Pseudodifferential analysis in {_Y63*-algebras [psi*-algebras] on transmission spaces, infinite solving ideal chains and K-theory for conformally compact spaces [Elektronische Ressource] / Jochen Alexander Ditsche

Sujets

• Mathematics

Informations

• Publié par: johannes_gutenberg-universitat_mainz
• Publié le: 01 janvier 2008
• Nombre de lectures: 5
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

∗Pseudodifferential analysis in Ψ -algebras on transmission
spaces, infinite solving ideal chains andK-theory for
conformally compact spaces
Dissertation
zur Erlangung des Grades
”Doktor
der Naturwissenschaften”
am Fachbereich ”Physik, Mathematik und Informatik”
der Johannes Gutenberg-Universit¨at
in Mainz
Jochen Alexander Ditsche
geb. in Du¨sseldorf
Mainz, den 9. August 2007Tag der mu¨ndlichen Pru¨fung: 15. M¨arz 2008
(D77) Dissertation, Johannes Gutenberg-Universit¨at MainzSummary
The present thesis is a contribution to the theory of algebras of pseudodifferential operators
on singular settings. In particular, we focus on the b-calculus and the calculus on conformally
compact spaces in the sense of Mazzeo and Melrose in connection with the notion of spectral
invariant transmission operator algebras.
We summarize results given by Gramsch, Ueberberg and Wagner [46] and Lauter [60] on the
∗construction of Ψ - and Ψ -algebras and the corresponding scales of generalized Sobolev spaces0
using commutators of certain closed operators and derivations.
1
∗ b 2In the case of a manifold with corners Z we construct a Ψ -completion A (Z, Ω ) of theb
1
0 b ∗2algebraofzeroorderb-pseudodifferentialoperatorsΨ (Z, Ω )inthecorrespondingC -closureb,cl
1 1
b 2 b
2 2B(Z, Ω ),→L(L (Z, Ω )). The construction will also provide that localised to the (smooth)
1b˚ 2interiorZ ofZ theoperatorsintheA (Z, Ω )canberepresentedasordinarypseudodifferentialb
operators.
∗InconnectionwiththenotionofsolvableC -algebras-introducedbyDynin[32]-wecalculate
1 1 1∗ 0 b E(F) b E(F) b
2 2 2the length of the C -closure of Ψ (F, Ω ,R ) inB(F, Ω ,R ) by localizingB(Z, Ω )
b,cl
Balong the boundary face F using the (extended) indical familiy I . Moreover, we discuss howFZ
one can ”localise” solving ideal chains in neighbourhoods U of arbitrary points p ∈ Z. Thisp
localisationprocesswillrecoverthesingularstructureofU ; further, theinducedlengthfunctionp
l is shown to be upper semi-continuous.p
∗ ∗We give construction methods for Ψ - and C -algebras admitting only infinite long solving
ideal chains. These algebras will first be realized as unconnected direct sums of (solvable)
∗C -algebras and then refined such that the resulting algebras have arcwise connected spaces
of one dimensional representations. In addition, we recall the notion of transmission algebras
on manifolds with corners (Z ) following an idea of Ali Mehmeti [3]. Thereby, we connecti i∈N
∞the underlyingC -function spaces using point evaluations in the smooth parts of the Z andi
use generalized Laplacians to generate an appropriate scale of Sobolev spaces. Moreover, it
is possible to associate generalized (solving) ideal chains to these algebras, such that to every
n∈N there exists an ideal chain of length n within the algebra.
Finally, we discuss the K-theory for algebras of pseudodifferential operators on conformally
compact manifolds X and give an index theorem for these operators. In addition, we prove
that the Dirac-operator associated to the metric of a conformally compact manifold X is not a
Fredholm operator.
iContents
1 Commutator methods 11
∗1.1 Basic results on Ψ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 11
∗1.2 Generating Ψ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
∗1.3 Generating Ψ -algebras by commutator methods . . . . . . . . . . . . . . . 21
1.4 Order shift algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
∗2 Ψ -algebras on manifolds with corners 29
2.1 Review of algebras of operators on manifolds with corners . . . . . . . . . 29
1∗ 0 b
22.2 Ψ -completions of Ψ (Z, Ω ) . . . . . . . . . . . . . . . . . . . . . . . . 34b,cl
2.3 b-Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
∗ ∗3 Localisation of C - and Ψ -chains 41
∗ ∗3.1 Representations of C - and Ψ -algebras . . . . . . . . . . . . . . . . . . . . 41
∗ ∗3.2 Solvable C - and Ψ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . 45
1b E(F)
23.3 The length ofB(F, Ω ,R ) . . . . . . . . . . . . . . . . . . . . . . . . . 47
1b
23.4 The local length ofB(Z, Ω ) . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Infinite solving series 59
4.1 Infinite ideals on direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Connecting the product algebra . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Transmission algebras with infinitely long ideal chains . . . . . . . . . . . . 67
4.3.1 Transmission spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.2 Ideal chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.3 A refined transmisson model . . . . . . . . . . . . . . . . . . . . . . 73
5 K-theory for conformally compact spaces 77
5.1 Algebras on conformally compact spaces . . . . . . . . . . . . . . . . . . . 77
∗5.2 Review of C -algebras of b-c-operators . . . . . . . . . . . . . . . . . . . . 81
5.3 K-theory for operators on conformally compact spaces . . . . . . . . . . . 87
5.4 Index of fully elliptic operators . . . . . . . . . . . . . . . . . . . . . . . . 95
5.5 The Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
∗5.6 K-theory for Ψ -algebras on conformally compact spaces . . . . . . . . . . 103
A K-Theory 105
A.1 K-theory for certain spectrally invariant algebras . . . . . . . . . . . . . . 105
B Boutet de Monvel’s algebra 111
B.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
iiiiv Contents
B.2 K-theory for Boutet de Monvels algebra . . . . . . . . . . . . . . . . . . . 113
B.3 The structure elements of the 0-calculus revisited . . . . . . . . . . . . . . 113
C Some remarks on the 0-calculus 119
1 12 b,c 1 b,c
2 2C.1 A remark on L (M, Ω ) andH (M, Ω ) . . . . . . . . . . . . . . . . . 119b,c
C.2 0-metrics that are not isometric . . . . . . . . . . . . . . . . . . . . . . . . 120
∗D Representations of hereditary C -algebras 123
D.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
D.2 Hereditary subalgebras and their spectrum . . . . . . . . . . . . . . . . . . 124
E Beals and Coifman-Mayer 127
E.1 Notations and prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . 127
E.2 Local classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
E.3 Global classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Bibliography 143
List of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153Conventions
• Sections are denoted by pairs of numbers like 3.4 and definitions, theorems, etc. by
triples of numbers, e.g. theorem 3.4.5 in section 3.4.
Equations are denoted by triples of numbers in parentheses like formula (3.4.5) in
section 3.4.
• Unless otherwise indicated, Banach algebras and spaces are always considered over
C.
• Unless otherwise indicated, functions are assumed to be complex valued in this
thesis.
n n• S(R ) denotes the Schwartz space on R , i.e. the space of rapidly decreasing
nfunctionsR −→C.
• LetE andF be Banach or Fr´echet spaces. ThenL(E,F) denotes the space of all
continuous linear mapsE −→F endowed with the usual operator norm.
• Let T be a topological space and let ϕ,ψ : T −→C be two mappings. We write
ϕ≺ψ, if ψ≡ 1 on suppϕ; in particular ϕ(x) =ϕ(x)ψ(x) holds for all x∈ suppϕ.
• ByR we denote the positive half axis [0,∞[⊆R.+
vIntroduction
The present thesis is a contribution to the theory of pseudodifferential operators on man-
ifolds with corners, conformally compact spaces and transmission spaces in connection
with spectral invariance:
1
∗ b
2(i) ForamanifoldwithcornersZ weshowhowtoconstructaΨ -completionA (Z, Ω )b
10 b
2of Ψ (Z, Ω ) using abstract construction concepts introduced by Gramsch, Ue-b,cl
1b
2berbergandWagner[46]. ThealgebraA (Z, Ω )willbedenseinthecorrespondingb
1 1∗ b 0 b
2 2C -closureB(Z, Ω )ofΨ (Z, Ω )andlocalizedintheinteriorofZ theoperatorsb,cl
1 1b b
2 2inA (Z, Ω )willbeordinarypseudodifferentialoperators. ThealgebraA (Z, Ω )b bT 1
s b
2itselfwillbecontainedinthespace L(H (Z, Ω ))ofalloperatorsoforder0withb
s
1s b
2respect to the scaleH (Z, Ω ) of b-Sobolev spaces.b
∗ ∗(ii) Wegivethedefinitionofsolvable Ψ -algebras andshowthatdenseΨ -subalgebrasA
∗ofC -algebrasB are solvable, provided they fulfil the operator theoretical condition
that if I E B is a closed ideal in B, then I∩A is dense in I (this will be called
property E ).0
1
∗ b l
2(iii) We calculate the length of theC -closureB(F, Ω ,R) of the parameter-dependent
10 b l
2calculus Ψ (F, Ω ,R) ofb-pseudodifferentialoperators on manifolds with cornersb,cl
F. To achieve this, we embed F into a larger manifold with cornersZ and use the
1 1b E(F) b∼2 2isomorphismB(F, Ω ,R ) B(Z, Ω )/I , whereI denotes the kernel of the= F F
indical family associated to F.
(iv) We define the notion of the local length l in p ∈ Z for certain classes of solvablep
1∗ b ∗
2C -algebras and show how to calculate it for B(Z, Ω ) using the C -subalgebra
1b
2B (Z, Ω ), where ϕ is an appropriate cut off function with ϕ≡ 1 in a neighbour-ϕ
hood