exhausting families of representations and spectra of ... filemotivationsfaithfull families full and...

Motivations Faithfull families Full and exhausting families Unbounded operators Parametric pseudodifferential operators

Exhausting families of representations and spectraof pseudodifferential operators

Nicolas Prudhon, joint work with Victor Nistor

June 30, 2015

First motivation : [Damak-Georgescu, Georgescu-Iftimovici,Georgescu-Nistor, Simon]In spectral theory of N-body Hamiltonians one associates to theLaplacian H a family of other operators Hφ, φ ∈ F , such that theessential spectrum Specess(H) of H is obtained in terms of theusual spectra Spec(Hφ) of Hφ as

Specess(H) = ∪φ∈F Spec(Hφ) . (1)

Sometimes the closure is not necessary.

Second motivation : [Georgescu-Nistor]Fredholm conditions for (pseudo)differential operators.M manifold, D differential operators on M of ordrer m.A procedure associates to (M,D) the following data:

(i) spaces Zα, α ∈ I (independent of D);

(ii) groups Gα, α ∈ I (independent of D); and

(iii) Gα-invariant differential operator Dα acting on Zα × Gα.

such that

D : Hs(M)→ Hs−m(M) is Fredholm⇔

D is elliptic and Dα is invertible for all α ∈ I .

If M is compact (without boundary), I is empty.If M is non-compact, the conditions on Dα are necessary.

Let A a C ∗-algebra and A its primitive ideal spectrum.

Definition

Let F be a set of representations of A. We say that the family Fis faithful if ρ := ⊕φ∈F φ : A→ ⊕φL(Hφ) is injective.

Proposition

Let F a family of representations of a C ∗-algebra A. The followingare equivalent :

F is faithful

∪φ∈F suppφ is dense in A

‖a‖ = supφ∈F ‖φ(a)‖ for all a ∈ A

The following property holds (A unital): a is invertible ⇔ φ(a)is invertible for all φ ∈ F and the set {‖φ(a)−1‖} is bounded.

Spec(a) = ∪φ∈F Spec(φ(a)) for any normal a ∈ A .

Definition

Proposition

F is faithful

Definition

Proposition

F is faithful

Definition

Proposition

F is faithful

Definition

Proposition

F is faithful

Invertiblity and spectra

We introduce now a more restrictive class of families ofrepresentations of A.

Definition

Let F be a set of representations of the C ∗-algebra A.

(i) We shall say that F is full if A = ∪φ∈F supp(φ).

(ii) We shall say that F is exhausting if, for any a ∈ A, thereexists φ ∈ F such that ‖φ(a)‖ = ‖a‖.

A family F is exhausting if, and only if, for any a ∈ A,

‖a‖ = maxφ∈F

‖φ(a)‖ .

Example

The set of all irreducible representations of a C ∗-algebra isexhausting.

Proposition

F full ⇒ F exhausting ⇒ F faithful.

Example

The set of all irreducible representations of a C ∗-algebra isexhausting.

Proposition

F full ⇒ F exhausting ⇒ F faithful.

Theorem

Let F be a set of representations of a unital C ∗-algebra A. Thefollowing are equivalent:

(i) F is exhausting.

(ii) An element a ∈ A is invertible if, and only if, φ(a) isinvertible in L(Hφ) for all φ ∈ F .

Theorem

Let F be a family of representations of a unital C ∗-algebra A.Then F is exhausting if, and only if,

Spec(a) = ∪φ∈F Spec(φ(a)) . (2)

for any a ∈ A.

Theorem

Let F be a set of representations of a unital C ∗-algebra A. Thefollowing are equivalent:

(i) F is exhausting.

(ii) An element a ∈ A is invertible if, and only if, φ(a) isinvertible in L(Hφ) for all φ ∈ F .

Theorem

Let F be a family of representations of a unital C ∗-algebra A.Then F is exhausting if, and only if,

Spec(a) = ∪φ∈F Spec(φ(a)) . (2)

for any a ∈ A.

Relation between full and exhausting families

Theorem

Let A be a C ∗-algebra. Let us assume that every π ∈ A has acountable base for its system of neighbourhoods. Then everyexhausting family F of representations of A is also full.Conversely, if every exhausting family F of representations of A isalso full, then every π ∈ A has a countable base for its system ofneighbourhoods.

Separable C ∗-algebras satisfy the assumptions of the previousproposition.However, there exist non separable C ∗-algebras whose spectrumhas a countable base of neighbourhoods.

Theorem

Separable C ∗-algebras satisfy the assumptions of the previousproposition.

However, there exist non separable C ∗-algebras whose spectrumhas a countable base of neighbourhoods.

Theorem

Separable C ∗-algebras satisfy the assumptions of the previousproposition.However, there exist non separable C ∗-algebras whose spectrumhas a countable base of neighbourhoods.

Abstract affiliated operators

Definition

Let A be a C ∗-algebra. An observable T affiliated to A is amorphism θT : C0(R)→ A of C ∗-algebras.

Definition

Let θT : C0(R)→ A be a self-adjoint operator affiliated to aC ∗-algebra A. The kernel of θT is of the form C0(U), for someopen subset of R. We define the spectrum SpecA(T ) as thecomplement of U in R. Explicitly,

SpecA(T ) = {λ ∈ R, h(λ) = 0, ∀h ∈ C0(R) such that θT (h) = 0 } .

Definition

Let A be a C ∗-algebra. An observable T affiliated to A is amorphism θT : C0(R)→ A of C ∗-algebras.

Definition

Let θT : C0(R)→ A be a self-adjoint operator affiliated to aC ∗-algebra A. The kernel of θT is of the form C0(U), for someopen subset of R. We define the spectrum SpecA(T ) as thecomplement of U in R. Explicitly,

SpecA(T ) = {λ ∈ R, h(λ) = 0, ∀h ∈ C0(R) such that θT (h) = 0 } .

Theorem

Let A be a unital C ∗-algebra and T an observable affiliated to A.Let F be a set of representations of A.

1 If F is exhausting, then

Spec(T ) = ∪φ∈F Spec(φ(T )) .

2 If F is faithful, then

Spec(T ) = ∪φ∈F Spec(φ(T )) .

The case of true operators

Definition

Let A ⊂ L(H) be a sub-C ∗-algebra of L(H). A (possiblyunbounded) self-adjoint operator T : D(T ) ⊂ H → H is said to beaffiliated to A if, for every continuous functions h on the spectrumof T vanishing at infinity, we have h(T ) ∈ A.

Remark

A self-adjoint operator T affiliated to A defines a morphismθT : C0(R)→ A, θT (h) := h(T ) such that Spec(T ) = Spec(θT ).Thus T defines an observable affiliated to A.

Definition

Let A ⊂ L(H) be a sub-C ∗-algebra of L(H). A (possiblyunbounded) self-adjoint operator T : D(T ) ⊂ H → H is said to beaffiliated to A if, for every continuous functions h on the spectrumof T vanishing at infinity, we have h(T ) ∈ A.

Remark

A self-adjoint operator T affiliated to A defines a morphismθT : C0(R)→ A, θT (h) := h(T ) such that Spec(T ) = Spec(θT ).Thus T defines an observable affiliated to A.

Recall that an unbounded operator T is invertible if, and only if, itis bijective and T−1 is bounded. This is also equivalent to0 /∈ Spec(θT ).

Theorem

Let A ⊂ L(H) be a unital C ∗-algebra and T a self-adjoint operatoraffiliated to A. Let F be a set of representations of A.

1 Let F be exhausting. Then T is invertible if, and only if φ(T )is invertible for all φ ∈ F .

2 Let F be faithful. Then T is invertible if, and only if φ(T ) isinvertible for all φ ∈ F and the set {‖φ(T )−1‖, φ ∈ F} isbounded.

The algebra of invariant pseudodifferential operators

M be a compact smooth Riemannian manifold

G be a Lie group with Lie algebra g := Lie(G ).

Ψ0(M × G )G the algebra of order 0, G -invariantpseudodifferential operators on M × G

Definition

A = Ψ0(M × G )G be the norm closure of Ψ0(M × G )G acting onL2(M × G ).

Definition

Notations

For any vector bundle E , we denote by S∗E the set of directions inits dual E ∗. If E is endowed with a metric, then S∗E can beidentified with the set of unit vectors in E ∗.

The symbols of operators in A are continuous functions on thequotient space

S∗(T (M × G ))/G = S∗(TM × TG )/G = S∗(TM × g) .

Proposition

The symbol mapA→ C(S∗(TM × g))

has kernel isomorphic to C ∗r (G )⊗K and we obtain the exactsequence

0 → C ∗r (G )⊗K → Ψ0(M × G )G → C(S∗(M × g)) → 0 ,

Note that the kernel of the symbol map will now have irreduciblerepresentations parametrised by Gr the temperate unitaryirreducible representations of G .

Notations

Proposition

0 → C ∗r (G )⊗K → Ψ0(M × G )G → C(S∗(M × g)) → 0 ,

Notations

Proposition

0 → C ∗r (G )⊗K → Ψ0(M × G )G → C(S∗(M × g)) → 0 ,

Let T ∈ Ψm(M × G )G .

Definition

T ] ∈ Ψm(M × G )G is the formal adjoint (defined using thecalculus of pseudodifferential operators).

T ∗ the Hilbert space adjoint of a (possibly unbouded) denselydefined operator.

All operators considered above are closed with minimal domain(the closure of the operators defined on C∞c (M × G )).

Let T ∈ Ψm(M × G )G be elliptic. Then T ∗ = T ]. Thus, if alsoT = T ], then T is self-adjoint and (T + ı)−1 ∈ C ∗r (G ), and henceit is affiliated to C ∗r (G ).

In other words, any elliptic, formally self-adjointT ∈ Ψm(M × G )G , m > 0, is actually self-adjoint.

The case G = Rn

Let us assume G = Rn, regarded as an abelian Lie group. Theexact sequence becomes

0→ C0(Rn)⊗K → Ψ0(M × Rn)Rn → C(S∗(TM × Rn))→ 0 .

Let S∗M := S∗(TM) ⊂ S∗(TM × Rn) correspond toT ∗M ⊂ T ∗M ×Rn. Then the closure of {φλ} in A is {φλ} ∪ S∗M.

It is customary to denote by

T (λ) := φλ(T )

for T a pseudodifferential operator in Ψ0(M × Rn)Rn.

The case G = Rn

T (λ) := φλ(T )

The case G = Rn

T (λ) := φλ(T )

The case G = Rn

Define T (λ) for m > 0.

We can either use

the Fourier transform or,

notice that the Laplacian ∆ is affiliated to the closure ofΨ0(M × Rn)R

n. This allows us to define ∆(λ). In general, we

write T = (1 + ∆)kS , with S ∈ Ψ0(M × Rn)Rn

and define

T (λ) = (1 + ∆)(λ)k S(λ).

The case G = Rn

Define T (λ) for m > 0. We can either use

the Fourier transform

and define

T (λ) = (1 + ∆)(λ)k S(λ).

The case G = Rn

n. This allows us to define ∆(λ).

In general, wewrite T = (1 + ∆)kS , with S ∈ Ψ0(M × Rn)R

nand define

T (λ) = (1 + ∆)(λ)k S(λ).

The case G = Rn

and define

T (λ) = (1 + ∆)(λ)k S(λ).

The case G = Rn

Proposition

Let F := {φλ, λ ∈ Rn} ∪ {ep, p ∈ S∗(TM × Rn) r S∗M}.(i) The family F is an exhausting family of representations of

Ψ0(M × Rn)Rn .

(ii) Let P ∈ Ψm(M × Rn)Rn, then

P : Hs(M ×Rn)→ Hs−m(M ×Rn) is invertible if, and only ifP(λ) : Hs(M)→ Hs−m(M) is invertible for all λ ∈ Rn andthe principal symbol of P is non-zero on all rays notintersecting S∗M.

(iii) If T ∈ Ψm(M × Rn)Rn, m > 0, is formally self-adjoint and

elliptic, then Spec(ep(T )) = ∅, and hence

Spec(T ) = ∪λ∈Rn Spec(T (λ)) .

The case G = Rn

Thank you !

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