism from the algebra of pseudo-differential operators into an algebra of functions, [14, Lemma 5.1], [13,Theorem 5.5], [32, pp. 54-55].Usually, it is defined in a manner inhospitable for operator theorists.However, in [30], a new approach to a principal symbol mapping on a certain C * -subalgebra Π in B(L 2 (R d )) is proposed; this mapping turns out to be a * -homomorphism from Π into a commutative C *algebra.The C * -algebra Π contains all classical compactly based pseudodifferential operators.This provides a very simple and algebraic approach to the theory.
Whereas our approach is more elementary than the classical approach, the C *algebra Π introduced in [30] (see also [20]) is much wider than the class of a classical compactly based pseudo-differential operators of order 0 on R d .The aim of this paper is to extend this C * -algebraic approach to the setting of smooth compact manifolds.
The C * -algebra Π in the Definition 1.1 below is the closure (in the uniform norm) of the * -algebra of all compactly supported classical pseudodifferential operators of order 0. However, we use an elementary definition of Π which does not involve pseudodifferential operators.The idea to consider this closure may be discerned yet in [3] (see Proposition 5.2 on p.512).For the recent development of this idea we refer to [30,20].
Let D k = ∂ i∂t k be the k−th partial derivative operator on R d (these are unbounded self-adjoint operators on be defined by the functional calculus.Let M f be the multiplication operator by the function f. )) be defined by setting Let A 1 = C + C 0 (R d ) and A 2 = C(S d−1 ).Let Π be the C * -subalgebra in B(L 2 (R d )) generated by the algebras π 1 (A 1 ) and π 2 (A 2 ).
According to [30], there exists an * -homomorphism such that sym(π 1 (f )) = f ⊗ 1, sym(π 2 (g)) = 1 ⊗ g.Here, A 1 ⊗ min A 2 is the minimal tensor product of the C * -algebras A 1 and A 2 (see Propositions 1.22.2 and 1.22.3 in [25]).Elements of A 1 ⊗ min A 2 are identified with continuous functions on R d × S d−1 .This * -homomorphism is called a principal symbol mapping.It properly extends the notion of the principal symbol of the classical pseudodifferential operator.
It is natural to ask whether C * -algebraic approach works in the general setting of smooth compact manifolds.It makes sense to de-manifoldize the question and reformulate it in a purely Euclidean fashion.We begin with the natural question on the properties of the C * -algebra Π.
Question 1.2.The natural unitary action of the group of diffeomorphisms on R d is defined as follows.Let Φ : R d → R d be a diffeomorphism.Let U Φ ∈ B(L 2 (R d )) be a unitary operator given by setting Here, J Φ is the Jacobian matrix of Φ.
Is the C * -algebra Π invariant under the action T → U −1 Φ T U Φ ?Does the *homomorphism sym behave equivariantly under this action?Theorem 3.5 provides a positive answer to Question 1.2 (under the additional requirement that Φ is affine outside of some ball).This additional assumption yields, in particular, that Φ extends to a diffeomorphism of the projective space P d (R).We emphasise that the Question 1.2 in full generality remains open.Furthermore, Theorem 3.11 proves an invariance of Π and equivariance of sym under local diffeomorphisms.
The resolution of Question 1.2 has opened an avenue for the definition of the C * -algebra Π X associated with an arbitrary compact smooth manifold X.This C *algebra has a remarkable property: it admits a * -homomorphism sym X : Π X → C(S * X), where S * X is the cosphere bundle of X (see the Subsection 2.7).If X = R d , then sym X coincides with the mapping sym above.Every classical order 0 pseudodifferential operator T on X belongs to Π X and its principal symbol in the sense of pseudodifferential operators equals sym X (T ).On the other hand, not every element of Π X is pseudodifferential (e.g. because principal symbol of a pseudodifferential operator is necessarily smooth, while that of element of Π X is only continuous).An approach to pseudodifferential calculi based on C * -algebras theory was first suggested by H.O. Cordes [8] (see [21] for the case of a closed manifold).
Below, we briefly describe the construction of Π X via the patching process (see more precise description in Subsection 7.2).
Let X be a compact smooth manifold with an atlas (U i , h i ) i∈I .We will fix a sufficiently good measure ν on X, given by a continuous positive density (see Definition 2.20).If T ∈ B(L 2 (X, ν)) is compactly supported in some chart (U i , h i ) (i.e., there exists φ ∈ C ∞ c (U i ) such that T = T M φ = M φ T ), then, by composing with h i , we can transfer T to an operator on L 2 (R d ).
Definition 1.3.Let X be a compact smooth manifold equipped with a continuous positive density ν and let T ∈ B(L 2 (X, ν)).We say that T ∈ Π X if (1) for every i ∈ I and for every φ ∈ C c (U i ), the operator M φ T M φ transferred to an operator on L 2 (R d ) belongs to Π; (2) for every ψ ∈ C(X), the operator [T, M ψ ] is compact.
Theorem 1.4.If X is a smooth compact manifold and if ν is a continuous positive density on X, then Π X is a C * -algebra and there exists (see Definition 7.8) a surjective * -homomorphism sym X : Π X → C(S * X) such that ker(sym X ) = K(L 2 (X, ν)).
In other words, we have a short exact sequence This short exact sequence first appeared in [3] (see Proposition 5.2 on p.512) and plays an important role in index theory (see, for instance, [5,Section 24.1.8]or [4,Section 2]).It is essentially equivalent to the fact that for any operator T ∈ Π X with principal symbol a ∈ C(S * X), For singular integral operators this result was proved by Gohberg [10] and Seeley [26].Proofs in the language of pseudodifferential operators have been given in [12,14].It should be noted that the definition given in [3] is somewhat imprecise (see [21], in particular, a discussion on p. 329).
As a corollary of Theorem 1.4, we provide a version of Connes Trace Theorem (see Theorem 1.5 below).As stated, it extends Theorem 1 in [7].Connes Trace Theorem is ubiquitous in Non-commutative Geometry.It serves as a ground for defining a general notion of the non-commutative integral and non-commutative Yang-Mills action (that is, Theorem 14 in [7] is taken as a definition in the noncommutative setting).
We now compare our Theorem 1.5 with various versions of Connes Trace Theorem available in the literature.Original proof of Connes was, according to [11] "somewhat telegraphic".For example, it was not mentioned in [7] that the manifold is Riemannian and that pseudodifferential operator featuring in Theorem 1 in [7] is classical.Two proofs are given in [11] (Theorem 7.18 on p.293) and both of them rely on the assumption of ellipticity of the underlying pseudo-differential operator (this assumption is redundant as demonstrated in our approach).Despite their critique of Connes exposition, the authors of [11] also do not mention the classicality of their pseudodifferential operator.Another two proofs are given in [2].As authors of [2] admit, their proofs are quite sketchy, however, they provide a correct statement.The advantage of our approach is threefold: (a) we consider a strictly larger class of operators (b) we consider a strictly larger class of traces (c) we work in a convenient category of C * -algebras (i.e., non-commutative topological spaces) and not in a category of classical pseudodifferential operators which does not have a natural counterpart in Non-commutative Geometry.
Theorem 1.5.Let ϕ be a normalised continuous trace on L 1,∞ .Let (X, G) be a compact Riemannian manifold and let ν be the Riemannian volume.
where λ is the Liouville measure on T * X and e −qX is the canonical weight of the Riemannian manifold (as defined in Subsection 2.8).
When T is a classical pseudodifferential operator, the right hand side coincides with Wodzicki residue of We refer the reader to the extensive discussion of this matter in [18].
One should note a sharp contrast between the setting of Theorem 1.5 and that of Theorem 1.4.Indeed, in the latter theorem, the (smooth compact) manifold is rather arbitrary, while in the former it is Riemannian.The Riemannian structure of X in Theorem 1.5 is needed in two places: (a) there is no natural measure on the cosphere bundle of an arbitrary smooth manifold (but such a measure arises naturally if the manifold is Riemannian) (b) Riemannian structure provides us with a natural second order differential operator (i.e, Laplace-Beltrami operator).In the setting of a general smooth manifoild, the second issue can be circumvented by replacing ∆ G with an arbitrary elliptic second order differential operator (whose resolvent falls into the ideal L d,∞ ).However, the lack of a natural measure on S * X prevents us from stating Theorem 1.5 in that generality.
We now briefly describe the structure of the paper.Section 2 collects known facts used further in the text.Theorems 3.5 and 3.11 in Section 3 assert equivariant behavior of the principal symbol mapping in Euclidean setting under the action of diffeomorphisms.Theorem 3.5 is proved in Section 5. Theorem 3.11 is proved in Section 6.Our main result, Theorem 1.4 is proved in Section 7 with the help of Globalisation Theorem from Subsection 7.1 (proved in Appendix A).Finally, Connes Trace Theorem on compact Riemannian manifolds (that is, Theorem 1.5) is proved in Section 8.

Preliminaries and notations
As usual, B(H) denotes the * -algebra of all bounded operators on the Hilbert space H and K(H) denotes the ideal of all compact operators in B(H).As usual, Euclidean length of a vector t ∈ R d is denoted by |t|.
We frequently use the equality

Principal ideals in B(H).
It is well known that every ideal in B(H) consists of compact operators.Undoubtedly, the most important ideals are the principal ones.Among them, a special role is played by the ideal L p,∞ , a principal ideal generated by the operator diag(((k + 1) − 1 p ) k≥0 ).We frequently use the following property (related to the Hölder inequality) of this scale of ideals We mention in passing that L p,∞ is quasi-Banach for every p > 0 (however, we do not need the quasi-norms in this text).
Definition 2.1.If I is an ideal in B(H), then a unitarily invariant linear functional ϕ : I → C is said to be a trace.
] for all T ∈ I and for all unitaries U ∈ B(H), and since the unitaries span B(H), it follows that traces are precisely the linear functionals on I satisfying the condition The latter may be reinterpreted as the vanishing of the linear functional ϕ on the commutator subspace which is denoted [I, B(H)] and defined to be the linear span of all commutators [T, S] : T ∈ I, S ∈ B(H).Note that ϕ(T 1 ) = ϕ(T 2 ) whenever 0 ≤ T 1 , T 2 ∈ I are such that the singular value sequences µ(T 1 ) and µ(T 2 ) coincide.For p > 1, the ideal L p,∞ does not admit a non-zero trace while for p = 1, there exists a plethora of traces on L 1,∞ (see e.g.[19]).An example of a trace on L 1,∞ is the Dixmier trace introduced in [9] that we now explain.
Example 2.2.Let ω be an extended limit.Then the functional Tr is additive and, therefore, extends to a trace on L 1,∞ .We call such traces Dixmier traces.These traces clearly depend on the choice of the functional ω on l ∞ .
An extensive discussion of traces, and more recent developments in the theory, may be found in [19] including a discussion of the following facts.
(2) All positive traces on L 1,∞ are continuous in the quasi-norm topology.
(3) There exist positive traces on L 1,∞ which are not Dixmier traces.(4) There exist traces on L 1,∞ which fail to be continuous.We are mostly interested in normalised traces ϕ : L 1,∞ → C, that is, satisfying ϕ(T ) = 1 whenever 0 ≤ T is such that µ(k, T ) = 1 k+1 for all k ≥ 0. Traces on L 1,∞ play a fundamental role in Non-commutative Geometry.For example, they allow to write Connes Character Formula (we refer the reader to Section 5.3 in [18] and references therein).
The importance of Sobolev spaces in the theory of differential operators can be seen e.g. from the fact that W 1,2 (R d ) is the domain of the self-adjoint tuple ∇.Also, W 2,2 (R d ) is the domain of the self-adjoint positive operator −∆.
We refer the reader to the books [1], [31] for further information on Sobolev spaces.
Further we need the following standard result (see e.g.p.322 in [31]).
Theorem 2.3.Sobolev space W m,2 (R d ), m ∈ Z + , is invariant under diffeomorphisms which are affine outside of some ball.
) (that is, bounded smooth function whose derivatives are also bounded functions), then the Calderon-Vaillancourt theorem (see e.g.unnumbered proposition on p.282 in [28]) the operator Op(p) defined by the formula (here then we say that Op(p) ∈ Ψ m (R d ).For m > 0, the class Ψ m (R d ) is defined by the same formula.The difference is that, for m > 0, operators in Ψ m (R d ) are no longer bounded as operators from The key property is that Moreover, by Theorem 2.5.1 in [24], we have The next lemma follows immediately from (2.4) and (2.5). then 2 is compact (see e.g.Theorem 4.1 in [27]).Thus, Differential operators of order m ≥ 0 with smooth bounded coefficients (all derivatives of the coefficients are also assumed bounded) belong to Ψ m (R d ).Indeed, it follows directly from (2.2) that (2.7) The following standard result is available, e.g. in Theorem 1.6.20 in [17].
Notation 2.8.Let X be a smooth d-dimensional manifold with atlas (U i , h i ) i∈I , where I is an arbitrary set of indices.
(1) We denote (2) We denote by Φ i,j : Ω i,j → Ω j,i the diffeomorphism given by the formula In the next fact, we recall the manifold structure of T * X. Fact 2.9.Let X be a d-dimensional manifold with an atlas {(U i , h i )} i∈I .Let T * X be the cotangent bundle of X and let π : T * X → X be the canonical projection.There exists an atlas {π −1 (U i ), H i } i∈I of T * X such that (1) for every i ∈ I, In the next fact, we identify functions on the T * X and their local representations.It is important that this identification preserves continuity.Fact 2.10.
then there exists a unique function F : T * X → C such that This action lifts down to an action on T * X (also denoted by σ λ ).A function on T * X invariant with respect to this action is called dilation invariant.Definition 2.12.Let X be a compact manifold.C * -algebra of all continuous dilation invariant functions on T * X\0 T * X (here, 0 T * X is the zero section of T * X) is denoted by C(S * X) and is called the algebra of continuous functions on the cosphere bundle of X.
2.8.Canonical weight of Riemannian manifold.If X is a smooth d-dimensional manifold, then T * X has a canonical symplectic structure.The corresponding Liouville measure λ on T * X satisfies the following property (see, for instance, [6]): (2.9) However, there is no canonical way to equip the cosphere bundle S * X of a smooth manifold X with a measure.The following class of measures is of particular interest: generates a measure on S * X by the Riesz-Markov theorem.However, there is no canonical way to select an integrable function w on T * X.This choice becomes possible if we assume in addition a Riemannian structure on X.
Let G be a Riemannian metric on X.For any i ∈ I, the components of the metric G in the chart (U i , h i ) give rise to a smooth mapping G i : (In what follows, GL + (d, R) stands for the set of all positive elements in GL(d, R).)For any i, j ∈ I such that U i ∩ U j = ∅, we have Here, Φ j,i are given in Notation 2.8.Notation 2.13.For every i ∈ I, let Ω i be as in Notation 2.8 and set It can be easily verified by a direct calculation that, for every i, j ∈ I, we have By Fact 2.10, there exists a function q X on T * X such that (2.10) The function q X is the square of the length function on T * X defined by the induced Riemannian metric on the cotangent bundle T * X.
Definition 2.14.The function e −qX on T * X is called the canonical weight of the Riemannian manifold (X, G).
plays a crucial role.It defines the natural measure on S * X.We note that the latter functional coincides (modulo a constant factor) with integration with respect to the kinematic density on S * X (see p.318 in [6]).
be a finite partition of unity.We call it good if each φ n is compactly supported in some chart.
Obviously, good partitions of unity exist only on compact manifolds.Definition 2.17.Let (X, G) be a compact Riemannian manifold.Let Ω i be as in Notation 2.8 and let g i : Here, i n ∈ I is chosen such that φ n is compactly supported in U in .
Though Definition 2.17 involves good partition of unity, the operator ∆ G does not actually depend on the particular choice of a good partition of unity.
Theorem 2.4 in [29] yields the following results.The first one is of conceptual importance.The second one is used in the proof of Theorem 1.5.
Theorem 2.18.Let (X, G) be a compact Riemannian manifold.Laplace-Beltrami operator admits a self-adjoint extension 2.10.Density on a manifold.Let B be the Borel σ-algebra on the manifold X.We need the notion of density on a manifold available, e.g. in [22, p.87].
Definition 2.20.Let ν be a countably additive measure on B. We assume that, for every i ∈ I, the measure ν • h −1 i on Ω i is absolutely continuous with respect to the Lebesgue measure on Ω i , and its Radon-Nikodym derivative a i is strictly positive and continuous on Ω i .
In this case, we say that ν is a continuous positive density on X.

Invariance of principal symbol under diffeomorphisms
In this section, we formulate a theorem which provides a partial positive answer to Question 1.2.This result is stated in two versions: Theorem 3.5 for diffeomorphisms of R d (which a core technical difficulty) and Theorem 3.11 for local diffeomorphisms (the result which would be actually used).(1) Let J Φ : R d → GL(d, R) be the Jacobian matrix of Φ; (2) Let unitary operator U : ) be defined by setting Φ is the adjoint to the Jacobi matrix.Here, for A ∈ GL(d, R), we set We frequently need the following compatibility lemma. where

It suffices to show that
. The latter equality is written as Replacing t with (Φ 1 • Φ 2 )(t), we need to verify that In other words, This is the chain rule property.
) is a smooth mapping.For every smooth mapping A : R d → GL(d, R), the mapping By Lemma 3.2, its inverse is Θ Φ −1 which is also a smooth mapping.
We are now ready to state the main result in this subsection.
If we view symbols as homogeneous functions We prove Theorem 3.5 in Section 5.
There are two reasons for us to require that Φ is affine outside of some ball.The first reason is that having equivariant behavior of the principal symbol under such diffeomorphisms is sufficient in the proof of Theorem 3.11 below.The second reason is that in the proof of Theorem 3.5 we conjugate the Laplacian with U Φ .Hence, it is of crucial importance that U Φ preserves the domain of Laplacian.Recall that the domain of Laplacian is a Sobolev spaces W 2,2 (R d ) and, by Theorem 2.3, U Φ leaves the domain of Laplacian invariant.

3.2.
Invariance under local diffeomorphisms.One may ask how does the algebra and the principal symbol mapping (locally) behaves under the change of coordinates.We need the following notations.
) is a smooth mapping.For every smooth mapping A : Ω → GL(d, R), the mapping is smooth.Thus, Θ Φ is smooth.By Lemma 3.9, its inverse is Θ Φ −1 which is also a smooth mapping.
If we view symbols as homogeneous functions on Theorem 3.11 is proved in Section 6 as a corollary of Theorem 3.5.

Conjugation of differential operators with U Φ
In this section, we examine the operators ) and show that they may be viewed as differential operators.
Lemma 4.1.Let Φ : R d → R d be a diffeomorphism such that Φ is affine outside of some ball.Mapping V Φ by setting Proof.This is a special case of Theorem 2.3 (or it can be verified by hands).Lemma 4.2.Let Φ : R d → R d be a diffeomorphism such that Φ is affine outside of some ball.We have ). Proof.By definition of U Φ (in Notation 3.1) and of V Φ (in Lemma 4.1), we have The assertion of the lemma now follows from the Definition 1.1.
Proof.Since Φ is a diffeomorphism, it follows that all those functions are smooth.Since Φ is affine outside of some ball, it follows that J Φ is constant outside of some ball.Thus, D k (|det(J Φ )| 1 2 ) = 0 outside of some ball.Using the definition of a Φ k , we now see that it vanishes outside of some ball.Using the definition of b Φ l and b Φ , we now see that it vanishes outside of some ball.Lemma 4.5.Let Φ : R d → R d be a diffeomorphism such that Φ is affine outside of some ball.We have Here, equalities are understood as equalities of differential operators acting from Proof.By the chain rule, we have Using the notations for V Φ (in Lemma 4.1) and for the multiplication operator, we can rewrite this formula as follows: Thus, where the last equality follows from the definition of a Φ k,l (in Notation 4.3) and the fact that Lemma 4.6.Let Φ : R d → R d be a diffeomorphism such that Φ is affine outside of some ball.We have Proof.By definition of U Φ (in Notation 3.1) and of V Φ (in Lemma 4.1), we have Since Φ is a diffeomorphism and since Φ is affine outside of some ball, it follows that h Φ is a smooth function on R d which is constant outside of some ball.It follows that M hΦ : W m,2 (R d ) → W m,2 (R d ).A combination of those mappings yields the assertion.By Lemma 4.6, we have Hence, we may view Lemma 4.7.Let Φ : R d → R d be a diffeomorphism such that Φ is affine outside of some ball.We have (1) Here, equalities are understood as equalities of linear operators acting from Proof.Repeating beginning of the proof of Lemma 4.6, we write It is immediate that Clearly, Combining (4.1) and (4.2), we obtain The equality (1) follows by combining Lemma 4.5, (4.3) and (4.4).
Taking the adjoint of (1), we write Thus, Clearly, Thus, By the definition on b Φ l and b Φ (in Lemma 4.4), we have Consider now the highest order term.Recalling Notation 4.3, we write This delivers (2).

Proof of Theorem 3.5
The proof of Theorem 3.5 is somewhat technical and is presented below in the series of lemmas.The strategy is as follows: (1) to show that every compact operator on L 2 (R d ) belongs to Π; (2) to show that the conjugation of , by U Φ belongs to Π; (4) to conclude the argument in Theorem 3.5; The following assertion is well-known (see e.g.Corollary 4.1.10in [9]).
Lemma 5.1.Let A be a C * -algebra.Let π : A → B(H) be an irreducible representation.One of the following mutually exclusive options holds: (1) π(A) does not contain any compact operator (except for 0); (2) π(A) contains every compact operator.
We now apply Lemma 5.1 to the C * -algebra A = Π and infer that Π contains the ideal K(L 2 (R d )).
Lemma 5.2.The algebra K(L 2 (R d )) is contained in Π and coincides with the kernel of the homomorphism sym.
Proof.Since Π contains π 1 (A 1 ), it follows (here, X ′ denotes the commutant of the set Clearly, π 2 (g n,k ) → sgn(D k ) as n → ∞ in weak operator topology.Thus, sgn(D k ) belongs to the weak closure of π 2 (A 2 ) and, hence, to the weak closure of Π.Therefore, For t ∈ R d , denote by ťk ∈ R d−1 the vector obtained by eliminating the k-th ) is such that π 1 (f ) commutes with sgn(D k ), then, for almost every ťk ∈ R d−1 , the function f ( ťk , •) commutes with the Hilbert transform.This easily implies that, for almost every ťk By Proposition II.6.1.8 in [5], representation id : Π → B(L 2 (R d )) is irreducible.
We now demonstrate that Π contains a non-zero compact operator.As proved above, for every non-zero f ∈ C ∞ c (R d ), there exists 1 ≤ k ≤ d such that π 1 (f ) does not commutes with sgn(D k ).Since π 2 (g n,k ) → sgn(D k ) as n → ∞ in weak operator topology, it follows that π 1 (f ) does not commute with π 2 (g n,k ) for some n, k.Thus, the operator [π 1 (f ), π 2 (g n,k )] is a non-zero compact operator, which belongs to Π.The first assertion of the lemma follows now from Lemma 5.1.
Let q : B(L )) be the canonical quotient map.Recall (see the proof of Theorem 3.3 in [20]) that sym is constructed as a composition sym = θ −1 • q, where θ −1 is some linear isomorphism (its definition and properties are irrelevant at the current proof).It follows that the kernel of sym coincides with the kernel of q, which is K(L 2 (R d )).Notation 5.3.Let Φ : R d → R d be a diffeomorphism such that Φ is affine outside of some ball.Denote Here, (a Φ k,l ) d k,l=1 and (b Φ l1,l2 ) d l1,l2=1 are as in Notation 4.3.The following two lemmas form the core of our computation.Lemma 5.4.Let Φ : R d → R d be a diffeomorphism such that Φ is affine outside of some ball.We have ) + Ψ −1 (R d ).
Fix a function ψ ∈ C ∞ c (R d ) such that ψ = 1 near 0 and such that (φ By the Leibniz rule, we have 3)).It follows now from (5.5) that (5.6) Denote for brevity the left hand side of (5.6) by T (so that T ∈ Ψ −1 (R d )).Due to the choice of ψ, we have that T = M ψ T. It follows now from (2.6) (applied with m = −1) that T is compact.In other words, we have Appealing to the definition of r Φ k and p Φ , we note that By Lemma 2.7 applied with q = (φ . Combining (5.7) and (5.8), we complete the proof.Lemma 5.6.Let Φ be a diffeomorphism such that Φ is affine outside of some ball.
Proof.Applying bounded Borel function to the tuple ∇, we obtain that Recall that (see e.g.Theorem 4.1 in [27]) . Since product of bounded and compact operators is compact, it follows that and Combining with Lemma 5.5, we obtain By Lemma 2.6, we have The assertion follows by combining the last two equations and Lemma 5.2.
Proof.Since sym is a * -homomorphism, it follows that sym([T 1 , T 2 ]) = 0.The assertion is now an immediate consequence of Lemma 5.2.
Proof.We prove the assertion by induction on m.For m = 1, there is nothing to prove.So, we only have to prove the step of induction.
Let us prove the assertion for m = 2.We have By Lemma 5.7, we have Therefore, This proves the assertion for m = 2.It remains to prove the step of induction.Suppose the assertion holds for m ≥ 2 and let us prove it for m + 1.Clearly, Using the inductive assumption, we obtain Using the assertion for m = 2, we obtain Combining the last two equations, we obtain This establishes the step of induction and, hence, completes the proof of the lemma.Lemma 5.9.Let Φ be a diffeomorphism which is affine outside of some ball.If Proof.Let Poly(S d−1 ) be the algebra of polynomials on In this notations, By Lemma 5.8, we have Since Thus, By Lemma 5.6, we have By Lemma 5.2, we have . By linearity, the same assertion holds if g ∈ Poly(S d−1 ).To prove the assertion in general, let g ∈ C(S d−1 ) and consider a sequence {g n } n≥1 ⊂ Poly(S d−1 ) such that g n → g in the uniform norm.We have Proof of Theorem 3.5.By the definition of the C * -algebra Π, for every T ∈ Π, there exists a sequence (T n ) n≥1 in the * -algebra generated by π 1 (A 1 ) and π 2 (A 2 ) such that T n → T in the uniform norm.We can write By Lemma 5.8, we have Denote for brevity, We have By Lemma 5.2, we have Suppose in addition that T is compactly supported.In particular, T = M φ T for some φ ∈ C ∞ c (R d ).Replacing S n with M φ S n and f n,l with φ • f n,l if necessary, we may assume without loss of generality that f n,l ∈ C ∞ c (R d ) for every n ≥ 1 and for every 1 ≤ l ≤ l n .
By Lemma 5.9, we have where Θ Φ is introduced in Notation 3.1.By Lemma 5.2, we have Since Π is a C * -algebra and since

Invariance of principal symbol under local diffeomorphisms
Theorem 3.11 is supposed to be a corollary of Theorem 3.5.To demonstrate this is indeed the case, we need an extension result for diffeomorphisms.
The following fundamental result is due to Palais [23] (see Corollary 4.3 there).
Theorem 6.1.Let Φ : R d → R d be a smooth mapping.Necessary and sufficient conditions for Φ to be a diffeomorphism are as follows: (1) for every t ∈ R d , we have det(J Φ (t)) = 0; (2) we have The next lemma is also due to Palais [23].We provide a proof for convenience of the reader.Note that B(t, r) is the open ball with radius r centered at t. Lemma 6.2.Let Ω ⊂ R d be an open set and let Φ : Ω → R d be a smooth mapping.If t ∈ Ω is such that det(J Φ (t)) = 0, then there exists a diffeomorphism Φ t : R d → R d such that (1) Φ t = Φ on B(t, r 1 (t)) with some r 1 (t) > 0; (2) Φ t is affine outside B(t, r 2 (t)) for some r 2 (t) < ∞; Proof.Without loss of generality, t = 0, Φ(0) = 0 and It is clear that Ψ r is well-defined smooth mapping for every sufficiently small r > 0.
A direct shows that det(J Ψr ) → 1 in the uniform norm as r → 0. In particular, for sufficiently small r > 0, det(J Ψr ) never vanishes.It follows from Theorem 6.1 that, for sufficiently small r > 0, Ψ r : R d → R d is a diffeomorphism.Choose any such r and denote it by r(0).Set Φ 0 = Ψ r(0) .This diffeomorphism obviously satisfies the required properties.
Proof.Indeed, since both sides are continuous in weak operator topology, it suffices to prove the assertion for the case when T is rank 1 operator. Let Since ξ 1 and ξ 2 are supported in B, it follows that expressions in the first and last displays coincide.This proves the assertion for every rank 1 operator T and, therefore, for every T.
Proof of Theorem 3.11.Let the operator T be supported on a compact set K ⊂ Ω.Let t ∈ K. Let diffeomorphism Φ t : R d → R d and numbers r 1 (t) and r 2 (t) be as in Lemma 6.2.
The collection {B(t, r 1 (t))} t∈K is an open cover of K.By compactness, one can choose a finite sub-cover.So, let {t n } N n=1 be such that We write (6.1) If n = 0, then T 0 is compact by Lemma 5.7.Clearly, Notation 3.6 and recall that a composition of bounded and compact operators is compact).Therefore, ( Ext Combining (6.1), (6.2) and (6.4), we obtain Now, combining (6.1), (6.3) and (6.5), we obtain sym Ext 7. Principal symbol on compact manifolds 7.1.Globalisation theorem.Globalisation theorem is a folklore.We provide its proof in Appendix A for convenience of the reader.
Definition 7.1.Let X be a compact manifold with an atlas {(U i , h i )} i∈I .Let B be the Borel σ-algebra on X and let ν be a countably additive measure on B. We say that {A i } i∈I are local algebras if (1) for every i ∈ I, A i is a * -subalgebra in B(L 2 (X, ν)); It is immediate that the mapping . The latter operator is compactly supported in Ω i .By Definition 2.20, exactly the same operator also belongs to B(L 2 (Ω i )) and, therefore, can be extended to an element Ext Ωi (W For the notion Ext Ωi we refer to Notation 3.7.Definition 7.5.Let X be a smooth compact manifold and let ν be a continuous positive density on X.For every i ∈ I, let Π i consist of the operators T ∈ B(L 2 (X, ν)) compactly supported in U i and such that For notation C(S * X) below we refer to Definition 2.12.For the notion sym, we refer to (1.1).Definition 7.6.Let X be a smooth compact manifold and let ν be a continuous positive density on X.For every i ∈ I, the mapping sym i : Π i → C(S * X) is defined by the formula That is, the collection {Π i } i∈I of * -algebras and the collection {sym i } i∈I of *homomorphisms satisfy the conditions in Theorem 7.4.Definition 7.8 below is the culmination of the paper.Having this definition at hands, we easily prove Theorem 1.4.Definition 7.8.Let X be a smooth compact Riemannian manifold and let ν be a continuous positive density on X.
(1) The domain Π X of the principal symbol mapping is the C * -algebra constructed in Theorem 7.4 from the collection {Π i } i∈I .
(2) The principal symbol mapping sym X : Π X → C(S * X) is the * -homomorphism constructed in Theorem 7.4 from the collection {sym i } i∈I .
Proof.It is immediate that Π i is a subalgebra in B(L 2 (X, ν)) and that sym i : Π i → C(S * X) is a homomorphism.We need to show that Π i is closed with respect to taking adjoints and that sym i is invariant with respect to this operation.Let T ∈ Π i and let us show that T * ∈ Π i .Recall that, due to the Condition 2.20, ν • h −1 i is absolutely continuous and that its density denoted by a i as well as its inverse a −1 i are assumed to be continuous in Ω i .The following equality 3 is easy to verify directly.
However, by Definition 7.5, the operator T is compactly supported in U i .Hence, the operator ( 3 The operators Ma i and M a −1 i are unbounded.The equality should be understood as LHSξ = RHSξ for every compactly supported ξ ∈ L 2 (Ω i ).
Indeed, for such ξ, we have Hence, the function ξ 2 = (W i T W −1 i ) * ξ 1 is compactly supported in Ω i .Hence, the function M a −1 i ξ 2 belongs to L 2 (Ω i ) and the right hand side of (7.1) makes sense.

Thus,
Since a i φ, a −1 i φ ∈ C c (R d ), it follows that every factor in the hand side of (7.2) belongs to Π. Hence, so is the expression on the left hand side.In other words, T * ∈ Π i .Thus, Π i is closed with respect to taking adjoints.
Recall that (by [30]) sym is a * -homomorphism.Applying sym to the equality (7.2), we obtain In the following lemma, Ξ Φi,j is defined according to the Notation 3.8.
Proof.Let V Φ ξ = ξ • Φ (provided that the image of the mapping Φ is contained in the domain of the function ξ).Since W j = V −1 Φi,j W i , it follows that (using the Notation 3.8) Combining the preceding paragraphs, we conclude that Denote for brevity ).The preceding display can be now re-written as ).Thus, . By Theorem 3.11, we have By Lemma 5.2, compact operators belong to Π. Therefore, T j ∈ Π and Proof of Theorem 7.
Hence, every F ∈ C ∞ (S * X) belongs to the A. In other words, C ∞ (S * X) ⊂ A. Since A is C * -subalgebra in C(S * X), it follows that A = C(S * X).Hence, sym X is surjective.

Proof of the Connes Trace Theorem
Lemma 8.1.Let g be as in Theorem 2. 19 By Theorem 2.5, we have 2 , the first factor belongs to L d r ,∞ by Theorem 1.4 in [15].When r = d 2 , the first factor belongs to L d r ,∞ by Theorem 1.3 in [15].When r < d 2 , the first factor belongs to L d r ,∞ by Theorem 1.1 in [15] (applied with r < d 2 ).The second factor belongs to Ψ 0 (R d ) and is, therefore, bounded.Lemma 8.2.Let g be as in Theorem 2.19.We have Proof.By definition, the principal symbol of 1 − ∆ g is By Theorem 2.5, we have and Op(p 1 ) are compactly supported from the left, it follows that so is Err 1 .Thus, Err 1 is a compact operator.Consequently, Err 1 ∈ Π and sym(Err 1 ) = 0. Let By Lemma 2.7, we have that Op(p 1 ) − T p2 is compact.So, our operator belongs to Π and its symbol equals that of T p2 , i.e. equals p 2 .
Lemma 8.3.Let (X, G) be a compact Riemannian manifold.Let ψ ∈ C ∞ (X) be compactly supported in the chart (U i , h ).Let ĝi : R d → GL + (d, R) be as in Theorem 2.19 and such that ĝi = g i in the neighborhood of the support of ψ where the first factor is in L d,∞ and the second factor is bounded.By Lemma 8.1, we have Combining the last three formulae, we complete the proof.
The same assertion holds for ∆ g , where g is as in Theorem 2.19 and for ψ ∈ Base of induction (i.e., the case n = 1) is obvious.It remains to prove step of induction.Suppose (8.1) holds for n and let us prove it for n + 1.We write The first term on the right hand side belongs to L d 2n+3 ,∞ by inductive assumption and Hölder inequality.Note that the operators 2 −n , are bounded.Hence, the second and third terms on the right hand side belong to L d 2n+3 ,∞ by Hölder inequality.This establishes the step of induction and, hence, proves the claim in Step 1.
Step 2: Note that is bounded, it follows that Taking adjoints, we obtain Lemma 8.6.Let (X, G) be a compact Riemannian manifold.Let T ∈ Π X be compactly supported in the chart (U i , h i ).Let ϕ be a continuous normalised trace on L 1,∞ .We have Here, q i is as in Notation 2.13.
Proof.Fix 0 ≤ ψ ∈ C ∞ (X) such that T = M ψ T M ψ and such that ψ is compactly supported in U i .
By the tracial property we have ).Since ϕ vanishes on L 2d 2d+1 ,∞ , it follows from Lemma 8.4 that ϕ(T (1 Since both operators T and A are compactly supported in the chart (U i , h i ), it follows that ϕ(T A ). Denote for brevity , where By Lemma 8.2, we have X i ∈ Π. the operator X i T i ∈ Π is compactly supported from the right.By Theorem 8.5 we have where m is the product of Lebesgue measure on R d and Haar measure on S d−1 .By Lemma 8.2, we have sym(X i T i )(t, s) = sym(T i )(t, s) By passing to spherical coordinates we obtain )e −qi(t,s) dtds.
Combining these 2 equalities, we complete the proof.
Proof of Theorem 1.5.Suppose first that T ∈ Π X is compactly supported in the chart (U i , h i ).By Lemma 8.6, we have A combination of these two equalities proves the assertion for T compacly supported in some U i .Let now T ∈ Π X be arbitrary.Let (φ n ) N n=1 be a fixed good partition of unity.We write By assumption, [T, M ψ ] is compact for every ψ ∈ C(X).In particular, T 0 is compact.Thus, Proof.It is established in Lemma A.1 that A is a unital * -subalgebra in B(L 2 (X, ν)).It suffices to show that A is closed in the uniform norm.
Let {T n } n≥1 ⊂ A and let T ∈ B(L 2 (X, ν)) be such that T n → T in the uniform norm.Let us show that T ∈ A.
Let i ∈ I and let φ ∈ C c (U i ).Take φ 0 ∈ C c (U i ) such that φφ 0 = φ.We have M φ T n M φ ∈ A i and, therefore, By assumption, M φ T n M φ → M φ T M φ in the uniform norm.Hence, M φ T M φ belongs to the closure of M φ0 A i M φ0 in the uniform norm.By Definition 7.1 (6), ), T ∈ A.
We now fix a good partition of unity and prove that the concrete map hom introduced in Lemma A.3 is * -homomorphism.Thus, hom is a * -homomorphism.

3. 1 .
Invariance under diffeomorphisms of R d .We need the following notations.Recall that GL(d, R) stands for the group of inverible real d × d matrices.Notation 3.1.Let Φ : R d → R d be a diffeomorphism.

1
A function f : R d ×S d−1 can be uniquely extended to a homogeneous function on R d ×(R d \{0}) by setting

Notation 3 . 6 .
Let H be a Hilbert space and let p ∈ B(H) be a projection.(1) If T ∈ B(H) is such that T = pT p, then we define the operator Rest p (T ) ∈ B(pH) by setting Rest p (H) = T | pH .(2) If T ∈ B(pH), then we define Ext p (T ) ∈ B(H) by setting Ext p (T ) = T • p.

Theorem 7 . 4 . 7 . 2 .
In the setting of Definitions 7.1, 7.3 and 7.2, we have (1) A is a unital C * -subalgebra in B(L 2 (X, ν)) which contains A i for every i ∈ I and K(L 2 (X, ν)); (2) there exists a * -homomorphism hom : A → B such that (a) hom = hom i on A i for every i ∈ I; (b) ker(hom) = K(L 2 (X, ν)); (3) * -homomorphism as in (2) is unique.Construction of the principal symbol mapping.Let B be the Borel σ-algebra on the manifold X and let ν : B → R be a continuous positive density.

7 .
Let X be a smooth compact manifold and let ν be a continuous positive density on (1) Collection {Π i } i∈I introduced in Definition 7.5 satisfies all the conditions in Definition 7.1; (2) Collection {sym i } i∈I introduced in Definition 7.6 satisfies all the conditions in Definition 7.3.

7 ( 1 )
. The condition (1) in Definition 7.1 is verified in Lemma 7.9.The condition (2) in Definition 7.1 is immediate.The condition (3) in Definition 7.1 is verified in Lemma 7.11.Let us verify the condition (4) in Definition 7.1.If i ∈ I and if T
the left hand side is positive.Therefore, hom in (M