Holdings: Elliptic theory and noncommutative geometry

Elliptic theory and noncommutative geometry: nonlocal elliptic operators

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Bibliographic Details
Main Authors:	Nazajkinskij, Vladimir E. 1955- (Author), Savin, Anton Ju (Author), Sternin, Boris Ju. 1939- (Author)
Format:	Book
Language:	English
Published:	Basel [u.a.] Birkhäuser 2008
Series:	Operator theory 183 : Advances in partial differential equations
Subjects:	Elliptischer Differentialoperator Nichtkommutative Geometrie
Online Access:	Inhaltsverzeichnis Inhaltsverzeichnis
Physical Description:	XII, 224 S. graph. Darst.
ISBN:	9783764387747 3764387742 9783764387754

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245	1	0	\|a Elliptic theory and noncommutative geometry \|b nonlocal elliptic operators \|c V. E. Nazaikinskii ; A. Yu. Savin ; B. Yu. Sternin
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Record in the Search Index

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adam_text	Contents Preface xi Introduction 1 I Analysis of Nonlocal Elliptic Operators 3 1 Nonlocal Functions and Bundles 5 1.1 Group Algebras and Crossed Products ................ 5 1.1.1 Group Algebras ......................... 5 1.1.2 ^-Crossed Products ...................... 7 1.1.3 Isomorphism Theorem ..................... 11 1.1.4 Smooth Crossed Products ................... 13 Motivation ........................... 13 Sufficient Condition for Locality ............... 14 Groups of Polynomial Growth ................. 15 Tempered Actions ....................... 16 Schweitzer theorem ....................... 17 1.2 Nonlocal Functions ........................... 18 1.2.1 Assumptions about the Group Γ ............... 18 1.2.2 Continuous Nonlocal Functions ................ 18 Nonlocal Functions as Operators in L2 ............ 19 Changes of Variables and Substitutions ........... 19 1.2.3 Smooth Nonlocal Functions .................. 20 1.3 Nonlocal Bundles ............................ 22 1.4 Case of Noncompact Spaces ...................... 22 1.4.1 Nonlocal Functions ....................... 23 1.4.2 Special Noncompact Manifolds ................ 24 2 Nonlocal Elliptic Operators 27 2.1 Scalar Operators ............................ 27 2.1.1 Symbol Algebras ........................ 27 vi Contents 2.1.2 Quantization of Symbols and Operators of Order Zero ... 27 Quantization of Continuous Symbols ............. 28 Quantization of Smooth Symbols ............... 29 2.1.3 Operators of Arbitrary Order ................. 30 2.2 Operators in Spaces of Sections of Bundles ............. 32 2.2.1 Operators of Order Zero .................... 32 2.2.2 Operators of Arbitrary Order ................. 34 2.2.3 Operators on Noncompact Manifolds ............. 35 2.3 Symbols as Nonlocal Functions on TM ............... 36 3 Elliptic Operators over C-Algebras 37 3.1 Hubert Modules and A-Index ..................... 38 3.1.1 Hubert Modules ........................ 38 3.1.2 Operators in Hubert Modules ................. 41 3.1.3 Fredholm Property and Index ................. 42 3.2 Pseudodifferential Operators over Λ ................. 44 3.2.1 Л -bundles and Section Spaces ................. 44 Л -bundles............................ 44 Space of Smooth Sections ................... 45 Space L2 ............................ 46 Sobolev Spaces ......................... 46 3.2.2 Symbols and Pseudodifferential Operators .......... 47 3.2.3 Ellipticity and the Л -Fredholm property ........... 50 3.3 Nonlocal Pseudodifferential Operators over Λ ............ 51 3.3.1 Operators in Trivial Л -Bundles................ 51 3.3.2 Operators in General Л -Bundles............... 52 3.3.3 Symbols in Trivial Bundles .................. 52 3.3.4 Symbols in Nontrivial Bundles ................ 54 3.3.5 Ellipticity and the Л -Fredholm Property ........... 54 II Homotopy Invariants of Nonlocal Elliptic Operators 57 4 Homotopy Classification 59 4.1 Ell-Group ................................ 59 4.2 Difference Construction ........................ 60 4.3 Isomorphism of Ell- and ir-Groups .................. 61 5 Analytic Invariants 63 5.1 Fredholm Index ............................. 63 5.2 C(r)-Index and Its Connection with the Fredholm Index ..... 63 Contents vii 6 Bott Periodicity 67 6.1 Preliminary Remarks .......................... 67 6.2 Exterior Product of Operators .................... 68 6.3 Euler Operator and the Bott Element ................ 69 6.4 Bott Mapping and the Periodicity Theorem ............. 70 6.5 Proof of the Periodicity Theorem ................... 71 7 Direct Image and Index Formulas in if-Theory 75 7.1 Direct Image Mapping in if-Theory for Embeddings ........ 75 7.1.1 Exterior Products ....................... 75 7.1.2 Normal Bundle ......................... 76 7.1.3 Definition of the Direct Image Mapping ........... 77 7.1.4 Л -Index Is Preserved by the Direct Image Mapping ..... 78 7.2 Index Formulas in iř-Theory ..................... 79 7.2.1 Direct Image in iř-Theory for the Projection into a Point . 79 7.2.2 Index Formulas ......................... 79 7.3 Proof of Theorem 7.7.......................... 79 7.3.1 Exterior Products of Operators ................ 80 7.3.2 Homotopy ............................ 82 7.3.3 Completion of the Proof .................... 86 8 Chern Character 89 8.1 Differential Forms and Graded Traces ................ 89 8.1.1 Noncommutative Differential Forms ............. 89 8.1.2 Graded Trace on the Algebra of Noncommutative Differential Forms ....................... 90 8.2 Chern Character of Projections .................... 92 8.3 Chern Character of Symbols ...................... 94 8.4 How to Compute the Chern Character ................ 95 8.4.1 Computation in Terms of the Symbol ............ 95 8.4.2 Computation in Terms of Connections ............ 96 Graded Trace .......................... 96 Chern Character of Projections ................ 96 Chern Character of Symbols ................. 97 9 Cohomological Index Formula 99 9.1 Todd Class ............................... 99 9.2 Index Theorem ............................. 101 9.2.1 Topological Index ....................... 101 9.2.2 Chern Character on the Group K0(A) ............ 102 9.2.3 Statement of the Theorem ................... 102 9.3 Vanishing Theorem ........................... 103 9.4 Proof of the Index Theorem ...................... 105 viii Contents 9.5 Proofs of Auxiliary Statements .................... 108 9.5.1 Multiplicativity of the Chern Character ........... 108 Case 1. Compact Manifold .................. 108 Case 2. Noncompact Manifold ................. 110 9.5.2 Chern Character of the Symbol of the Euler Operator . . . 114 10 Cohomological Formula for the Л -Index 115 10.1 Noncommutative Differential Forms ................. 116 10.2 Graded Traces over A°° ........................ 117 10.2.1 Cohomology of Groups ..................... 118 10.2.2 Construction of Traces ..................... 119 10.2.3 Examples ............................ 119 Traces of Degree 0....................... 119 Trace of Degree 1 for the Group Z .............. 120 Trace of Degree 2 for the Group Z2 .............. 120 10.3 Graded Traces over C°°(X) χ Γ.................... 120 10.4 Chern Character and the Index Formula ............... 122 10.4.1 Chern Character ........................ 122 10.4.2 Index Formula ......................... 123 10.4.3 Proofs of Auxiliary Statements ................ 123 11 Index of Nonlocal Operators over C-Algebras 127 11.1 Classification of Nonlocal Elliptic Operators ............. 128 11.1.1 Ell-Group ............................ 128 11.1.2 Difference Construction .................... 128 11.1.3 Isomorphism of the Groups Ell and if ............ 128 11.1.4 Künneth Formula and Classification Modulo Torsion .... 129 11.2 Chern Character and the Index Theorem .............. 130 11.2.1 Chern Character ........................ 130 11.2.2 Index Theorem ......................... 131 Case 1. Product of Elements of Even ii-Groups ....... 131 Case 2. Product of Elements of Odd A -Groups ....... 132 III Examples 137 12 Index Formula on the Noncommutative Torus 139 12.1 Operators on the Noncommutative Torus .............. 139 12.2 Index Computation ........................... 140 12.2.1 Reduction to the Two-Dimensional Torus .......... 140 12.2.2 Index of Operators on the 2-Torus .............. 141 12.3 Special Cases .............................. 142 Contents їх 13 An Application of Higher Traces 145 13.1 Index with Values in Odd AT-Groups................. 145 13.2 Odd Index Formula ........................... 146 13.2.1 Suspension........................... 147 13.2.2 Index Theorem......................... 147 Step 1. Reduction to the Cosphere Bundle and the Chern- Simons Character .................. 148 Step 2. Desuspension and Integration over S1 ........ 148 13.3 Example of Л -Index Computation .................. 149 14 Index Formula for a Finite Group Γ 153 14.1 Trajectory Symbol ........................... 153 14.2 Index Formula ............................. 154 IV Appendices 157 A ^-Algebras 161 A.I Basic Notions .............................. 161 A. 1.1 Definitions and Examples ................... 161 A.I. 2 Unital Algebras and Units ................... 162 A.1.3 Homomorphisms, Ideals, Quotient Algebras, and Extensions 163 A. 1.4 Commutative C-Algebras ................... 165 A. 1.5 Spectrum and Functional Calculus .............. 165 A.1.6 Local C-Algebras ....................... 167 A.l.ľ Positive Elements ........................ 167 A. 1.8 Projections in C-Algebras .................. 168 A.2 Representations of C-Algebras .................... 169 A.2.1 Basic Definitions ........................ 169 A.2.2 Existence of Representations ................. 172 A.2.3 Representations of Ideals and Quotient Algebras ...... 173 A.2.4 Primitive Ideals ......................... 174 A.2.5 Algebras of Type I ....................... 175 A.3 Tensor Products and Nuclear Algebras ................ 176 A.3.1 Minimal and Maximal Tensor Products ........... 176 A.3.2 Nuclear Algebras ........................ 177 A.3.3 Primitive Ideals in the Tensor Product ............ 178 B Ä-Theory of Operator Algebras 179 B.I Covariant A -Theory .......................... 179 B.I.I Topological A -Theory ..................... 179 B.1.2 Group K0(A) .......................... 182 B.1.3 Group Α ,(Λ) .......................... 184 B.1.4 Bott Periodicity ........................ 186 x Contents В. 1.5 Long Exact Sequence in if-Theory .............. 187 Homomorphism di ....................... 187 Homomorphism do ....................... 188 B.1.6 Stability of K-Groups..................... 188 B.I. 7 K-groups of Local C-Algebras ................ 189 B.2 K-Homology .............................. 189 B.2.1 ii-Homology of a Topological Space ............. 189 Even Groups .......................... 191 Odd Groups ........................... 191 B.2. 2 if-Homology of Operator Algebras: Definitions ....... 192 Fredholm Modules ....................... 192 Dual Algebras ......................... 193 Extensions of C-Algebras ................... 195 Equivalence of Different Definitions .............. 196 B.2.3 Suspension and Bott Periodicity ............... 197 B.2. 4 Long Exact Sequence in if-Homology ............ 198 B.2.5 Stability ............................. 199 B.2. 6 Duality between if-Homology and if-Theory of Operator Algebras ............................. 199 Even Groups .......................... 199 Odd Groups ........................... 199 С Cyclic Homology and Cohomology 201 C.I Cyclic Cohomology ........................... 201 C.I.I Enveloping Differential Algebra ................ 201 C.I. 2 Graded Traces and Cyclic Functionals ............ 203 C.1.3 Cochains, Cyclic Cochains, and the Hochschild Differential 204 C.I. 4 Cyclic Cohomology and Hochschild Cohomology ...... 204 C.1.5 Example ............................. 205 C.1.6 Cup Product in Cyclic Cohomology ............. 206 C.I. 7 Periodicity in Cyclic Cohomology ............... 207 C.1.8 Example ............................. 209 C.2 Cyclic Homology ............................ 209 C.2.1 Definition of Cyclic Homology ................. 209 C.2.2 Periodicity ........................... 210 C.2.3 Pairing between Cyclic Homology and Cohomology ..... 211 C.2.4 Morita Invariance ....................... 211 C.2. 5 Chern Characters in Homology ................ 211 Concise Bibliographical Remarks 213 Bibliography 217 Index 222
adam_txt	Contents Preface xi Introduction 1 I Analysis of Nonlocal Elliptic Operators 3 1 Nonlocal Functions and Bundles 5 1.1 Group Algebras and Crossed Products . 5 1.1.1 Group Algebras . 5 1.1.2 ^-Crossed Products . 7 1.1.3 Isomorphism Theorem . 11 1.1.4 Smooth Crossed Products . 13 Motivation . 13 Sufficient Condition for Locality . 14 Groups of Polynomial Growth . 15 Tempered Actions . 16 Schweitzer theorem . 17 1.2 Nonlocal Functions . 18 1.2.1 Assumptions about the Group Γ . 18 1.2.2 Continuous Nonlocal Functions . 18 Nonlocal Functions as Operators in L2 . 19 Changes of Variables and Substitutions . 19 1.2.3 Smooth Nonlocal Functions . 20 1.3 Nonlocal Bundles . 22 1.4 Case of Noncompact Spaces . 22 1.4.1 Nonlocal Functions . 23 1.4.2 Special Noncompact Manifolds . 24 2 Nonlocal Elliptic Operators 27 2.1 Scalar Operators . 27 2.1.1 Symbol Algebras . 27 vi Contents 2.1.2 Quantization of Symbols and Operators of Order Zero . 27 Quantization of Continuous Symbols . 28 Quantization of Smooth Symbols . 29 2.1.3 Operators of Arbitrary Order . 30 2.2 Operators in Spaces of Sections of Bundles . 32 2.2.1 Operators of Order Zero . 32 2.2.2 Operators of Arbitrary Order . 34 2.2.3 Operators on Noncompact Manifolds . 35 2.3 Symbols as Nonlocal Functions on TM . 36 3 Elliptic Operators over C-Algebras 37 3.1 Hubert Modules and A-Index . 38 3.1.1 Hubert Modules . 38 3.1.2 Operators in Hubert Modules . 41 3.1.3 Fredholm Property and Index . 42 3.2 Pseudodifferential Operators over Λ . 44 3.2.1 Л -bundles and Section Spaces . 44 Л -bundles. 44 Space of Smooth Sections . 45 Space L2 . 46 Sobolev Spaces . 46 3.2.2 Symbols and Pseudodifferential Operators . 47 3.2.3 Ellipticity and the Л -Fredholm property . 50 3.3 Nonlocal Pseudodifferential Operators over Λ . 51 3.3.1 Operators in Trivial Л -Bundles. 51 3.3.2 Operators in General Л -Bundles. 52 3.3.3 Symbols in Trivial Bundles . 52 3.3.4 Symbols in Nontrivial Bundles . 54 3.3.5 Ellipticity and the Л -Fredholm Property . 54 II Homotopy Invariants of Nonlocal Elliptic Operators 57 4 Homotopy Classification 59 4.1 Ell-Group . 59 4.2 Difference Construction . 60 4.3 Isomorphism of Ell- and ir-Groups . 61 5 Analytic Invariants 63 5.1 Fredholm Index . 63 5.2 C(r)-Index and Its Connection with the Fredholm Index . 63 Contents vii 6 Bott Periodicity 67 6.1 Preliminary Remarks . 67 6.2 Exterior Product of Operators . 68 6.3 Euler Operator and the Bott Element . 69 6.4 Bott Mapping and the Periodicity Theorem . 70 6.5 Proof of the Periodicity Theorem . 71 7 Direct Image and Index Formulas in if-Theory 75 7.1 Direct Image Mapping in if-Theory for Embeddings . 75 7.1.1 Exterior Products . 75 7.1.2 Normal Bundle . 76 7.1.3 Definition of the Direct Image Mapping . 77 7.1.4 Л -Index Is Preserved by the Direct Image Mapping . 78 7.2 Index Formulas in iř-Theory . 79 7.2.1 Direct Image in iř-Theory for the Projection into a Point . 79 7.2.2 Index Formulas . 79 7.3 Proof of Theorem 7.7. 79 7.3.1 Exterior Products of Operators . 80 7.3.2 Homotopy . 82 7.3.3 Completion of the Proof . 86 8 Chern Character 89 8.1 Differential Forms and Graded Traces . 89 8.1.1 Noncommutative Differential Forms . 89 8.1.2 Graded Trace on the Algebra of Noncommutative Differential Forms . 90 8.2 Chern Character of Projections . 92 8.3 Chern Character of Symbols . 94 8.4 How to Compute the Chern Character . 95 8.4.1 Computation in Terms of the Symbol . 95 8.4.2 Computation in Terms of Connections . 96 Graded Trace . 96 Chern Character of Projections . 96 Chern Character of Symbols . 97 9 Cohomological Index Formula 99 9.1 Todd Class . 99 9.2 Index Theorem . 101 9.2.1 Topological Index . 101 9.2.2 Chern Character on the Group K0(A) . 102 9.2.3 Statement of the Theorem . 102 9.3 Vanishing Theorem . 103 9.4 Proof of the Index Theorem . 105 viii Contents 9.5 Proofs of Auxiliary Statements . 108 9.5.1 Multiplicativity of the Chern Character . 108 Case 1. Compact Manifold . 108 Case 2. Noncompact Manifold . 110 9.5.2 Chern Character of the Symbol of the Euler Operator . . . 114 10 Cohomological Formula for the Л -Index 115 10.1 Noncommutative Differential Forms . 116 10.2 Graded Traces over A°° . 117 10.2.1 Cohomology of Groups . 118 10.2.2 Construction of Traces . 119 10.2.3 Examples . 119 Traces of Degree 0. 119 Trace of Degree 1 for the Group Z . 120 Trace of Degree 2 for the Group Z2 . 120 10.3 Graded Traces over C°°(X) χ Γ. 120 10.4 Chern Character and the Index Formula . 122 10.4.1 Chern Character . 122 10.4.2 Index Formula . 123 10.4.3 Proofs of Auxiliary Statements . 123 11 Index of Nonlocal Operators over C-Algebras 127 11.1 Classification of Nonlocal Elliptic Operators . 128 11.1.1 Ell-Group . 128 11.1.2 Difference Construction . 128 11.1.3 Isomorphism of the Groups Ell and if . 128 11.1.4 Künneth Formula and Classification Modulo Torsion . 129 11.2 Chern Character and the Index Theorem . 130 11.2.1 Chern Character . 130 11.2.2 Index Theorem . 131 Case 1. Product of Elements of Even ii-Groups . 131 Case 2. Product of Elements of Odd A'-Groups . 132 III Examples 137 12 Index Formula on the Noncommutative Torus 139 12.1 Operators on the Noncommutative Torus . 139 12.2 Index Computation . 140 12.2.1 Reduction to the Two-Dimensional Torus . 140 12.2.2 Index of Operators on the 2-Torus . 141 12.3 Special Cases . 142 Contents їх 13 An Application of Higher Traces 145 13.1 Index with Values in Odd AT-Groups. 145 13.2 Odd Index Formula . 146 13.2.1 Suspension. 147 13.2.2 Index Theorem. 147 Step 1. Reduction to the Cosphere Bundle and the Chern- Simons Character . 148 Step 2. Desuspension and Integration over S1 . 148 13.3 Example of Л -Index Computation . 149 14 Index Formula for a Finite Group Γ 153 14.1 Trajectory Symbol . 153 14.2 Index Formula . 154 IV Appendices 157 A ^-Algebras 161 A.I Basic Notions . 161 A. 1.1 Definitions and Examples . 161 A.I. 2 Unital Algebras and Units . 162 A.1.3 Homomorphisms, Ideals, Quotient Algebras, and Extensions 163 A. 1.4 Commutative C-Algebras . 165 A. 1.5 Spectrum and Functional Calculus . 165 A.1.6 Local C-Algebras . 167 A.l.ľ Positive Elements . 167 A. 1.8 Projections in C-Algebras . 168 A.2 Representations of C-Algebras . 169 A.2.1 Basic Definitions . 169 A.2.2 Existence of Representations . 172 A.2.3 Representations of Ideals and Quotient Algebras . 173 A.2.4 Primitive Ideals . 174 A.2.5 Algebras of Type I . 175 A.3 Tensor Products and Nuclear Algebras . 176 A.3.1 Minimal and Maximal Tensor Products . 176 A.3.2 Nuclear Algebras . 177 A.3.3 Primitive Ideals in the Tensor Product . 178 B Ä-Theory of Operator Algebras 179 B.I Covariant A'-Theory . 179 B.I.I Topological A'-Theory . 179 B.1.2 Group K0(A) . 182 B.1.3 Group Α',(Λ) . 184 B.1.4 Bott Periodicity . 186 x Contents В. 1.5 Long Exact Sequence in if-Theory . 187 Homomorphism di . 187 Homomorphism do . 188 B.1.6 Stability of K-Groups. 188 B.I. 7 K-groups of Local C-Algebras . 189 B.2 K-Homology . 189 B.2.1 ii-Homology of a Topological Space . 189 Even Groups . 191 Odd Groups . 191 B.2. 2 if-Homology of Operator Algebras: Definitions . 192 Fredholm Modules . 192 Dual Algebras . 193 Extensions of C-Algebras . 195 Equivalence of Different Definitions . 196 B.2.3 Suspension and Bott Periodicity . 197 B.2. 4 Long Exact Sequence in if-Homology . 198 B.2.5 Stability . 199 B.2. 6 Duality between if-Homology and if-Theory of Operator Algebras . 199 Even Groups . 199 Odd Groups . 199 С Cyclic Homology and Cohomology 201 C.I Cyclic Cohomology . 201 C.I.I Enveloping Differential Algebra . 201 C.I. 2 Graded Traces and Cyclic Functionals . 203 C.1.3 Cochains, Cyclic Cochains, and the Hochschild Differential 204 C.I. 4 Cyclic Cohomology and Hochschild Cohomology . 204 C.1.5 Example . 205 C.1.6 Cup Product in Cyclic Cohomology . 206 C.I. 7 Periodicity in Cyclic Cohomology . 207 C.1.8 Example . 209 C.2 Cyclic Homology . 209 C.2.1 Definition of Cyclic Homology . 209 C.2.2 Periodicity . 210 C.2.3 Pairing between Cyclic Homology and Cohomology . 211 C.2.4 Morita Invariance . 211 C.2. 5 Chern Characters in Homology . 211 Concise Bibliographical Remarks 213 Bibliography 217 Index 222
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id	DE-604.BV023354678
illustrated	Illustrated
index_date	2024-07-02T21:06:32Z
indexdate	2024-07-09T21:16:41Z
institution	BVB
isbn	9783764387747 3764387742 9783764387754
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-016538238
oclc_num	633862834
open_access_boolean
owner	DE-355 DE-BY-UBR DE-824 DE-19 DE-BY-UBM DE-703 DE-11 DE-83
owner_facet	DE-355 DE-BY-UBR DE-824 DE-19 DE-BY-UBM DE-703 DE-11 DE-83
physical	XII, 224 S. graph. Darst.
publishDate	2008
publishDateSearch	2008
publishDateSort	2008
publisher	Birkhäuser
record_format	marc
series	Operator theory
series2	Operator theory
spelling	Nazajkinskij, Vladimir E. 1955- Verfasser (DE-588)121444643 aut Elliptic theory and noncommutative geometry nonlocal elliptic operators V. E. Nazaikinskii ; A. Yu. Savin ; B. Yu. Sternin Basel [u.a.] Birkhäuser 2008 XII, 224 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Operator theory 183 : Advances in partial differential equations Elliptischer Differentialoperator (DE-588)4140057-4 gnd rswk-swf Nichtkommutative Geometrie (DE-588)4171742-9 gnd rswk-swf Elliptischer Differentialoperator (DE-588)4140057-4 s Nichtkommutative Geometrie (DE-588)4171742-9 s DE-604 Savin, Anton Ju. Verfasser aut Sternin, Boris Ju. 1939- Verfasser (DE-588)120484536 aut Operator theory 183 : Advances in partial differential equations (DE-604)BV000000970 183 http://d-nb.info/988365510/04 Inhaltsverzeichnis Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016538238&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Nazajkinskij, Vladimir E. 1955- Savin, Anton Ju Sternin, Boris Ju. 1939- Elliptic theory and noncommutative geometry nonlocal elliptic operators Operator theory Elliptischer Differentialoperator (DE-588)4140057-4 gnd Nichtkommutative Geometrie (DE-588)4171742-9 gnd
subject_GND	(DE-588)4140057-4 (DE-588)4171742-9
title	Elliptic theory and noncommutative geometry nonlocal elliptic operators
title_auth	Elliptic theory and noncommutative geometry nonlocal elliptic operators
title_exact_search	Elliptic theory and noncommutative geometry nonlocal elliptic operators
title_exact_search_txtP	Elliptic theory and noncommutative geometry nonlocal elliptic operators
title_full	Elliptic theory and noncommutative geometry nonlocal elliptic operators V. E. Nazaikinskii ; A. Yu. Savin ; B. Yu. Sternin
title_fullStr	Elliptic theory and noncommutative geometry nonlocal elliptic operators V. E. Nazaikinskii ; A. Yu. Savin ; B. Yu. Sternin
title_full_unstemmed	Elliptic theory and noncommutative geometry nonlocal elliptic operators V. E. Nazaikinskii ; A. Yu. Savin ; B. Yu. Sternin
title_short	Elliptic theory and noncommutative geometry
title_sort	elliptic theory and noncommutative geometry nonlocal elliptic operators
title_sub	nonlocal elliptic operators
topic	Elliptischer Differentialoperator (DE-588)4140057-4 gnd Nichtkommutative Geometrie (DE-588)4171742-9 gnd
topic_facet	Elliptischer Differentialoperator Nichtkommutative Geometrie
url	http://d-nb.info/988365510/04 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016538238&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV000000970
work_keys_str_mv	AT nazajkinskijvladimire elliptictheoryandnoncommutativegeometrynonlocalellipticoperators AT savinantonju elliptictheoryandnoncommutativegeometrynonlocalellipticoperators AT sterninborisju elliptictheoryandnoncommutativegeometrynonlocalellipticoperators

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