CUBO A Mathematical Journal
Vol.11, No¯ 05, (51–56). December 2009

K-Theory of an Algebra of Pseudodifferential Operators

on a Noncompact Manifold

Patrícia Hess and Severino T. Melo

Instituto de Matemática e Estatística,

Universidade de São Paulo,

Rua do Matão 1010, 05508-090 São Paulo, Brazil

emails: phess@ime.usp.br,

toscano@ime.usp.br

ABSTRACT

Let A denote the C*-algebra of bounded operators on L
2(R × S1) generated by: all multiplica-

tions a(M ) by functions a ∈ C∞(S1), all multiplications b(M ) by functions b ∈ C([−∞, +∞]),
all multiplications by 2π-periodic continuous functions, Λ = (1 − ∆R×S1 )

−1/2
, where ∆R×S1 is

the Laplacian operator on L
2(R × S1), and ∂tΛ, ∂xΛ, for t ∈ R and x ∈ S

1
. We compute the

K-theory of A and of its quotient by the ideal of compact operators.

RESUMEN

Denotemos A la C*-algebra de operadores acotados sobre L2(R × S1) gerados por todas las
multiplicaciones a(M ) por funciones a ∈ C∞(S1), todas las multiplicaciones b(M ) por fun-
ciones b ∈ C([−∞, +∞]), todas las multiplicaciones por funciones continuas 2π-periódicas,
Λ = (1 − ∆R×S1 )

−1/2
, donde ∆R×S1 es el operador de Laplace sobre L

2(R × S1), y ∂tΛ, ∂xΛ,
para t ∈ R y x ∈ S

1
. Nosotros calculamos la K-teoria de A y su cuociente por el ideal de los

operadores compactos.

Key words and phrases: K-Theory, pseudodifferential operators.

Math. Subj. Class.: 46L80, 47G30.



52 P. Hess and S.T. Melo CUBO
11, 5 (2009)

1 Introduction

Let B denote an n-dimensiomal compact Riemannian manifold. Write Ω = R×B1 and let ∆Ω

denote its Laplacian. Define A as the C*-algebra of bounded operators on L2(Ω) generated by:

(i) all a(Mx), operators of multiplication by a ∈ C
∞

(B), [a(Mx)f ](t, x) = a(x)f (t, x) ;

(ii) all b(Mt), operators of multiplication by b ∈ C([−∞, +∞]), [b(Mt)f ](t, x) = b(t)f (t, x) ;

(iii) every multiplication by 2π-periodic continuous functions;

(iv) Λ = (1 − ∆Ω)
−1/2

;

(v) 1
i

∂
∂t

Λ, t ∈ R;

(vi) LΛ, where L ia a first order differential operator on B.

A contains a class of classical pseudodifferential operators, including all A = LΛN , where

L is a differential operator of order N with coefficients approaching periodic functins at infinity.

A : L2(Ω) → L2(Ω) is Fredholm if and only if so is L : HN (Ω) → L2(Ω) and they have the same

index.

The structure of A is given in [4], so we will mention some of those results. The principal

symbol extends to a ∗-homomorphism σ : A → Cb(S
∗
Ω), where Cb(S

∗
Ω) denotes the algebra of

continuous funtions on the co-sphere bundle of Ω. Let E denote the kernel of σ and KΩ := K(L
2
(Ω))

the ideal of compact operators on L2(Ω).

Theorem 1. There is a ∗-isomorphism

Ψ :
E

KΩ
−→ C(S1 ×{−1, +1},KZ×B). (1)

In fact, E is the commutator ideal of A. Composing the canonical projection E → E/KΩ

with Ψ, we can extend the map E → C(S1 × {−1, +1},KZ×B) and obtain a ∗-homomorphism

γ : A→ C(S1 ×{−1, +1},L(L2(Z × B))).

From now on, we will just work in the case that the manifold B is the circle S1. Therefore, the

class of generators LΛ can be replaced by the operator 1
i

∂
∂β

Λ, β ∈ S1, since S1 has trivial tangent

bundle.

Definition 2. Let D be a C*-algebra and I its commutator ideal. We call symbol space of D the

compact space M such that we have D/I ∼= C(M ) by the Gelfand map.

With this definition, we observe that the map σ is the composition A → A/E → C(MA),

where MA is the symbol space of A. Given A ∈A, we say that the symbol of A is the continuous

function σA ∈ C(MA) associated to the class A in A/E by the Gelfand isomorphism.



CUBO
11, 5 (2009)

K-Theory of an Algebra of Pseudodifferential Operators ... 53

Theorem 3. The symbol space of A is the set

MA = {(t, x, (τ, ξ), e
iθ

); t ∈ [−∞, +∞], x ∈ S1, (τ, ξ) ∈ R2 : τ 2 + ξ2 = 1,

θ ∈ R e θ = t se |t| < ∞},

and the symbols of the generators (or classes of them) of A are given below as functions on

(t, x, (τ, ξ), eiθ ) in the same order that they appear in the definition:

a(x), b(t), eijθ, 0, τ, ξ. (2)

For each ϕ ∈ R, let Uϕ be the operator on L
2
(S1) given by Uϕf (z) = z

−ϕf (z), z ∈ S1, and

let Yϕ := FdUϕF
−1
d , where Fd : L

2
(S1) → ℓ2(Z) is the discrete Fourier transform. So, Yϕ is an

unitary operator such that, for all k ∈ Z, (Yku)j = uj+k, and YϕYω = Yϕ+ω.

Consider the following functions defined on R and taking values in the algebra of bounded

operators on L2(S1):

B4(τ ) = Λ̃(τ ) , B5(τ ) = −τ Λ̃(τ ) , B6(τ ) =
1

i

∂

∂β
Λ̃(τ ), τ ∈ R, β ∈ S1, (3)

where Λ̃(τ ) = (1 + τ 2 − ∆S1 )
−1/2.

Proposition 4. For each generator A of A, γA(e
2πiϕ,±1) is given respectively by:

a(Mx) , b(±∞) , Y−j , YϕBi(ϕ − Mj)Y−ϕ , i = 4, 5, 6,

where Bi(ϕ−Mj ) ∈L(ℓ
2
(Z, L2(S1))), i = 4, 5, 6, are the operators of multiplicaion by the sequences

Bi(ϕ − j) ∈ L
2
(S

1
), and Bi’s are given in (3).

In this note, we announce results about the calculus of the K-theory of A. Their proofs will

appear in a forthcoming paper.

2 C*-subalgebras of A

Now, we define two C*-subalgebras of A and mention some facts about their structure. With

the results mentioned here, we can compute their K-theory in the next section.

Let A† the C*-algebra of bounded operators on L2(Ω) generated by the class of operators of

multiplication by a ∈ C∞(S1), Λ, 1
i
∂tΛ and

1
i
∂β Λ, and let A

⋄ denote the C*-algebra generated

by the same operators of A† and operators of multiplication by 2π-periodic continuous functions.

Based on Cordes, [1], and Melo, [4], we can prove the following result.

Theorem 5. Let E† and E⋄ denote the commutator ideals of A† and A⋄, respectively. Then

E† ∼= C0(R,K(L
2
(S

1
))) and E⋄ ∼= C(S1,K(L2(Z × S1))).



54 P. Hess and S.T. Melo CUBO
11, 5 (2009)

The ∗-isomorphism E⋄ ∼= C(S1,K(L2(Z × S1))) can be extended to a ∗-homomorphism γ′ :

A⋄ → C(S1,L(L2(Z × S1))) given by γ′A(z) = γA(z,±1) for every A ∈A
⋄ ⊂A.

Knowing these isomorphisms, we are able to describe the symbol spaces of both C*-subalgebras

of A.

Theorem 6. The symbol space of A† is homeomorphic to S1 × S1 and the symbol space of A⋄ is

homeomorphic to S1 × S1 × S1. Moreover, the symbol of an operator in A† or A⋄ coincides with

the symbol of same operator regarded as an element of A.

We can see A† as a C*-subalgebra of L(L2(R × S1)) as well. So, if we conjugate A† with

F ⊗ IS1 , where F : L
2
(R) → L2(R) is the Fourier transform and IS1 is the identity on L

2
(S

1
), we

obtain an algebra that can be viewed as a subalgebra of Cb(R,L(L
2
(S

1
))). Let B† be this algebra.

The idea is to define an action, and then, to show that that A⋄ has a crossed product structure.

Consider α the following translation-by-one automorphism:

[α(B)](τ ) = B(τ − 1), τ ∈ R, L ∈B†.

Let B⋄ the algebra A⋄ conjugated with the Fourier transform. Analogously to [5, Theorem 8],

we obtain the next result.

Theorem 7. Let α be as described above. We have:

B⋄ ∼= B
†
⋊αZ

where B† ⋊α Z is the envelopping C*-algebra([2], 2.7.7) of the Banach algebra with involution

ℓ1(Z,B†) of all summable Z-sequences in B†, equipped with the product

(A · B)(n) =
∑

k∈Z

Akα
k
(Bn−k), n ∈ Z, A = (Ak)k∈Z, B = (Bk)k∈Z,

and involution A∗(n) = αn(A∗
−n).

3 K-Theory

In this section, we compute the K-groups of the algebras that we were defined in this work.

The main idea is to compute the connecting maps in the K-theory six-term exact sequence

associated to the C*-algebra short exact sequence induced by the principal symbol

0 −→ E = kerσ
i
−→ A

π
−→

A

E
∼= Imσ −→ 0, (4)

where i is the inclusion and π is the canonical projection.

But before we begin the computation involving the largest algebra, we start with A† and

A⋄using the same idea as above.



CUBO
11, 5 (2009)

K-Theory of an Algebra of Pseudodifferential Operators ... 55

Studying the K-theory six-term exact sequence induced by the sequence

0 −→ E†
i
−→ A†

π
−→

A†

E†
−→ 0,

we just obtain partial results about the K-theory of A†, because we do not have a explicit description

of the range of the exponential map.

The same thing happens with the exponential map in the sequence involving A⋄. To compute

the index map δ⋄1 : K1(A
⋄/E⋄) → K0(E

⋄
), it was necessary to use the Fedosov index formula [3].

Theorem 8. δ⋄1 is surjective.

From Theorem 7, we have B⋄ ∼= B†⋊αZ. Then the K-groups of B
† and B⋄ fit in the Pimsner-

Voiculescu exact sequence, [6, theorem 2.4]:

K0(B
†
)

id∗−α
−1

∗

−→ K0(B
†
)

ϕ∗◦i∗
−→ K0(B

⋄
)

↑ ↓

K1(B
⋄
)

ϕ∗◦i∗
←− K1(B

†
)

id∗−α
−1

∗

←− K1(B
†
)

(5)

Now, we have enough information to conclude that

K0(A
†
) ∼= Z , K1(A

†
) ∼= Z

2 , Ki(A
⋄
) ∼= Z

3 , i = 0, 1.

The next step is to work with A. In the begining of this section the sequence induced by the

symbol was mentioned. We will rewrite it, quotiented by the ideal of compact operators KΩ :

0 −→
E

KΩ

i
−→

A

KΩ

π
−→

A

E
−→ 0, (6)

where i is the inclusion and π is the canonical projection.

By the isomorphism (1), we know that E/KΩ can be viewed as two copies of C(S
1,K(L2(Z ×

S
1
))). Then,

Ki

(
E

KΩ

)
∼= Z ⊕ Z, i = 0, 1.

We know the range of σ by Theorem 3. With the help of [5, Proposition 3], we can state the

next result.

Proposition 9. K0(A/E) ∼= K1(A/E) ∼= Z
6.

From sequence (6), we have:

Z
2 ∼= K0(E/KΩ)

i∗
−→ K0(A/KΩ)

π∗
−→ K0(A/E) ∼= Z

6

δ1 ↑ ↓ δ0

Z
6 ∼= K1(A/E)

π∗
←− K1(A/KΩ)

i∗
←− K1(E/KΩ) ∼= Z

2

(7)



56 P. Hess and S.T. Melo CUBO
11, 5 (2009)

The image of δ1 is determined the same way as the image of δ
⋄
1 .

Theorem 10. δ1 is surjective.

Corollary 11. K0(A/KΩ) has no torsion.

Theorem 12. Given the short exact sequence

0 −→KΩ
i
−→ A

π
−→

A

KΩ
−→ 0,

where i is the inclusion and π is the canonical projection, the following statements hold:

(i) δ1 : K1(
A

KΩ
) → Z is surjective (there exists a matrix of operators in A with Fredholm index

equal to one);

(ii) K0(A) ∼= K0(A/KΩ) ∼= Z
5
;

(iii) K1(A) ∼= Z
4 and K1(A/KΩ) ∼= Z

5.

Received: December, 2008. Revised: April, 2009.

References

[1] Cordes, H.O., On the two-fold symbol chain of a c*-algebra of singular integral operators

on a polycylinder, Revista Matemática Iberoamericana, 2(1 y 2):215–234, 1986.

[2] Dixmier, J., C*-algebras, North Holland, New York, 1977.

[3] Fedosov, B.V., Index of an elliptic system on a manifold, Funct. Anal. and its Application,

4(4):312–320, 1970.

[4] Melo, S.T., A comparison algebra on a cylinder with semi-periodic multiplications, Pacific J.

Math., 142(2):281–304, 1990.

[5] Melo, S.T. and Silva, C.C., K-theory of pseudodifferential operators with semi-periodic

symbols, K-theory, 37:235–248, 2006.

[6] Pimsner, M. and Voiculescu, D., Exact sequences for k-groups and ext-groups of certain

cross-product c*-algebras, J. Operator Theory, 4(1):93–118, 1980.


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