Non-commutative residue of projections in Boutet de Monvel's calculus
aa r X i v : . [ m a t h . A P ] S e p NON-COMMUTATIVE RESIDUE OF PROJECTIONS INBOUTET DE MONVEL’S CALCULUS
ANDERS GAARDE
Abstract.
Using results by Melo, Nest, Schick, and Schrohe on the K -theory of Boutet de Monvel’s calculus of boundary value problems, weshow that the non-commutative residue introduced by Fedosov, Golse,Leichtnam, and Schrohe vanishes on projections in the calculus.This partially answers a question raised in a recent collaborationwith Grubb, namely whether the residue is zero on sectorial projectionsfor boundary value problems: This is confirmed to be true when thesectorial projections is in the calculus. Introduction
Boutet de Monvel [2] constructed a calculus, often called the Boutet deMonvel calculus (or algebra), of pseudodifferential boundary operators on amanifold with boundary. It includes the classical differential boundary valueproblems as well of the parametrices of the elliptic elements:Let X be a compact n -dimensional manifold with boundary ∂X ; we con-sider X as an embedded submanifold of a closed n -dimensional manifold e X .Denote by X ◦ the interior of X . Let E and F be smooth complex vectorbundles over X and ∂X , respectively, with E the restriction to X of a bundle e E over e X .An operator in Boutet de Monvel’s calculus — a (polyhomogeneous)Green operator — is a map A acting on sections of E and F , given bya matrix(1.1) A = P + + G KT S : C ∞ ( X, E ) C ∞ ( X, E ) ⊕ → ⊕ C ∞ ( ∂X, F ) C ∞ ( ∂X, F ) , where P is a pseudodifferential operator ( ψ do) on e X with the transmissionproperty and P + is its truncation to X :(1.2) P + = r + P e + , r + restricts from e X to X ◦ , e + extends by 0. Mathematics Subject Classification. G is a singular Green operator, T a trace operator, K a Poisson operator,and S a ψ do on the closed manifold ∂X . See [2], Grubb [6], or Schrohe [13]for details.Fedosov, Golse, Leichtnam, and Schrohe [4] extended the notion of non-commutative residue known from closed manifolds (cf. Wodzicki [14], [15],and Guillemin [9]) to the algebra of Green operators. The noncommutativeresidue of A from (1.1) was defined to be(1.3) res X ( A ) = Z X Z S ∗ x X tr E p − n ( x, ξ ) − dS ( ξ ) dx + Z ∂X Z S ∗ x ′ ∂X (cid:2) tr E (tr n g ) − n ( x ′ , ξ ′ ) + tr F s − n ( x ′ , ξ ′ ) (cid:3) − dS ( ξ ′ ) dx ′ . Here tr E and tr F are traces in Hom( E ) and Hom( F ), respectively; − dS ( ξ )(resp. − dS ( ξ ′ )) denotes the surface measure divided by (2 π ) n (resp. (2 π ) n − );tr n g is the normal trace of g ; and the subscripts − n and 1 − n indicate thatwe only consider the homogeneous terms of degree − n resp. 1 − n . Also, asign error in [4] has been corrected, cf. Grubb and Schrohe [8, (1.5)].It is well-known [14] that on a closed manifold, the noncommutativeresidue of a classical ψ do projection (or idempotent) is zero. In the presentpaper we wish to show that the same holds in the case of Green operators.We will use K -theoretic arguments (in a C ∗ -algebra setting) to effectivelyreduce the problem to the known case of closed manifolds.In our recent collaboration with Grubb [5] we studied certain spectralprojections: For the realization B = ( P + G ) T of an elliptic boundary valueproblem { P + + G, T } of order m > θ and ϕ , one can define the sectorial projection Π θ,ϕ ( B ). It is a (not necessarilyself-adjoint) projection whose range contains the generalized eigenspace of B for the sector Λ θ,ϕ = { re iω | r > , θ < ω < ϕ } and whose nullspacecontains the generalized eigenspace for Λ ϕ,θ +2 π . It was considered earlier byBurak [3], and in the boundary-less case by Wodzicki [14] and Ponge [12].In general this operator is not in Boutet de Monvel’s calculus, but weshowed that it has a residue in a slightly more general sense. The questionwas posed whether this residue vanishes.The question of the non-commutative residue of projections is particularlyinteresting in the context of zeta-invariants as discussed by Grubb [7] and in[5]: The basic zeta value C ,θ ( B ) for the realization B of a boundary valueproblem is defined via a choice of spectral cut in the complex plane; the ESIDUE OF PROJECTIONS IN BOUTET DE MONVEL’S CALCULUS 3 difference in the basic zeta value based on two spectral cut angles θ and ϕ is then given as the non-commutative residue of the corresponding sectorialprojection:(1.4) C ,θ ( B ) − C ,ϕ ( B ) = 2 πim res X (Π θ,ϕ ( B )) . Our results here show that the dependence of C ,θ ( B ) upon θ is trivialwhenever the projection Π θ,ϕ ( B ) lies in Boutet de Monvel’s calculus.It should be noted that the litterature in functional analysis and PDE-theory often uses “projection” as a synonym for idempotent, while C ∗ -algebraists furthermore require that projections are self-adjoint; we will tryto avoid confusion by explicitly using the term “ ψ do projection” for theidempotent operators here.2. Preliminaries and notation
We employ Blackadar’s [1] approach to K -theory: A pre- C ∗ -algebra B is called local if it, as a subalgebra of its C ∗ -completion B , is closed underholomorphic function calculus (and all of its matrix algebras must have thisproperty as well). Let M ∞ ( B ) denote the direct limit of the matrix algebras M m ( B ), m ∈ N . Define IP ∞ ( B ) = Idem( M ∞ ( B )) — resp. IP m ( B ) =Idem( M m ( B )) — to be the set of all — resp. all m × m — idempotentmatrices with entries from B . Define the relation ∼ on IP ∞ ( B ) by(2.1) x ∼ y if there exist a, b ∈ M ∞ ( B ) such that x = ab and y = ba. If B has a unit we define K ( B ) to be the Grothendieck group of the semi-group V ( B ) = IP ∞ ( B ) / ∼ . If B has no unit, we consider the scalar mapfrom the unitization — indicated with a tilde as in e B or B ∼ — of B to thecomplex numbers s : e B → C defined by s ( b + λ e B ) = λ , and then define K ( B ) as the kernel of the induced map s ∗ : K ( e B ) → K ( C ).A fact that we shall use several times is that if B is local, then [1, p. 28](2.2) V ( B ) ∼ = V ( B ) , and hence K ( B ) ∼ = K ( B ) . Combined with the standard picture of K this implies that(2.3) K ( B ) = { [ x ] − [ y ] | x, y ∈ IP m ( B ) , m ∈ N } in the case where B is unital, and(2.4) K ( B ) = { [ x ] − [ y ] | x, y ∈ IP m ( e B ) with x ≡ y mod M m ( B ) , m ∈ N } in the non-unital case [1]. ANDERS GAARDE
Let A denote the set of Green operators as in (1.1) of order and classzero; it defines a ∗ -subalgebra of the bounded operators on the Hilbert space H = L ( X, E ) ⊕ H − ( ∂X, F ); we will denote by A its C ∗ -closure in B ( H ). A is local with A = A , cf. Melo, Nest, and Schrohe [10], so K ( A ) ∼ = K ( A ).Note that the K -theory of A is independent of the specific bundles [10,Section 1.5], so for simplicity we study explicitly in this paper only thesimplest trivial case E = X × C and F = ∂X × C . K denotes the subalgebra of smoothing operators, K its C ∗ -closure (theideal of compact operators). We let I denote the set of elements in A ofthe form(2.5) ϕP ψ + G KT S ! with ϕ, ψ ∈ C ∞ c ( X ◦ ), P a ψ do on e X of order zero, and G, K, T , and S ofnegative order and class zero. I will be the C ∗ -closure of I in A .The noncommutative residue defined in [4] is a trace — a linear map thatvanishes on commutators — res : A → C , and therefore induces a grouphomomorphism res ∗ : K ( A ) → C such that(2.6) res ∗ ([ A ] ) = res X ( A )for any idempotent A ∈ A . Our goal is to prove the vanishing of res ∗ , whichobviously implies that res X ( A ) = 0 for any idempotent A .The quotient map q : A → A / K induces an isomorphism q ∗ : K ( A ) → K ( A / K ) [10]. The isomorphisms K ( A ) ∼ = K ( A ) ∼ = K ( A / K ) allow us toextend the noncommutative residue: For each [ A + K ] in K ( A / K ) there isan A ∈ IP ∞ ( A ) such that q ∗ [ A ] = [ A + K ] , and we then define(2.7) f res ∗ [ A + K ] = res ∗ [ A ] = res X ( A ) . The map f res ∗ is really just res ∗ ◦ q − ∗ , and is thus a group homomorphism K ( A / K ) → C . 3. K-theory and the residue
We employ results from Melo, Schick, and Schrohe [11], in particularthe fact that “each element in K ( A / K ) can be written as the sum of twoelements, one in the range of m ∗ and one in the range of s ′′ , thus in therange of i ∗ ” (bottom of page 11). In other words(3.1) K ( A / K ) = q ∗ m ∗ K ( C ( X )) + i ∗ K ( I / K ) . ESIDUE OF PROJECTIONS IN BOUTET DE MONVEL’S CALCULUS 5
Here m : C ( X ) → A sends f to the multiplication operator f
00 0 ! and i is the inclusion I / K → A / K ; m ∗ and i ∗ are then the corresponding inducedmaps in K . We will in general suppress i and i ∗ to simplify notation.We will show that f res ∗ vanishes on both terms in the right hand side of(3.1). The following lemma treats the first of these terms: Lemma 1. f res ∗ vanishes on q ∗ m ∗ K ( C ( X )) .Proof. Recall that a multiplication operator is in particular a Green operatorwhose noncommutative residue is zero.Let f ∈ IP m ( C ∞ ( X )); m ( f ) acts by multiplication with a smooth (ma-trix) function and therefore lies in IP m ( A ). Then q ∗ m ∗ [ f ] = q ∗ [ m ( f )] =[ m ( f ) + K ] , and according to (2.7)(3.2) f res ∗ ( q ∗ m ∗ [ f ] ) = res ∗ [ m ( f )] = res X ( m ( f )) = 0 . Since C ∞ ( X ) is local in C ( X ) [1, 3.1.1-2], any element of K ( C ( X )) can bewritten as [ f ] − [ g ] for some f, g ∈ IP m ( C ∞ ( X )), cf. (2.3). The lemmafollows from this. (cid:3) We now turn to the second term of (3.1); our strategy is to show thatthe elements of K ( I / K ) correspond to ψ dos with symbols supported in theinterior of X . This allows us to construct certain projections for which thenoncommutative residue is given as the residue of a projection on the closedmanifold e X .The principal symbol induces an isomorphism I / K ∼ = C ( S ∗ X ◦ ) [10, The-orem 1]. We will denote the induced isomorphism in K by σ ∗ , i.e.,(3.3) σ ∗ : K ( I / K ) ∼ = −→ K ( C ( S ∗ X ◦ )) . Like in Lemma 1 we wish to consider smooth functions instead of merelycontinuous functions; the following shows that instead of C ( S ∗ X ◦ ), it suf-fices to look at smooth functions (symbols) compactly supported in theinterior:The algebra C ∞ c ( S ∗ X ◦ ), equipped with the sup-norm, is a local C ∗ -algebra [1, 3.1.1-2] with completion C ( S ∗ X ◦ ). It follows from (2.2) thatthe injection C ∞ c ( S ∗ X ◦ ) → C ( S ∗ X ◦ ) induces an isomorphism(3.4) K ( C ∞ c ( S ∗ X ◦ )) ∼ = K ( C ( S ∗ X ◦ )) . We now show that each compactly supported symbol in K ( C ∞ c ( S ∗ X ◦ ))gives rise to a ψ do projection Π + on X which is in fact the truncation of a ANDERS GAARDE ψ do projection on e X . This will allow us to calculate the residue of Π + fromthe residue of a projection on the closed manifold e X . Lemma 2.
Let p ( x, ξ ) ∈ IP m ( C ∞ c ( S ∗ X ◦ ) ∼ ) . There is a zero-order ψ doprojection Π acting on C ∞ ( X, C m ) , such that its symbol is constant on e X \ X ,its truncation Π + is an idempotent in M m ( I ∼ ) , and (3.5) σ ∗ q ∗ ([Π + ] ) = [ p ] . Proof.
By definition of the unitization of C ∞ c ( S ∗ X ◦ ), we can write p as asum(3.6) p ( x, ξ ) = α ( x, ξ ) + β, with α ∈ M m ( C ∞ c ( S ∗ X ◦ )) and β ∈ M m ( C ). Note that β itself is idempo-tent, since p = β outside the support of α .We extend α by zero to obtain a smooth function on the closed manifold S ∗ e X denoted e α ( x, ξ ). We get a ψ do symbol (also denoted e α ( x, ξ )) of orderzero on e X by requiring e α to be homogeneous of degree zero in ξ . Let e p ( x, ξ ) = e α ( x, ξ ) + β .We now have an idempotent ψ do-symbol e p on e X ; we then construct a ψ do projection on e X that has e p as its principal symbol.In [7, Chapter 3], Grubb constructed an operator that, for a suitablechoice of atlas on the manifold, carries over to the Euclidean Laplacianin each chart, modulo smoothing operators. Hence, choose that particularatlas on e X and let D denote this particular operator, i.e., with scalar symbol d ( x, ξ ) = | ξ | . Define the auxiliary second order ψ do C = OP( c ( x, ξ )), withsymbol c ( x, ξ ) given in the local coordinates of the specified charts as(3.7) c ( x, ξ ) = (2 e p ( x, ξ ) − I ) d ( x, ξ ) . Since e p is idempotent, the eigenvalues of 2 e p − I are ±
1, cf. (A.2), so C is anelliptic second order operator and c ( x, ξ ) − λ is parameter-elliptic for λ oneach ray in C \ R .Then we can define the sectorial projection, cf. [12], [5], Π = Π θ,ϕ ( C ) withangles θ = − π , ϕ = π ,(3.8) Π = i π Z Γ θ,ϕ λ − C ( C − λ ) − dλ. Π is a ψ do projection [12] on e X with symbol π given in local coordinates by(3.9) π ( x, ξ ) = i π Z C ( x,ξ ) q ( x, ξ, λ ) dλ, ESIDUE OF PROJECTIONS IN BOUTET DE MONVEL’S CALCULUS 7 where q ( x, ξ, λ ) is the symbol with parameter for a parametrix of c ( x, ξ ) − λ ,and C ( x, ξ ) is a closed curve encircling the eigenvalues of c ( x, ξ ) — theprincipal symbol of C — in the { Re z > } half-plane.The eigenvalues of c ( x, ξ ) = (2 e p ( x, ξ ) − I ) | ξ | are ±| ξ | , so we can choose C ( x, ξ ) as the boundary of a small ball B ( | ξ | , r ) around + | ξ | .Hence, the principal symbol of π ( x, ξ ) is π ( x, ξ ) = i π Z C ( x,ξ ) q − ( x, ξ, λ ) dλ = i π Z ∂B ( | ξ | ,r ) [(2 e p ( x, ξ ) − I ) | ξ | − λ ] − dλ = e p ( x, ξ ) , (3.10)according to Lemma 4. So Π is a ψ do projection with principal symbol e p ( x, ξ ), as desired.Observe that for x outside the support of e α , we have c ( x, ξ ) = (2 β − I ) | ξ | and q ( x, ξ, λ ) = q − ( x, ξ, λ ) = ((2 β − I ) | ξ | − λ ) − so π ( x, ξ ) = π ( x, ξ ) = β there. (We cannot be sure that the full symbol of π equals e p insidethe support, since coordinate-dependence will in general influence the lowerorder terms of the parametrix.) In particular, π ( x, ξ ) is constant equal to β for x ∈ e X \ X .Now consider the truncation Π + . We have(3.11) (Π + ) = (Π ) + − L (Π , Π) = Π + − L (Π , Π) , where the singular Green operator L ( P, Q ) is defined as (
P Q ) + − P + Q + for ψ dos P and Q . Since π ( x, ξ ) equals the constant matrix β in a neighborhoodof the boundary ∂X it follows, cf. [6, Theorem 2.7.5], that L (Π , Π) = 0, so(Π + ) = Π + .Since the symbol of Π − β is compactly supported within X ◦ , we canwrite Π + = ϕP ψ + β for some ϕ, ψ, P , as in (2.5); hence Π + is in M m ( I ∼ ).Technically, Π + lies in the algebra where the boundary bundle F is the zero-bundle, but inserting zeros into Π + ’s matrix form will clearly allow us toaugment it to the present case with F = ∂X × C .Finally we take a look at (3.5): Since Π + is an idempotent in M m ( I ∼ )it defines a K -class [Π + ] in K ( I ∼ ). Then q ∗ [Π + ] defines a class in K ( I / K ∼ ), a class defined solely by its principal symbol. Since the principalsymbol is exactly the idempotent p ( x, ξ ) we obtain (3.5) by definition. (cid:3) Theorem 3.
The noncommutative residue of any projection in (the normclosure of ) the Boutet de Monvel calculus is zero.
ANDERS GAARDE
Proof.
As mentioned, it suffices to show that res ∗ vanishes on K ( A ) ∼ = K ( A ). In turn, according to equation (3.1) and Lemma 1, we only need toshow that f res ∗ vanishes on K ( I / K ).So let ω ∈ K ( I / K ). Employing (2.4), (3.3), and (3.4) we can find p, p ′ in IP m ( C ∞ c ( S ∗ X ◦ ) ∼ ) such that(3.12) σ ∗ ω = [ p ] − [ p ′ ] . Now, for p , p ′ we use Lemma 2 to find corresponding ψ dos Π, Π ′ with thespecific properties mentioned there. By (3.5) and (3.12) we see that(3.13) q ∗ [Π + ] − q ∗ [Π ′ + ] = σ − ∗ (cid:0) [ p ] − [ p ′ ] (cid:1) = ω. Using equation (2.7) we now see that(3.14) f res ∗ ω = res X (Π + ) − res X (Π ′ + ) . Here(3.15) res X (Π + ) = Z X Z S ∗ x X tr π − n ( x, ξ ) − dS ( ξ ) dx. By construction, π ( x, ξ ) is constant equal to β outside X ; in particular π − n ( x, ξ ) is zero for x ∈ e X \ X and therefore(3.16) Z X Z S ∗ x X tr π − n ( x, ξ ) − dS ( ξ ) dx = Z e X Z S ∗ x e X tr π − n ( x, ξ ) − dS ( ξ ) dx. In other words(3.17) res X (Π + ) = res e X (Π) , where the latter is the noncommutative residue of a ψ do projection on aclosed manifold. It is well-known [14], [15] that the latter always vanishes.Likewise we obtain res X (Π ′ + ) = 0 and finally(3.18) f res ∗ ω = 0as desired. (cid:3) In [5], it was an open question whether the residue is zero on a sectorialprojection for a boundary value problem. This theorem answers that ques-tion in the positive for the cases where the sectorial projection lies in the C ∗ -closure of A .It is not, at this time, clear for which boundary value problems this istrue. We showed in [5] that there certainly are boundary value problemswhere the sectorial projection is not in A ; whether or not they lie in A issomething we intend to return to in a future work. ESIDUE OF PROJECTIONS IN BOUTET DE MONVEL’S CALCULUS 9 A. Appendix
Lemma 4.
Let M ∈ IP m ( C ) . Let d > and let ∂B ( d, r ) denote the closedcurve in the complex plane along the boundary of the ball with center d andradius < r < d . Then (A.1) i π Z ∂B ( d,r ) [(2 M − I ) d − λ ] − dλ = M. Proof.
A direct computation shows that, for λ = ± d ,(A.2) [(2 M − I ) d − λ ] − = Md − λ − I − Md + λ . The result in (A.1) then follows from the residue theorem. (cid:3)
Acknowledgements
The author is grateful to Gerd Grubb and Ryszard Nest for several helpfuldiscussions.
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