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Analysis on Vector Bundles over Noncommutative Tori

Thesis by

Jim Tao

In Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy in Mathematics

CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California

2019

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© 2019 Jim Tao

ORCID: 0000-0002-0751-9273

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ACKNOWLEDGEMENTS

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ABSTRACT

Noncommutative geometry is the study of noncommutative algebras, especially C∗-algebras, and their geometric interpretation as topological spaces. One C∗ -algebra particularly important in physics is the noncommutative n-torus, the irra-tional rotation C∗-algebra AΘ with n unitary generators U1, . . . ,Un which satisfy

UkUj = e2πiθj,kUjUk and U∗j = U−1j , where Θ ∈ Mn(R) is skew-symmetric with

upper triangular entries that are irrational and linearly independent over Q. We

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PUBLISHED CONTENT AND CONTRIBUTIONS

(a) Farzad Fathizadeh, Franz Luef, and Jim Tao. A twisted local index formula for curved noncommutative two tori. arXiv e-prints, art. arXiv:1904.03810, Apr 2019.

J. T. contributed new rearrangement lemmas, proven using techniques from analytic combinatorics, and Mathematica code that used Mathematica’s sym-bolic pattern language to automate lengthy, complicated index theory calcu-lations.

(b) Jim Tao. The theory of pseudo-differential operators on the noncommutative n-torus. InJournal of Physics Conference Series, volume 965 ofJournal of Physics Conference Series, page 012042, February 2018. doi: 10.1088/1742-6596/965/1/012042.

J. T. participated in proving the results and writing this paper.

(c) Jim Tao. The theory of pseudo-differential operators on the noncommutative n-torus. arXiv e-prints, art. arXiv:1704.02507, Apr 2017.

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TABLE OF CONTENTS

Copyright page . . . ii

Acknowledgements . . . iii

Abstract . . . iv

Published Content and Contributions . . . v

Table of Contents . . . vi

Chapter I: The Pseudo-differential Calculus on Noncommutative Tori . . . 1

1.1 Introduction . . . 1

1.2 Preliminaries . . . 3

1.3 Asymptotic formula for the symbol of the adjoint of a pseudo-differential operator . . . 7

1.4 Asymptotic formula for the symbol of a product of two pseudo-differential operators . . . 10

1.5 Sobolev spaces on the noncommutativentorus . . . 14

1.6 The pseudo-differential calculus on f.g. projective modules over the noncommutativentorus . . . 21

Chapter II: A Twisted Local Index Formula for Curved Noncommutative Two Tori . . . 31

2.1 Introduction . . . 31

2.2 Preliminaries . . . 33

2.3 Noncommutative geometric spaces and index theory . . . 38

2.4 Calculation of a noncommutative local formula for the index . . . 53

Chapter III: Summary Conclusion . . . 62

Appendix A: Rearrangement Lemmas . . . 63

A.1 Proof of Lemma 2.4.1 and Corollary 2.4.2 . . . 63

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C h a p t e r 1

THE PSEUDO-DIFFERENTIAL CALCULUS ON

NONCOMMUTATIVE TORI

Published under licence in Journal of Physics: Conference Series by IOP Publishing Ltd.

Available on arXiv underhttp://arxiv.org/licenses/nonexclusive-distrib/1.0/

1.1 Introduction

The methods of spectral geometry are useful for investigating the metric aspects of noncommutative geometry [11, 14, 16, 20] and in these contexts require extensive use of pseudo-differential operators. In the foundational paper [13], Connes showed that, by direct analogy with the theory [38, 61, 69] of pseudo-differential operators onRn, one may derive a similar pseudo-differential calculus on noncommutativen

tori Tnθ. With the development of this calculus came many results concerning the local differential geometry of noncommutative tori for n = 2,4, as shown in the groundbreaking paper [23] in which the Gauss–Bonnet theorem onT2θis proved and

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of the state implementing the conformal perturbation of the metric.

Certain details of the proofs in the derivation of the calculus in [13] were omitted, such as the evaluation of oscillatory integrals, so we make it the objective of this paper to fill in all the details. After reproving in more detail the formula for the symbol of the adjoint of a pseudo-differential operator and the formula for the symbol of a product of two pseudo-differential operators, we define the corresponding analog of Sobolev spaces for which we prove the Sobolev and Rellich lemmas. We then extend these results to finitely generated projective right modules over the noncommutative ntorus.

We list these results below.

Theorem 1.1.1. Suppose P is a pseudo-differential operator with symbolσ(P) =

ρ= ρ(ξ)of orderM. Then the symbol of the adjoint P∗ is of order M and satisfies

σ(P∗) ∼Í `∈Zn≥0

∂`δ`[(ρ(ξ))]

`1!···`n! .

Theorem 1.1.2. Suppose that P is a pseudo-differential operator with symbol

σ(P)= ρ= ρ(ξ)of orderM1, andQis a pseudo-differential operator with symbol σ(Q) = φ = φ(ξ) of order M2. Then the symbol of the product QP is of order

M1 + M2 and satisfies σ(QP) ∼ Í`∈Zn≥0

1

`1!···`n!∂

`φ(ξ)δ`ρ(ξ), where ` := Î j∂

`j

j andδ` :=Î

jδ `j

j .

Theorem 1.1.3. Fors > k+1, Hs ⊆ Akθ.

Theorem 1.1.4. Let{aN} ∈ A∞θ be a sequence. Suppose that there is a constantC

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converges inHt.

Theorem 1.1.5. (a) For a pseudo-differential operatorPwithr×rmatrix valued

symbolσ(P)= ρ= ρ(ξ), the symbol of the adjointP∗satisfies

σ(P∗) ∼ Õ

(`1,...,`n)∈(Z≥0)n

∂`1

1 · · ·∂

`n

n δ`11· · ·δn`n(ρ(ξ))∗

`1!· · ·`n!

.

(b) IfQis a pseudo-differential operator withr×rmatrix valued symbolσ(Q)=

ρ0 = ρ0(ξ)

, then the product PQ is also a pseudo-differential operator and

has symbol

σ(PQ) ∼ Õ

(`1,...,`n)∈(Z≥0)n

∂`1

1 · · ·∂

`n

n (ρ(ξ))δ `1

1 · · ·δ

`n

n (ρ0(ξ))

`1!· · ·`n! .

Theorem 1.1.6. Fors > k+1, Hs ⊆ e(Aθk)r.

Theorem 1.1.7. Let{ ®aN} ∈ e(A∞θ )rbe a sequence. Suppose that there is a constant

C so that|| ®aN||s ≤ C for all N. Let s > t. Then there is a subsequence{ ®aNj} that

converges inHt.

1.2 Preliminaries

Fix some skew symmetricn×nmatrixθwith upper triangular entries inR\Qthat are

linearly independent overQ. Consider the irrational rotationC∗-algebra Aθ withn

unitary generatorsU1, . . . ,Unwhich satisfyUkUj =e2πiθj,kUjUk andU∗j =U−1j (Aθ

is the universal suchC∗-algebra with these generators and relations, so we opt not to use a GNS construction). Let {αs}s∈Rn be a n-parameter group of automorphisms

given byÎ jU

mj

j 7→ e is·mÎ

jU mj

j . We define the subalgebra A k

θ ofCk elements of

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and we define the subalgebra A∞θ of smooth elements of Aθto be thosea ∈ Aθsuch that the mappingRn → Aθ given bys 7→αs(a)is smooth. An alternative definition

of the subalgebraA∞θ of smooth elements is the elements inAθthat can be expressed

by an expansion of the formÍ

m∈Znam Î

jU mj

j , where the sequence{am}m∈Zn is in

the Schwartz spaceS(Zn)in the sense that, for allα∈Zn,

sup

m∈Zn (Ö

j

|mj|αj|am|)< ∞.

Define the trace τ : Aθ → Cby τ(ÎjU mj

j ) = 0 for mj not all zero and τ(1) = 1

and define an inner product h·,·i : Aθ× Aθ → Cby ha,bi = τ(b∗a) with induced norm || · || : Aθ → R≥0. Let Dj = −i∂j and define derivationsδj by the relations

δj(Uj) = Uj and δj(Uk) = 0 for j , k. For convenience, denote ∂` := Îj∂ `j

j ,

δ` := Î jδ

`j

j , and D

` := Î

jD `j

j . We define a map ψ : ρ 7→ Pρ assigning a

pseudo-differential operator on A∞θ to a symbolρ∈C∞(Rn,A∞θ ).

Definition 1.2.1. For ρ ∈ C∞(Rn,A∞θ ), let Pρ be the pseudo-differential operator sending arbitrarya ∈ A∞θ to

Pρ(a):= 1

(2π)n ∫

Rn ∫

Rn

e−is·ξρ(ξ)αs(a)dsdξ.

The integral above does not converge absolutely; it is an oscillatory integral. We define oscillatory integrals below as in [61].

Definition 1.2.2. Letq be a nondegenerate real quadratic form on Rn, a be aC∞

complex-valued function defined onRnsuch that the functions(1+|x|2)−m/2∂αa(x)

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xα∂βϕ(x)are bounded onRnfor all pairsα, β ∈Zn≥0. Suppose further thatϕ(0)= 1.

Then the limit

lim

→0

eiq(x)a(x)ϕ(x)dx

exists, is independent ofϕ(as long asϕ(0)=1), and is equal to∫eiq(x)a(x)dxwhen a ∈ L1. When a< L1, we continue to denote this limit by∫eiq(x)a(x)dx, and have an estimate

eiq(x)a(x)dx

≤ Cq,m max |α|≤m+n+1inf

{U ∈R:|(1+|x|2)−m/2∂αa| ≤ Ua.e.}

whereCq,m depends only on the quadratic formqand the orderm.

As shown in [61], oscillatory integrals behave essentially like absolutely convergent integrals in that one can still make changes of variables, integrate by parts, differen-tiate under the integral sign, and interverse integral signs. Given certain conditions on ρ, Pρ(a)satisfies the conditions of the oscillatory integral, and we can evaluate Pρ(a).

Definition 1.2.3. An element ρ = ρ(ξ) = ρ(ξ1, . . . , ξn)ofC∞(Rn,A∞θ )is a symbol of ordermif and only if for all non-negative integersi1, . . . ,in,j1, . . . ,jn

||δi1

1 · · ·δ

in

n(∂ j1

1 · · ·∂

jn

n ρ)(ξ)|| ≤Cρ(1+ |ξ|)m−|j|.

Example 2.6(i) of [61] gives a convenient formula for evaluating the oscillatory integrals that appear in our calculation ofPρ(a).

Proposition 1.2.4. Suppose that, for some m, a is aC∞ complex-valued function

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allα∈Zn

≥0. Then

1

(2π)n ∫

Rn ∫

Rn

e−iy·ηa(y)dydη= 1

(2π)n ∫

Rn ∫

Rn

e−iy·ηa(η)dydη= a(0).

We apply Proposition 1.2.4 to get a basic result.

Lemma 1.2.5. Let a = Í

m∈Znam Î

jU mj

j be an arbitrary element of A

θ and let

ρ∈C∞(Rn,A∞θ )be a symbol of orderM. Then

Pρ(a)= Õ m∈Zn

ρ(m)am n Ö

j=1 Umj j.

Proof. First consider the casea=Î jU

mj

j . We get

Pρ© ­

« n Ö

j=1 Umj

®

¬

= (1)n

Rn ∫

Rn

e−is·ξρ(ξ)αs©­ «

n Ö

j=1 Umj

®

¬

dsdξ

= (1)n

Rn ∫

Rn

e−is·ξρ(ξ)eis·m

n Ö

j=1

Umj jdsdξ

= 1

(2π)n ∫

Rn ∫

Rn

e−is·(ξ−m)ρ(ξ)dsdξ

n Ö

j=1 Umj j

= 1

(2π)n ∫

Rn ∫

Rn

e−is·ηρ(η+m)dsdη

n Ö

j=1 Umj j

= ρ(m) n Ö

j=1 Umj j,

as desired, having substitutedη= ξ−mand applied the result of Proposition 1.2.4. Now consider the general casea= Í

m∈ZnamÎjU

mj

j . Sinceαsis an automorphism

on Aθ, we get

Pρ(a)= Õ

m∈Zn

ρ(m)am n Ö

j=1 Umj j,

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1.3 Asymptotic formula for the symbol of the adjoint of a pseudo-differential

operator

Here we prove the formula for the symbol of the adjoint for the noncommutativen torus, adapting the proof of Lemma 1.2.3 of [38] to the noncommutativentorus.

Theorem 1.3.1. Suppose P is a pseudo-differential operator with symbolσ(P) =

ρ= ρ(ξ)of orderM. Then the symbol of the adjointP∗ is of orderM and satisfies

σ(P∗) ∼ Õ `∈Zn≥0

∂`δ`[(ρ(ξ))] `1!· · ·`n!

.

Proof. Leta,b∈ A∞θ . We have

hPρ(a),bi= τ

b∗ 1

(2π)n ∫

Rn ∫

Rn

e−is·ξρ(ξ)αs(a)dsdξ

= 1

(2π)n ∫

Rn ∫

Rn

e−is·ξτ(b∗ρ(ξ)αs(a))dsdξ

= 1

(2π)n ∫

Rn ∫

Rn

e−is·ξτ(α−s(ρ(ξ)∗b)∗a)dsdξ

= τ 1

(2π)n ∫

Rn ∫

Rn

e+is·ξα−s(ρ(ξ)∗b)dsdξ ∗

a

= ha,P∗ρ(b)i,

where

P∗ρ(b)= 1 (2π)n

Rn ∫

Rn

e+is·ξα−s(ρ(ξ)∗b)dsdξ = (1)n

Rn ∫

Rn

e+is·ξα−s(ρ(ξ)∗)α−s(b)dsdξ

= Õ

m,k

1

(2π)n ∫

Rn ∫

Rn

e+is·ξe+is·mρm(ξ)∗e−is·kbkdsdξU

−mn

n · · ·U

−m1

1 U

k1

1 · · ·U

kn

n

= Õ

m,k

1

(2π)n ∫

Rn ∫

Rn

e−is·ηρm((k−m) −η)∗bkdsdηU

−mn

n · · ·U

−m1

1 U

k1

1 · · ·U

kn

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= Õ m,k

ρm(k−m)∗bkUn−mn· · ·U

−m1

1 U

k1

1 · · ·U

kn

n

= Õ

m,k

(ρm(k−m)U1m1· · ·Unmn)∗(bkU1k1· · ·Unkn),

so

σ(P∗ρ)(ξ)=       Õ m

ρm(ξ−m) n Ö

j=1 Umj j

      ∗ =       1

(2π)n ∫

Rn ∫

Rn

e−i x·yÕ

m

ρm(ξ−y)αx©­ «

n Ö

j=1 Umj

®

¬

dxdy

      ∗ = 1

(2π)n ∫

Rn ∫

Rn

e−i x·yαx(ρ(ξ−y))dxdy ∗

.

We have

ρ(ξ−y)= Õ

|`|<N1

(−y)`

`! (∂`ρ)(ξ)+RN1(ξ,y)

αx(ρ(ξ−y))= Õ

|`|<N1

Õ

m (−y)`

`! (∂`ρm)(ξ)ei x·m©­ «

n Ö

j=1 Umj

®

¬

+αx(RN1(ξ,y)),

where

RN1(ξ,y)= N1

Õ

|`|=N1

(−y)`

`!

∫ 1

0

(1−γ)N1−1(∂`ρ)(ξ yγ)

so the corresponding symbol is

σ(P∗ρ)(ξ)=       Õ

|`|<N1

Õ

m

(−m)`

`! (∂`ρm)(ξ) m Ö

j=1

Umj j +TN1(ξ,y)

      ∗ = Õ

|`|<N1

∂`δ`[(ρ(ξ))]

`! +TN1(ξ,y)

,

where

TN1(ξ,y)=

1

(2π)n ∫

Rn ∫

Rn

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It remains to show that this symbol is of orderM. Obviously,

Õ

N≤`<N1

∂`δ`[(ρ(ξ))] `! ∈SM

−N,

so we need to show that the remainder is of order M−N1. Note that σ(P∗ρ)(ξ) − Õ

|`|<N1

∂`δ`[(ρ(ξ))]

`! =TN1(ξ,y)

.

Integrating by parts, we get

Rn ∫

Rn

e−i x·y(−y)

`

`! αx

∫ 1

0

(1−γ)N1−1(∂`ρ)(ξyγ)

dxdy

= `!1 ∫

Rn ∫

Rn

e−i x·y(−Dx)`αx

∫ 1

0

(1−γ)N1−1(∂`ρ)(ξyγ)

dxdy

= `!1 ∫ Rn

Rn

e−i x·y(−δ)`αx

∫ 1

0

(1−γ)N1−1(∂`ρ)(ξyγ)

dxdy

= `!1 ∫ Rn

Rn

e−i x·y

∫ 1

0

(1−γ)N1−1(−δ)

x((∂`ρ)(ξ− yγ))dγdxdy = `!1

Rn ∫

Rn

e−i x·y(−1)` ∫ 1

0

(1−γ)N1−1α

x((δ`∂`ρ)(ξ−yγ))dγdxdy,

where, for arbitrarya =Í

mamÎjU mj

j ,

(−Dx)`αx(a)=(−Dx)`αx©­ « Õ m am n Ö

j=1 Umj

®

¬

=(−Dx)` Õ

m

amei x·m n Ö

j=1 Umj j

m

am(−m)`ei x·m n Ö

j=1 Umj j

=(−δ)`αx(a).

Sinceρ ∈SM and|`|= N1we haveδ`∂`ρ ∈SM−N1. We get the boundedness of `N1−M(ξ)

Rn ∫

Rn

e−i x·yαx(RN1(ξ,y))dxdy

because∂`RN1(y, ξ)is the rest of indexN1in the Taylor expansion of∂

`(ξyγ)for

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1.4 Asymptotic formula for the symbol of a product of two pseudo-differential

operators

Next we prove the formula for the product or composition of symbols for the noncommutativentorus, adapting the proof of Theorem 7.1 of [69].

Theorem 1.4.1. Suppose that P is a pseudo-differential operator with symbol

σ(P)= ρ= ρ(ξ)of orderM1, andQis a pseudo-differential operator with symbol σ(Q) = φ = φ(ξ) of order M2. Then the symbol of the product QP is of order

M1+M2and satisfies

σ(QP) ∼ Õ `∈Zn ≥0

1 `1!· · ·`n!

∂`φ(ξ)δ`ρ(ξ),

where∂` :=Î j∂

`j

j andδ

` := Î jδ

`j

j .

Proof. We want to show that ifρ:Rn → A∞θ is of orderM1andφ:Rn→ A∞θ is of

orderM2,Pφ◦Pρ= Pµ, where µis of orderM1+M2and has asymptotic expansion

µ∼Õ `

1 `!∂

`φ(ξ)δ`ρ(ξ).

Let {ϕk} be the partition of unity constructed in Theorem 6.1 of [69] and define φk(ξ):= φ(ξ)ϕk(ξ). We have

Pφk(a)= 1

(2π)n ∫

Rn ∫

Rn

eis·ξφk(ξ)αs(a)dsdξ.

Summing overk from zero to infinity and applying Fubini’s Theorem, we get ∞

Õ

k=0

Pφk(a)= 1

(2π)n ∫

Rn ∫

Rn

e−is·ξ ∞

Õ

k=0

φk(ξ)αs(a)dsdξ

= 1

(2π)n ∫

Rn ∫

Rn

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so

Pφ(a)=

Õ

k=0 Pφk(a)

and the convergence of the series is absolute and uniform for alla ∈ A∞θ. We want to compute the symbol ofPφ◦Pρ, but issues with convergence of integrals make it

so we need to compute the symbol ofPφk ◦Pρ. Let a∈ A ∞

θ be arbitrary. Applying

Pφk ◦Pρwe get

Pφk(Pρ(a))= 1

(2π)n ∫

Rn ∫

Rn

e−is·ξφk(ξ)αs(Pρ(a))dsdξ

= 1

(2π)n ∫

Rn ∫

Rn

e−is·ξφk(ξ)αs

1

(2π)n ∫

Rn ∫

Rn

e−it·ηρ(η)αt(a)dtdη

dsdξ

= 1

(2π)2n ∫ Rn ∫ Rn ∫ Rn ∫ Rn

e−is·ξ−it·ηφk(ξ)αs(ρ(η))αs+t(a)dtdη

dsdξ

= 1

(2π)2n ∫ Rn ∫ Rn ∫ Rn ∫ Rn

e−i x·ξ−i(y−x)·ηφk(ξ)αx(ρ(η))αy(a)dydη

dxdξ

= 1

(2π)2n ∫ Rn ∫ Rn ∫ Rn ∫ Rn

e−i x·(ξ−η)−iy·ηφk(ξ)αx(ρ(η))αy(a)dydη

dxdξ

= 1

(2π)2n ∫ Rn ∫ Rn ∫ Rn ∫ Rn

e−i x·σ−iy·τφk(σ+τ)αx(ρ(τ))αy(a)dydτ

dxdσ

= 1

(2π)n ∫

Rn ∫

Rn

e−iy·τ

1

(2π)n ∫

Rn ∫

Rn

e−i x·σφk(σ+τ)αx(ρ(τ))dxdσ

αy(a)dydτ

= 1

(2π)n ∫

Rn ∫

Rn

e−iy·τµk(τ)αy(a)dydτ,

where

µk(τ)=

1

(2π)n ∫

Rn ∫

Rn

e−i x·σφk(σ+τ)αx(ρ(τ)))dxdσ,

having done the changes of variables(x,y) = (s,s+t) and(σ, τ) = (ξ−η, η)and applied Proposition 1.2.4. This suggests that

Pφ(Pρ(a))= 1 (2π)n

Rn ∫

Rn

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where µ(τ) = Í∞

k=0µk(τ). We need to show thatµis a symbol inSM1+M2 and has

our desired asymptotic expression. Defineµk by

µk(ξ)=

1

(2π)2

Rn ∫

Rn

e−i x·yφk(ξ+ y)αx(ρ(ξ))dxdy

for all ξ ∈ Rn. By Taylor’s formula with integral remainder given in Theorem 6.3

of [69], we get

φk(ξ+y)= Õ

|`|<N1

y`

`!(∂`φk)(ξ)+RN1(y, ξ),

where

RN1(y, ξ)= N1

Õ

|`|=N1

y`

`!

∫ 1

0

(1−γ)N1−1(∂`φ

k)(ξ+γy)dγ (1.1)

for ally, ξ ∈R2. Substituting back into our expression for µk(ξ)we get

µk(ξ)=

1

(2π)n ∫

Rn ∫

Rn

e−i x·y Õ |`|<N1

y`

`!(∂`φk)(ξ)αx(ρ(ξ))dxdy+T

(k)

N1(ξ),

where

TN(k)

1(ξ)=

1

(2π)n ∫

Rn ∫

Rn

e−i x·yRN1(y, ξ)αx(ρ(ξ))dxdy.

Expressingρ(ξ)asρ(ξ)=Í

mρm(ξ)Înj=1U mj

j , we see that

αx(ρ(ξ))= Õ

m

ρm(ξ)ei x·m n Ö

j=1 Umj j

so

µk(ξ) −T

(k)

N1(ξ)=

Õ

|`|<N1

Õ

m

1

(2π)n ∫

Rn ∫

Rn

e−i x·yy

`

`!(∂`φk)(ξ)ρm(ξ)ei x·m n Ö

j=1

Umj jdxdy

= Õ

|`|<N1 1

`!(∂`φk)(ξ) Õ

m

ρm(ξ) n Ö

j=1

Umj j 1

(2π)n ∫

Rn ∫

Rn

e−i x·(y−m)y`dxdy

= Õ

|`|<N1 1

`!(∂`φk)(ξ) Õ

m

ρm(ξ) n Ö

j=1

Umj jm`

= Õ

|`|<N1 1

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Let µ(ξ)= Í∞

k=0µk(ξ). It remains to show that µis of orderM1+ M2. Obviously,

Õ

N≤|`|<N1

y`

`!(∂`φ)(ξ)(δ`ρ)(ξ) ∈SM1+M2 −N,

so we just need to show that the remainder is of orderM1+M2−N1. Note that µ(ξ) − Õ

|`|<N1

y`

`!(∂

`φ)(ξ)(δ`ρ)(ξ)= Õ∞ k=0

TN(k)

1(ξ),

where

TN(k)

1(ξ)=

1

(2π)n ∫

Rn ∫

Rn

e−i x·yRN1(y, ξ)αx(ρ(ξ))dxdy

and

RN1(y, ξ)= N1

Õ

|`|=N1

y`

`!

∫ 1

0

(1−γ)N1−1(∂`φ

k)(ξ+γy)dγ.

Integrating by parts, we get

Rn ∫

Rn

e−i x·yy

`

`!

∫ 1

0

(1−γ)N1−1(∂`φ

k)(ξ+γy)dγαx(ρ(ξ))dxdy = `!1 ∫

Rn ∫

Rn

e−i x·yy`

∫ 1

0

(1−γ)N1−1(∂`(φ

k)(ξ+γy)dγαx(ρ(ξ))dxdy = `!1 ∫

Rn ∫

Rn

e−i x·y

∫ 1

0

(1−γ)N1−1(∂`(φ

k)(ξ+γy)dγD`xαx(ρ(ξ))dxdy = `!1

Rn ∫

Rn

e−i x·y

∫ 1

0

(1−γ)N1−1(∂`(φ

k)(ξ+γy)dγδ`αx(ρ(ξ))dxdy = `!1 ∫

Rn ∫

Rn

e−i x·y

∫ 1

0

(1−γ)N1−1(∂`(φ

k)(ξ+γy)dγαx((δ`ρ)(ξ))dxdy,

where for arbitrarya=Í

mamÎjU mj

j we have (Dx)`αx(a)=(Dx)`αx©­

« Õ m am n Ö

j=1 Umj

®

¬

=(Dx)` Õ

m

amei x·m n Ö

j=1 Umj j

m

amm`ei x·m n Ö

j=1 Umj j

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Since|`| = N1, we have ∂`(φk) ∈ SM2−N1 andδ`ρ ∈ SM1. We get the boundedness

of

µN1−M1−M2(ξ)

Rn ∫

Rn

e−i x·yRN1(y, ξ)αx(ρ(ξ))dxdy

since δ`ρ ∈ SM1 and ∂`R

N1(y, ξ) is the rest of index N1 in Taylor’s expansion of

∂`φ

k(ξ+γy)for which on has∂`φk ∈SM2−N1.

1.5 Sobolev spaces on the noncommutativentorus

We develop a theory of Sobolev spaces on the noncommutative n-torus. Sobolev spaces on noncommutative tori were originally introduced by Spera [66], but have also been written about by various other researchers [40, 52, 57, 58, 65, 70]. We constructHs, the Sobolev spaces, in terms of the Hilbert inner product, andCk, the analog of k times continuously differentiable functions, in terms of the C∗ norm. Letλ(ξ)=(1+ξ12+· · ·+ξn2)1/2. Consider the following inner product onA∞θ .

Definition 1.5.1. Define the Sobolev inner producth·,·is : A∞θ × A∞θCby

ha,bis := hPλs(a),Pλs(b)i =

Õ

m

(1+|m1|2+· · ·+|mn|2)sbmam.

Note that for s = 0 this agrees withh·,·i. This inner product induces the following norm.

Definition 1.5.2. Define the Sobolev norm|| · ||s : A∞θR≥0by

||a||2s := hPλs(a),Pλs(a)i =

Õ

m

(1+|m1|2+· · ·+|mn|2)s|am|2.

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Definition 1.5.3. Define the Sobolev space Hs to be the completion of A∞θ with respect to|| · ||s.

We can prove that a pseudo-differential operator of orderd ∈Rcontinuously maps HsintoHs−d. However, we must first prove the case wheres= d.

Theorem 1.5.4. Suppose ρ ∈ Sd. Then ||Pρ(a)||0 ≤ C||a||d for some constant

C > 0anddefines a bounded operatorPρ :Hd →H0.

Proof. Note that {Î jU

mj

j : m ∈ Z

n} is an orthogonal basis with respect to h·,·is, which is orthonormal in the case s = 0. We have ||ÎjU

mj

j ||

2 0 = 1,

||ρm(ξ)Î jU

mj

j ||

2

0 = |ρm(ξ)|

2, and||ρ(ξ)||2 0 =

Í

m|ρm(ξ)|2. Sinceρis of orderd, we

have||ρ(ξ)||0 ≤Cρ(1+|ξ|)d, and since(1− |ξ|)2 0 gives us(1+|ξ|)2 2(1+|ξ|2), we have

||ρ(ξ)||02 ≤C2ρ(1+|ξ|)2d ≤ Cρ22d(1+|ξ|2)d.

Letkρ :=Cρ22d. Then we have

Õ

m

|ρm(ξ)|2≤ kρ(1+|ξ|2)d.

Let es,m := (1+ |m1|2+ · · ·+ |mn|2)−s/2ÎjU mj

j and Es := {es,m | m ∈ Z

n}. By

definition we have Es orthonormal with respect to h·,·is. It suffices to prove this

theorem for the casea= ed,m by the orthonormality ofEdsince

||Pρ(a)||02=Õ m

|am|2||Pρ(ed,m)||02

and

||a||d2= Õ m

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Since||ed,m||2d =1, it suffices to show that

||Pρ(ed,m)||02 ≤ K

for some constantK > 0. We have

||Pρ(ed,m)||20 = ||ρ(m)ed,m||02

= ||ρ(m)(1+|m1|2+· · ·+|mn|2)−d/2 n Ö

j=1 Umj j||02

= Õ k

ρk(m) n Ö

j=1

Ukjj(1+|m1|2+· · ·+|mn|2)−d/2 n Ö

j=1 Umj j

2 0

=(1+|m1|2+· · ·+|mn|2)−d Õ k

ρk(m) n Ö

j=1 Ujkj

n Ö

j=1 Umj j

2 0

=(1+|m1|2+· · ·+|mn|2)−d Õ k

ρk(m)w(m,k) n Ö

j

Umj j+kj

2 0

=(1+|m1|2+· · ·+|mn|2)−d Õ k,`

ρk−m(m)w(m,k−m) n Ö

j=1 Ukjj

2 0

=(1+|m1|2+· · ·+|mn|2)−d Õ

k

km(m)|2

=(1+|m1|2+· · ·+|mn|2)−d Õ

k,`

|ρk(m)|2

≤ (1+|m1|2+· · ·+|mn|2)−dkρ(1+|m1|2+· · ·+|mn|2)d= kρ,

where

w(k,m):=

n Ö

j=1 Umj j

n Ö

j=1 Ukj

­

« n Ö

j=1

Umj j+kjª ®

¬

∈S1 ⊂ C

so our desired constant isK = kρ=Cρ22d and we are done.

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Lemma 1.5.5. For any a ∈ A∞θ and s,t ∈ R, a ∈ Hs if and only if Pλt(a) ∈ Hs−t

with||a||s = ||Pλt(a)||st.

Proof. Suppose thata∈ Hs orPλt(a) ∈ Hs−t. Then

||Pλt(a)||2st =

Õ

m

(1+|m1|2+· · ·+|mn|2)s−tλ2t(m)|am|2

m

(1+|m1|2+· · ·+|mn|2)s−t(1+|m1|2+· · ·+|mn|2)t|am|2

m

(1+|m1|2+· · ·+|mn|2)s|am|2

= ||a||s

so we know thata∈ Hs andPλt(a) ∈ Hs−t.

Then the general case follows quite easily.

Corollary 1.5.6. Suppose ρ ∈ Sd. Then ||Pρ(a)||s−d ≤ C||a||s for some constant

C > 0anddefines a bounded operatorPρ :Hs → Hs−d.

Proof. By Lemma 1.5.5, we have ||Pρ(a)||s−d = ||Pλs−d(Pρ(a))||0. By Theorem 1.4.1, the symbolσ(Pλs−d ◦Pρ)is of orderd+(s−d)=s, so Theorem 1.5.4 gives us||Pλs−d(Pρ(a))||0≤ C||a||s for some constantC > 0.

We can also define an analog of theCk norm for the noncommutativentorus.

Definition 1.5.7. Define theCk norm|| · ||∞,k : A∞θ →R≥0as follows:

||a||∞,k :=

Õ

|`|≤k

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where theC∗ norm|| · ||C∗is given by

||a||C2∗ :=sup{|λ|:a ∗

a−λ·1 not invertible}.

Since, for arbitrarya =Í

mamÎjU mj

j ,

D`sαs(a)=(−i∂s)` Õ

m

eis·mam n Ö

j=1 Umj j

m

m`eis·mam n Ö

j=1 Umj j

m

δ`eis·mam n Ö

j=1 Umj j

= δ`αs(a)

we haveAθk =Ck.

We can easily prove an analog of the Sobolev lemma as follows.

Theorem 1.5.8. Fors > k+1, Hs ⊆ Akθ.

Proof. First consider the case k = 0. Note that || · ||∞,0 = || · ||C∗ so for arbitrary amÎjU

mj

j we have ||am

n Ö

j=1

Umj j||2,0= sup{|λ|: |am|2−λ·1 not invertible}= |am|2

and for arbitrarya= Í

mamÎjU mj

j we have ||a||2,0≤ Õ

m ||am

n Ö

j=1

Umj j||2,0= Õ m

|am|2= ||a||20

by the triangle inequality. We have

a= Õ

m

amλs(m)λ−s(m) n Ö

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so by the Cauchy-Schwarz inequality we get

||a||02≤ ||a||2sÕ m

(1+|m1|2+· · ·+|mn|2)−s.

Since 2s > 2,(1+|m1|2+· · ·+|mn|2)−sis summable overm ∈Znand||a||02 ≤C||a||s.

Thus we get||a||∞,0 ≤ ||a||0 ≤ C||a||sandHs ⊆ A0θ.

Now supposek > 0. Using what we’ve proven for the previous case, we have

||δ`(a)||∞,0 ≤C||δ`(a)||s−|`|

=C||Õ m

m`am n Ö

j=1

Umj j||s−|`|

<C||Õ m

(1+|m1|2+· · ·+ |mn|2)|`|am n Ö

j=1

Umj j||s−|`|

=C||Pλ|`|(a)||s−|`|

=C||a||s

for|`| ≤ k sinces− |`| ≥ s−k > 1. Therefore,

||a||∞,k =

Õ

|`|≤k

||δ`(a)||∞,0 ≤

Õ

|`|≤k

C||a||s ≤ C||a||s(k +1)(k+2)/2

and we getHs ⊆ Akθ.

We get the following corollary.

Corollary 1.5.9. Ù

s∈R

Hs = A∞θ .

Proof. Supposea ∈ Ñ s∈RH

s. Then for anyk

Z≥0, a ∈Hk+2, so by the theorem we just proved,a∈ Aθk. Consequentlya ∈ A∞θ , soÑ

s∈RH

s A

θ.

Suppose a ∈ A∞θ. Then since Hs is the completion of A∞θ with respect to || · ||s, A∞θ ⊆ Hs for alls ∈R, and A∞θ ⊆ Ñ

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We can also prove an analog of the Rellich lemma for the noncommutativentorus.

Theorem 1.5.10. Let{aN} ∈ A∞θ be a sequence. Suppose that there is a constant

C so that||aN||s ≤ C for all N. Let s > t. Then there is a subsequence{aNj} that

converges inHt.

Proof. Letes,m :=(1+|m1|2+· · ·+|mn|2)−s/2Înj=1U mj

j andEs := {es,m |m ∈Z n}.

Esis an orthonormal basis with respect to h·,·is, so we can writeaN := ÍkaN,kes,k.

Then

|aN,k|2 ≤ Õ

k

|aN,k|2 ≤ C2

and |aN,k| ≤ C. Applying the Arzela-Ascoli theorem to {aN,k} for some fixed k, we can get a subsequence {aNj,k} of {aN,k} such that for any > 0 there exists M() ∈ N such that |aNi,k − aNj,k| < whenever i,j ≥ M(). Do this for all

|k1|2 + · · · + |kn|2 ≤ r, replacing {aN} with {aNj} each time. Then we get a subsequence{aNj}of{aN}such that for any > 0 there existsM() ∈ Nsuch that, for all|k1|2+· · ·+|kn|2 ≤ r,|aNi,k−aNj,k|< wheneveri,j ≥ M(). Now consider the sum

||aNi −aNj|| 2

t = Õ

k

|aNi,k−aNj,k|

2(1+|k

1|2+· · ·+|kn|2)t−s.

Decompose it into two parts: one where |k1|2+ · · ·+ |kn|2 > r2 and one where |k1|2+· · ·+|kn|2 ≤r. On|k1|2+· · ·+|kn|2 >r2we estimate

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so that

Õ

|k1|2+···+|kn|2≥r2

|aNi,k −aNj,k|

2(1+|k

1|2+· · ·+|kn|2)t−s

< (1+r2)t−sÕ k

|aNi,k−aNj,k|

2 2C2(1+ r2)t−s.

If > 0 is given, we chooserso that 2C2(1+r2)t−s < . The remaining part of the

sum is over|k1|2+· · ·+|kn|2 ≤ r2and can be bounded above by0:= −2C2(1+r2)t−s

ifi,j ≥ M(√n0/(2r+1))because a ball of radiusrcentered at the origin is contained in a cube of side length 2r that has(2r +1)n lattice points. Then the total sum is bounded above by, and we are done.

1.6 The pseudo-differential calculus on f.g. projective modules over the

non-commutativentorus

We can generalize these results to arbitrary finitely generated projective right mod-ules over the noncommutative n torus following p. 553 of [16], which considers finitely generated projective modules over an arbitrary unital ∗-algebra. Let E be a finitely generated projective right A∞θ -module. Since E is a finitely generated projective right A∞θ -module, we can writeE as a direct summandE = e(A∞θ )r of a free module(A∞θ )r with direct complementF = (ide)(A

θ )r, where the idempotent

e ∈ Mr(A∞θ )is self-adjoint. We make frequent use of the liftι: E → (A∞θ )r sending ®

a ∈E toa®∈E ⊆ (A∞θ )r. Consider anr×r matrix valued symbol ρ=(ρj,k)where ρj,k :Rn→ A∞θ are scalar symbols and ρj,k ∈Sd. Define the operatorPρ :E → E as follows: Pρ( ®a) := (2π)−n∫

Rn ∫

Rne

−is·ξρ(ξ)α

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actually easier to work with h·,·i := h·,·i(A

θ)r, applying ι when necessary. Since Pρ( ®a)j = (2π)−n

Rn ∫

Rne

−is·ξÍr

k=1ρj,k(ξ)αs(ak)dsdξ,Lemma 1.2.5 generalizes to E as follows after applying it to each component: Pρ( ®a) = Í

mρ(m) ®amÎnj=1U

mj j .

Theorems 1.3.1 and 1.4.1 generalize as follows.

Theorem 1.6.1. (a) For a pseudo-differential operatorPwithr×rmatrix valued

symbolσ(P)= ρ= ρ(ξ), the symbol of the adjointP∗satisfies

σ(P∗) ∼ Õ

(`1,...,`n)∈(Z≥0)n

∂`1

1 · · ·∂

`n

n δ `1

1 · · ·δ

`n

n (ρ(ξ))∗

`1!· · ·`n!

.

(b) IfQis a pseudo-differential operator withr×rmatrix valued symbolσ(Q)=

ρ0 = ρ0(ξ), then the product

PQ is also a pseudo-differential operator and

has symbol

σ(PQ) ∼ Õ

(`1,...,`n)∈(Z≥0)n

∂`1

1 · · ·∂

`n

n (ρ(ξ))δ`11· · ·δn`n(ρ0(ξ))

`1!· · ·`n! .

Proof. First let’s prove part (a). Letρbe anr×rmatrix valued symbol of order M anda,® b®∈ E. We have

hPρ( ®a),®bi =τ

®

b∗ 1

(2π)n ∫

Rn ∫

Rn

e−is·ξρ(ξ)αs( ®a)dsdξ

=τ 1

(2π)n ∫

Rn ∫

Rn

e+is·ξα−s(ρ(ξ)∗b®)dsdξ ∗

®

a

= h ®a,P∗ρ(®b)i,

where

P∗ρ(®b)j = 1 (2π)n

Rn ∫

Rn

e+is·ξ

r Õ

k=1

α−s(ρ(ξ)∗j,kbk)dsdξ

= 1

(2π)n ∫

Rn ∫

Rn

e+is·ξ

r Õ

k=1

α−s(ρ(ξ)∗j,k)α−s(bk)dsdξ

m,p "

ρm(p−m) n Ö

h=1 Umh

h #∗

bp n Ö

h=1 Uph

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so

σ(P∗ρ)(ξ)= "

Õ

m

ρm(ξ−m) n Ö

h=1 Umh

h #∗

= "

1

(2π)n ∫

Rn ∫

Rn

e−i x·yÕ

m

ρm(ξ−y)αx n Ö

h=1 Umh

h !

dxdy

#∗

=

1

(2π)n ∫

Rn ∫

Rn

e−i x·yαx(ρ(ξ− y))dxdy ∗

.

The rest of the proof reduces to ther =1 case, applying it to each entry inρ=(ρj,k).

We proceed to part (b). Let ρbe anr ×r matrix valued symbol of orderm1andφ be anr×r matrix valued symbol of orderm2. Let{ϕk}be a partition of unity and defineφk(ξ):= φ(ξ)ϕk(ξ). Leta®∈ E. We have

Pφk(Pρ( ®a))= 1

(2π)n ∫

Rn ∫

Rn

e−is·ξφk(ξ)αs(Pρ( ®a))dsdξ = (1)n

Rn ∫

Rn

e−is·ξφk(ξ)αs

1

(2π)n ∫

Rn ∫

Rn

e−it·ηρ(η)αt( ®a)dtdη

dsdξ

= 1

(2π)2n ∫ Rn ∫ Rn ∫ Rn ∫ Rn

e−is·ξ−it·ηφk(ξ)αs(ρ(η))αs+t( ®a)dtdη

dsdξ

= 1

(2π)2n ∫ Rn ∫ Rn ∫ Rn ∫ Rn

e−i x·ξ−i(y−x)·ηφk(ξ)αx(ρ(η))αy( ®a)dydη

dxdξ

= 1

(2π)2n ∫ Rn ∫ Rn ∫ Rn ∫ Rn

e−i x·(ξ−η)−iy·ηφk(ξ)αx(ρ(η))αy( ®a)dydη

dxdξ

= 1

(2π)2n ∫ Rn ∫ Rn ∫ Rn ∫ Rn

e−i x·σ−iy·ηφk(σ+τ)αx(ρ(τ))αy( ®a)dydτ

dxdσ

= 1

(2π)2n ∫

Rn ∫

Rn

e−iy·τ

Rn ∫

Rn

e−i x·σφk(σ+τ)αx(ρ(τ))dxdσ

αy( ®a)dydτ

= 1

(2π)n ∫

Rn ∫

Rn

e−iy·τλk(τ)αy( ®a)dydτ,

whereλk(τ)= (21π)n

Rn ∫

Rne

−i x·σφ

k(σ+τ)αx(ρ(τ))dxdσ so

Pφ(Pρ( ®a))= 1 (2π)n

Rn ∫

Rn

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whereλ(τ) = Í∞

k=0λk(τ). Let λk(ξ) := (21π)n

Rn ∫

Rne

−i x·yφ

k(ξ+ y)αx(ρ(ξ))dxdy.

Since

λk(ξ)α,γ =

1

(2π)n ∫

Rn ∫

Rn

e−i x·y

r Õ

β=1

φk(ξ+ y)α,βαx(ρ(ξ)β,γ)dxdy

= r Õ

β=1 1

(2π)n ∫

Rn ∫

Rn

e−i x·yφk(ξ+ y)α,βαx(ρ(ξ)β,γ)dxdy,

the rest of the proof reduces to ther = 1 case, applying it to each summand in the above sum.

Letλ(ξ)=(1+ξ12+· · ·+ξ2

n)1

/2id

E. Consider the following inner product onE.

Definition 1.6.2. Define the Sobolev inner producth·,·is : E×E →Cbyh ®a,b®is := hPλs( ®a),Pλs(®b)i = Íj,m(1+|m1|2+· · ·+|mn|2)sbj,maj,m.

Note that for s = 0 this agrees with h·,·iE. Since h·,·is,E = h·,·is,(A∞θ)r|E×E, it’s actually easier to work with h·,·is := h·,·is,(A

θ)r, applying ιwhen necessary. This inner product induces the following norm.

Definition 1.6.3. Define the Sobolev norm|| · ||s,E :E →R≥0by

|| ®a||2s,E := hPλs( ®a),Pλs( ®a)i =

Õ

j,m

(1+|m1|2+· · ·+|mn|2)s|aj,m|2.

Since || · ||s,E = || · ||s,(A∞θ)r|E, it’s actually easier to work with || · ||s := || · ||s,(A∞ θ)r, applying ιwhen necessary. Using this norm, we can define the analog of Sobolev spaces onE.

Definition 1.6.4. Define the Sobolev space Hs to be the completion of E with

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We can prove that a pseudo-differential operator of orderd ∈Rcontinuously maps HsintoHs−d. However we must first prove the case wheres = d.

Theorem 1.6.5. Suppose ρ is a matrix valued symbol of order d. Then, for any

®

a ∈ E, ||Pρ( ®a)||0 ≤ C|| ®a||d for some constant C > 0 anddefines a bounded operatorPρ :Hd → H0.

Proof. Let {vj : j ∈ {1, . . . ,r}} be the standard basis of (A∞θ )r considered as a

finitely generated free right A∞θ -module. Let {fj = evj : j ∈ {1, . . . ,r}} be a

generating set of the eigenvalue 1 eigenspace ofeconsidered as an endomorphism of

(A∞θ )r. Note that{fjÎgU

mg

g :m ∈Zn,j ∈ {1, . . . ,r}}spansE as aC-vector space,

and fjÎgU mg

g and fkÎgU

ng

g are orthogonal. Consider the C-vector subspace

spanned by {fjÎgU

mg

g ,m ∈ Zn} for fixed j. It has countable dimension. Let

{fjaj,m : m ∈ I} be a basis of this subspace with index set I ⊆ Zn, where aj,m ∈

A∞θ . We can apply Gram–Schmidt with respect to h·,·i and get an orthonormal basis Fj := {fj,m : m ∈ I}. The subspaces for different j are still orthogonal,

so F := {fj,m : m ∈ I,j ∈ {1, . . . ,r}} is an orthogonal basis of E considered as a C-vector space, normalized with respect to h·,·i. We have ||fj,m||02 = 1, ||ρ(ξ)h,jfj,m||02 = ||ρ(ξ)h,j||2, and ||ρ(ξ)h,jfj||02 = Íh||ρ(ξ)h,j||02. Since ρh,j is of

order d, we have ||ρ(ξ)h,j||0 ≤ Cρ(1 + |ξ|)d, and since (1 − |ξ|)2 ≥ 0 gives us

(1+|ξ|)2 ≤ 2(1+|ξ|2), we have||ρ(ξ)h,j||2

0 ≤C 2

ρ(1+|ξ|)2d ≤Cρ22d(1+|ξ|2)d. Let

kρ := C2ρ2d. Then we have ||ρ(ξ)h,j||2

0 ≤ kρ(1+ |ξ|

2)d. Let f

s,j,m := fj,m/||fj,m||s

and

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By definition, Fs is an orthonormal basis with respect to h·,·is. It suffices to prove

this theorem for the case a® = fd,j,m by the orthogonality of Fd since ||Pρ( ®a)||02 = Í

j,m|aj,m|2||Pρ(fd,j,m)||02 and || ®a||2d = Í

j,m |aj,m|2||fd,j,m||d2. Since ||fd,j,m||2d = 1, it

suffices to show that||Pρ(fd,j,m)||02 ≤ K for some constantK > 0. We have

||Pρ(fd,j,m)||02= ||ρ(m)fd,j,m||20 =

ρ(m)fj,m/||fj,m||d 2 0 = Õ h

ρ(m)h,jfj,m/||fj,m||d 2 0 = 1

||fj,m||2d Õ h

ρ(m)h,jfj,m 2 0 ≤ 1

||fj,m||2d Õ

h

||ρ(m)h,j||02

≤ 1

||fj,m||2dr kρ||fj,m||

2

d =r kρ

so our desired constant isK =r kρ=rCρ22dand we are done.

For the general cases, dwe need to prove a lemma saying that|| · ||s = || · ||s−t◦Pλt.

Lemma 1.6.6. For anya®∈ E ands,t ∈R,a®∈Hs if and only ifPλt( ®a) ∈ Hs−twith

|| ®a||s = ||Pλt( ®a)||st.

Proof. Let {vj : j ∈ {1, . . . ,r}} be the standard basis of (A∞θ )r considered as a

finitely generated free right A∞θ -module. Let {vj,m = vjÎgU

mg

g : m ∈ Zn,j ∈

{1, . . . ,r}}be an orthonormal basis of(A∞θ )r considered as a

C-vector space.

Sup-pose thata®∈Hs orPλt( ®a) ∈Hs−t. Then

||Pλt( ®a)||2st =

Õ

j,m

||vj,m||2s−tλ2 t(

m)|aj,m|2= Õ

j,m

||vj,m||2s|aj,m|2 = || ®a||s2

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Then the general case follows quite easily.

Corollary 1.6.7. Supposeρis a matrix valued symbol of orderd. Then

||Pρ( ®a)||s−d ≤C|| ®a||s

for some constantC > 0anddefines a bounded operator Pρ: Hs → Hs−d.

Proof. By Lemma 1.6.6, we have ||Pρ( ®a)||s−d = ||Pλs−d(Pρ( ®a))||0. By Proposition 1.6.1(b), the matrix valued symbol σ(Pλs−d ◦Pρ) is of order d +(s − d) = s, so Theorem 1.6.5 gives us||Pλs−d(Pρ( ®a))||0 ≤C|| ®a||s for some constantC > 0.

We can also define an analog of theCk norm onE.

Definition 1.6.8. Define the Ck norm || · ||∞,k : E → R≥0 as follows: || ®a||∞,k :=

Í

|`|≤k ||δ`( ®a)||C∗where theC∗norm|| · ||C∗is given by|| ®a||2

C∗ :=sup{|λ| :a®∗a®−λ· 1 not invertible}.

Since, for arbitrarya®=Í

j,m fjÎgU

mg

g aj,m,

D`sαs( ®a)=(−i∂s)` Õ

j,m

eis·mfj n Ö

g=1

Ugmgaj,m = Õ

j,m

m`eis·mfj n Ö

g=1

Ugmgaj,m

j,m

δ`eis·mfj n Ö

g=1

Ugmgaj,m = δ`αs( ®a)

we havee(Akθ)r =Ck. We can easily prove an analog of the Sobolev lemma onEas follows.

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Proof. First consider the case k = 0. Note that || · ||∞,0 = || · ||C∗ so for ar-bitrary fjÎng=1U

mg

g aj,m we have ||fjÎng=1U mg

g aj,m||∞2,0 = sup{|λ| : |aj,m| 2 λ·

1 not invertible} = |aj,m|2 and for arbitrary a® = Í

j,m fjÎng=1U mg

g aj,m we have

|| ®a||2,

0 ≤

Í

j,m||fjÎng=1U mg

g aj,m||2∞,0 ≤

Í

j,m|aj,m|2 = || ®a||2

0 by the triangle in-equality. We have a® = Í

j,mλs(m)fjÎng=1U mg

g aj,mλ−s(m) so by the

Cauchy-Schwarz inequality we get || ®a||2

0 ≤ || ®a|| 2

s Í

m(1+ |m1|2 + · · · + |mn|2)−s. Since

2s > 2, (1+ |m1|2+· · ·+|mn|2)−s is summable overm ∈ I and j ∈ {1, . . . ,r} so || ®a||2

0 ≤ C|| ®a||s. Thus we get|| ®a||∞,0 ≤ || ®a||0 ≤ C|| ®a||s andH

s e(A0

θ)r.

Now supposek > 0. Using what we’ve proven for the previous case, we have

||δ`( ®a)||∞,0 ≤ C||δ`( ®a)||s−|`| =C

Õ j,m

m`fj n Ö

g=1

Ugmgaj,m

s−|`|

<C Õ j,m

(1+|m1|2+· · ·+|mn|2)|`|fj n Ö

g=1

Ugmgaj,m

s−|`|

= C||Pλ|`|( ®a)||s−|`| =C|| ®a||s

for |`| ≤ k since s− |`| ≥ s− k > 1. Therefore, || ®a||∞,k = Í|`|≤k||δ`( ®a)||∞,0 ≤

Í

|`|≤kC|| ®a||s ≤ C|| ®a||s(k+1)(k +2)/2 and we getHs ⊆ e(Akθ)r.

We get the following corollary.

Corollary 1.6.10. Ñ

s∈RHs =e(A∞θ )r.

Proof. Supposea ∈ Ñ s∈RH

s. Then for anyk

Z≥0, a® ∈Hk+2, so by the theorem we just proved,a®∈ e(Akθ)r. Consequentlya®∈e(A∞θ )r, soÑ

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Suppose a ∈ e(A∞θ )r. Then since Hs is the completion of e(A

θ )r with respect to || · ||s,e(A∞θ )r ⊆ Hs for alls ∈R, ande(A∞θ )r ⊆ Ñs∈RH

s.

We can also prove an analog of the Rellich lemma onE.

Theorem 1.6.11. Let{ ®aN} ∈e(A∞θ )rbe a sequence. Suppose that there is a constant

C so that|| ®aN||s ≤ C for all N. Let s > t. Then there is a subsequence{ ®aNj} that

converges inHt.

Proof. Let{vj : j ∈ {1, . . . ,r}}be the standard basis of(A∞θ )r considered as a free

A∞θ -module. Let{fj = evj : j ∈ {1, . . . ,r}}be a generating set of the eigenvalue 1

eigenspace ofeconsidered as an endomorphism of(A∞θ)r. Note that{f jÎgU

mg

g :

m ∈Zn,j ∈ {1, . . . ,r}}spansE as aC-vector space, and fjÎgU

mg

g and fkÎgU

ng

g

are orthogonal. Consider the C-vector subspace spanned by {fjÎgU

mg

g ,m ∈ Zn}

for fixed j. It has countable dimension. Let {fjaj,m : m ∈ I} be a basis of this

subspace with index setI ⊆ Znwhereaj,m ∈ A∞θ . We can apply Gram–Schmidt with

respect to h·,·i and get an orthonormal basisFj := {fj,m : m ∈ I}. The subspaces

for different j are still orthogonal, so F := {fj,m : m ∈ I,j ∈ {1, . . . ,r}} is an orthogonal basis of E considered as a C-vector space, normalized with respect to

h·,·i. Letes,j,m :=(1+|m1|2+· · ·+|mn|2)−s/2fj,mand

Es := {es,j,m :m∈ I,j ∈ {1, . . . ,r}}.

By definition, Es is an orthogonal basis with respect to h·,·is, so we can write ®

aN := Íh,kes,h,kaN,h,k. Then |aN,h,k|2 ≤ Íh,k|aN,h,k|2 ≤ C2 and |aN,h,k| ≤ C.

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a subsequence {aNj,h,k} of {aN,h,k} such that for any > 0 there exists M() ∈

N such that |aNi,h,k − aNj,h,k| < whenever i,j ≥ M(). Do this for all g ∈

{0,1}, 1 ≤ h ≤ r, and |k1|2+· · ·+ |kn|2 ≤ R2, replacing {aN} with {aNj} each time. Then we get a subsequence {aNj} of {aN} such that for any > 0 there exists M() ∈ N such that, for all 1 ≤ h ≤ r and |k1|2 + · · · + |kn|2 ≤ R2, |aNi,h,k−aNj,h,k| < wheneveri,j ≥ M(). Now consider the sum||aNi −aNj||

2

t = Í

h,k|aNi,h,k−aNj,h,k|

2(1+ |k

1|2+· · ·+|kn|2)t−s.Decompose it into two parts: one

where |k1|2 + · · · + |kn|2 > R2 and one where |k1|2 + · · · + |kn|2 ≤ R2. On |k1|2 + · · · + |kn|2 > R2 we estimate (1 + |k1|2 + · · · + |kn|2)t−s < (1 + R2)t−s

so that Í h

Í

|k1|2+···+|kn|2≥R2|aNi,h,k − aNj,h,k|

2(1+ |k

1|2 + · · · + |kn|2)t−s < (1 +

R2)t−sÍ

h,k|aNi,h,k −aNj,h,k|

2 2rC2(1+R2)t−s. If >0 is given, we choose Rso

that 2rC2(1+R2)t−s < . The remaining part of the sum is over|k1|2+· · ·+|kn|2 ≤ R2

and can be bounded above by0:= −2rC2(1+R2)t−s ifi,j M(√n0/(

2R+1))

because ann-ball of radiusRcentered at the origin is contained in ann-cube of side length 2Rthat has(2R+1)nlattice points. Then the total sum is bounded above by , and we are done.

Acknowledgement

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C h a p t e r 2

A TWISTED LOCAL INDEX FORMULA FOR CURVED

NONCOMMUTATIVE TWO TORI

Available on arXiv underhttp://arxiv.org/licenses/nonexclusive-distrib/1.0/

2.1 Introduction

The celebrated Atiyah-Singer index theorem provides a local formula that calculates the analytical (Fredholm) index of the Dirac operator on a compact spin manifold, with coefficients in a vector bundle, in terms of topological invariants, namely characteristic classes [1–4]. The theorem has deep applications in mathematics and theoretical physics. For instance, it implies important theorems such as the Gauss-Bonnet theorem and the Riemann-Roch theorem, and it is used in physics to count linearly the number of solutions of fundamental partial differential equations.

In noncommutative differential geometry [15, 16], an analog of the Dirac operator is used to encode the metric information and to use ideas from spectral geometry and Riemannian geometry in studying a noncommutative algebra, viewed as the algebra of functions on a space with noncommuting coordinates. That is, the data

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ordinary manifold.

The identity of the analytical index and the topological index defined by the Connes-Chern character was established in the noncommutative setting in [15]. Moreover, a local formula for the Connes-Chern character, which is suitable for explicit cal-culations, was derived in [11]. However, as we shall elaborate further in §2.3, the notion of a spectral triple is suitable for type II algebras in Murray-von Neumann classification of algebras, and not for type III cases. The remedy brought forth in [21] for studying the latter is the notion of atwisted spectral triple: they have shown that a twisted version of spectral triples can incorporate type III cases and examples that arise in noncommutative conformal geometry, and that the index pairing and the coincidence of the analytic and the topological index continues to hold in the twisted case. However, a general local formula for the index in the twisted case has not yet been found, except for the special case of twists afforded by scaling automorphism in [55]. In fact, the difficulty of the problem of finding a general local index formula for twisted spectral triples raises the need for treating more examples.

The main result of the present article is a local formula for the index of the Dirac operatorD+e,σ of a twisted spectral triple on the opposite algebra of the noncommu-tative two torus T2θ, which is twisted by a general noncommutative vector bundle

represented by an idempotente. The Dirac operator that we consider is associated with a general metric in the canonical conformal class on T2θ. Our approach is

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[13] forC∗-dynamical systems. It should be noted that, following the Gauss-Bonnet theorem proved in [23] for the canonical conformal class on T2θ and its extension

in [30] to general conformal classes, study of local geometric invariants of curved metrics on noncommutative tori has received remarkable attention in recent years [19, 22, 25, 32, 33, 49, 51]. For an overview of these developments one can refer to [34, 48].

This article is organized as follows. In §2.2 we provide background material about the noncommutative two torus, its pseudodifferential calculus, and heat kernel meth-ods. In §2.3 we explain our construction of a vector bundle over the noncommutative two torus using a twisted spectral triple withσ-connections and derive the symbols of the operators necessary for the index calculation. In §2.4 we calculate the explicit local terms that give the index. The appendix contains proofs of new rearrangement lemmas, which overcome the new challenges posed in our calculations due to the presence of an idempotent as well as a conformal factor in our calculations in the noncommutative setting.

2.2 Preliminaries

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Noncommutative two torus

For a fixed irrational numberθ, the noncommutative two torusT2θis a noncommuta-tive manifoldwhose algebra ofcontinuousfunctionsC(T2

θ)is the universal (unital)

C∗-algebra generated by two unitary elementsUandVthat satisfy the commutation relation

VU =e2πiθUV.

The ordinary two torusT2 =(R/2πZ)2acts onC(T2θ)by

αs(UmVn)= eis·(m,n)UmVn, s ∈T2, m,n∈Z. (2.1)

Since this action, in principle, comes from translation bys ∈T2written in the Fourier mode, its infinitesimal generatorsδ1, δ2 :C∞(T2θ) →C∞(T2θ)are the derivations that are analogs of partial differentiations on the ordinary two torus and are given by the defining relations

δ1(U)=U, δ1(V)=0, δ2(U)= 0, δ2(V)=V.

The spaceC∞(T2θ)of thesmoothelements consists of all elements inC(T2θ)that are smooth with respect to the action α given by (2.1), which turns out to be a dense subalgebra ofC(T2θ)that can alternatively be described as the space of all elements of the formÍ

m,n∈Zam,nUmVnwith rapidly decaying complex coefficients am,n. The analog of integration is provided by the linear functionalτ :C(T2θ) → Cwhich is the bounded extension of the linear functional that sends any smooth element with thenoncommutative Fourier expansionÍ

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volume form of a flat canonical metric onT2θ. In §2.3, we will explain how one can

consider a general metric in the conformal class of the flat canonical metric on T2θ

by means of a positive invertible elemente−h, wherehis a self-adjoint element in C∞(T2θ).

Heat kernel expansion

We will explain in §2.3 that our method of calculation of a noncommutative local index formula, which is the main result of this article, is based on the McKean-Singer index formula [54] and calculation of the relevant terms in small time heat kernel expansions. Here we elaborate on the derivation of the small heat kernel expansion for an elliptic differential operator of order 2 onC(T2θ). The main tool that will be used is the pseudodifferential calculus developed in [13] forC∗-dynamical systems (see also [41, 67]). This calculus, in the case ofT2θ, associates to apseudodifferential symbol ρ:R2→ C∞(T2θ)a pseudodifferential operator Pρ:C∞(T2θ) → C∞(T2θ)by the formula

Pρ(a)= 1 (2π)2

R2 ∫

R2

e−is·ξρ(ξ)αs(a)ds dξ, a ∈C∞(T2θ).

For example, any differential operator given by a finite sum of the formÍ

aj1,j2δ

j1

1 δ

j2

2

withaj1,j2 ∈ C

(

T2θ)is associated with the polynomial Íaj1,j2ξ

j1

1 ξ

j2

2 . In general a smooth map ρ:R2 →C∞(T2θ)is a pseudodifferential symbol of orderm∈ Zif for any non-negative integersi1,i2,j1,j2there is a constantCsuch that

||∂j1

1 ∂

j2

2 δ

i1

i2

2ρ(ξ)|| ≤ C(1+|ξ|)

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where ∂1 and ∂2 are respectively the partial differentiations with respect to the coordinatesξ1andξ2ofξ ∈R2. We denote the space of symbols of ordermbySm.

A symbol ρof ordermisellipticif ρ(ξ)is invertible for larger enoughξ and there is a constantcsuch that

||ρ(ξ)−1|| ≤ c(1+|ξ|)−m.

In the case of differential operators, one can see that the symbol is elliptic if its leading part is invertible away from the origin.

An important feature of an elliptic operator is that it admits aparametrix, namely an inverse in the algebra of pseudodifferential operators modulo infinitely smoothing operators. The latter are the operators whose symbols belong to the intersection of all symbolsS−∞ = ∩mZSm. This illuminates the importance of pseudodifferential calculus for solving elliptic differential equations.

For our purposes in this article, it is crucial to illustrate that given a positive elliptic differential operator 4 of order 2 on C∞(T2

θ), one can use the pseudodifferential

calculus to derive a small time asymptotic expansion for the trace of the heat kernel exp(−t4). Note that the pseudodifferential symbol of4is of the form

σ4(ξ)= p2(ξ)+p1(ξ)+p0(ξ),

where each pk is homogeneous of order k inξ. Since the eigenvalues of4 are real

and non-negative, using the Cauchy integral formula and a clockwise contourγthat goes around the non-negative real line, one can write

exp(−t4)= 1

2πi

γe

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Since 4 −λ is an elliptic operator, its parametrix Rλ will approximate (4 −λ)−1

appearing above in the integrand. Since4 −λis of order 2, one can write for the symbol ofRλ:

σRλ(ξ, λ,4)=b0(ξ, λ,4)+b1(ξ, λ,4)+b2(ξ, λ,4)+· · · ,

where eachrj is of order−2− j.

Then, by considering the ellipticity of 4 −λ and the following composition rule, which gives an asymptotic expansion for the symbol of the composition of two pseudodifferential operators,

σP1P2 ∼

Õ

i1,i2∈Z≥0

1 i1!i2!∂

i1

1∂

i2

2σP1δ

i1

i2

2σP2,

one can find a recursive formula for therj. That is, it turns out that

b0(ξ, λ,4)=(p2(ξ) −λ)−1, (2.3)

and for anyn∈Z≥1,

bn(ξ, λ,4)=− © ­ ­ ­

«

Õ

i1+i2+j+2−k=n

0≤j<n,0≤k≤2 1 i1!i2!

∂i1

1∂

i2

2bj(ξ, λ,4)δ

i1

i2

2(pk(ξ))

ª ® ® ®

¬

b0(ξ, λ,4). (2.4)

By calculating these terms, one can replace (4 −λ)−1 in (2.2) by Rλ and find an approximation of exp(−t4).

Then, calculating the trace of this approximation, one can derive a small time asymptotic expansion for Tr(exp(−t4)). That is, one finds that there are elements

a2n(4)=

1 2πi

R2 ∫

γe

−λ

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such that ast →0+, for anya ∈C∞(T2θ):

Tr(aexp(−t4)) ∼t−1 ∞

Õ

n=0

τ(a a2n(4))tn. (2.6)

The termsa2n(4)are local geometric invariants associated with the operator4when

it is a natural geometric operator such as a Laplacian, or the square of a twisted Dirac operator as in this article.

2.3 Noncommutative geometric spaces and index theory

Spectral triples

Geometric spaces are defined in terms of spectral data in noncommutative geometry [16]. A noncommutative geometric space is a spectral triple (A,H,D), where A is an involutive algebra represented by bounded operators on a Hilbert space H, and Dis the analog of the Dirac operator. That is, D is an unbounded self-adjoint operator on its domain which is dense in the Hilbert space H, and the spectrum of D has similar properties to the spectrum of the Dirac operator on a compact Riemannian spinc manifold, or even more generally, that of an elliptic self-adjoint differential operator of order 1 on a compact manifold. Thespectral dimensionof such a triple is the smallest positive real numberd such that|D|−d is in the domain

of theDixmier traceTrω[26] (or one can say that for any > 0, the operator|D|−d−

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in the operatorD.

The main classical example of this setup is the triple C∞(M),L2(S),Dg

, where C∞(M)is the algebra of smooth complex-valued functions on a compact Riemannian manifold with a spinc structureS, L2(S)is the Hilbert space of the L2-spinors, and Dg is the Dirac operator associated with the metric, which acts on its domain in

L2(S)and squares to a Laplace-type operator, see for example [5] for details about the Dirac operator. SinceSis a vector bundle overM, the algebraC∞(M)acts naturally by bounded operators on the Hilbert spaceL2(S), namely: (f·s)(x):= f(x)s(x), f ∈

C∞(M),s ∈ L2(S),x ∈ M. More importantly, the latter action interacts in a bounded manner with the Dirac operatorDgin the sense that the commutator ofDgwith the

action of each f ∈C∞(M)is a bounded operator as[Dg, f]= Dgf − f Dg =c(df),

where c(df) denotes the Clifford multiplication on S by the de Rham differential of f. By construction, the Dirac operator Dg depends heavily on the Riemannian

metricgonM. It is known from classical facts in spectral geometry that the spectral dimension of the spectral triple C∞(M),L2(S),Dg is equal to the dimension of

the manifold M, and that the important local curvature related information can be detected in the small time asymptotic expansions of the form

Trf exp(−t Dg2) ∼t→0+ t−dimM/2 ∞

Õ

j=0 tj

M

f(x)a2j(x)dx, f ∈C∞(M),

(2.7) where the densities a2j(x)dx are uniquely determined by the Riemann curvature

tensor and its contractions and covariant derivatives, cf. [20].

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in studying noncommutative algebras, is fully illustrated in [18] by showing that the Dirac operatorDgcontains the full metric information. That is, it is shown that any

spectral triple(A,H,D)whose algebraAis commutative, and satisfies suitable con-ditions, is equivalent to the spectral triple C∞(M),L2(S),Dg

of a Riemannian spinc manifold. In fact, following the Gauss-Bonnet theorem for the noncommutative two torus proved in [23] and its extension in [30], the calculation and conceptual un-derstanding of the local curvature terms in noncommutative geometry has attained remarkable attention in recent years [6, 19, 22, 25, 27, 32, 33, 49–51]. In these studies, the noncommutative local geometric invariants are detected in the analogs of the small time asymptotic expansion (2.7) written for spectral triples.

Twisted spectral triples

It turns out that the notion of a spectral triple (A,H,D) is suitable for studying algebras that possess a non-trivial trace, and a twisted notion of spectral triples is proposed in [21] to incorporate algebras that do not have this property. The reason is that if(A,H,D)is a spectral triple with spectral dimensiond, then using the Dixmier trace Trω, one can define the linear function φ : A → C by φ(a) = Trω(a|D|−d), which turns out to be a trace, under the minimal regularity assumption that the commutators [|D|,a] are also bounded for any a ∈ A. The main reason for the trace property φ(ab) = φ(ba),a,b ∈ A, is that for any a ∈ A, the commutator

References

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