Phillips N.Ch. — Equivariant K-Theory and Freeness of Group Actions on C*-Algebras :: Электронная библиотека попечительского совета мехмата МГУ

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Phillips N.Ch. — Equivariant K-Theory and Freeness of Group Actions on C*-Algebras

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Название: Equivariant K-Theory and Freeness of Group Actions on C*-Algebras

Автор: Phillips N.Ch.

Аннотация:

This book is based on the author's Ph.D. thesis written at the University of California at Berkeley. However, it incorporates substantial revisions and many additional results. The research reported here has been partially supported by a National Science Foundation Graduate Fellowship, by the Mathematical Sciences Research Institute (NSF Grant 8120790). and by a National Science Foundation Postdoctoral Fellowship. All of this support is gratefully acknowledged. Thanks are also due to Marsha Colby, who converted an enormous number of sometimes rather illegible handwritten pages into the computer files from which this manuscript was produced, and who, with rare exceptions, was able to persuade a University of California computer to print many symbols that it had never been intended to make.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1987

Количество страниц: 371

Добавлена в каталог: 17.04.2008

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Предметный указатель

Contractible C*-algebra      54 61 134 218
Contractible space      2
Contractible space, locally      189
Covariant representation      16 17 43 51—52 135 283
Covering dimension      189
Crossed product      18
Crossed product by $F_{n}$       209—211 213—215 219 221
Crossed product by $S^{1}$ (circle group)      264 283
Crossed product by $Z/2Z\times Z/2Z$       137—138
Crossed product by compact abelian group      50—51 55 132 164 250
Crossed product by compact group      3 12—13 19 38 41—45 48—50 53—57 63 66 144 196—199 210—211 236—245 248 284 325—326
Crossed product by discrete group      164
Crossed product by finite abelian group      4 60 257 259—261 307 310 315—319
Crossed product by finite group      287 294 297—298 305
Crossed product by locally compact abelian group      4 212—213 236
Crossed product by locally compact group      17—18 34 45 51—52 55 209
Crossed product by R      201 209 210 213—214 219 227—228
Crossed product by Z      190 201 209—214 219 227—228 263 283
Crossed product by Z/2Z      135—136 273 300
Crossed product by Z/3Z      263
Crossed product by Z/4Z      305
Crossed product by Z/nZ      201—202 264—265
Crossed product by Z/pZ (p prime)      258
Crossed product commutes with direct limits      34
Crossed product, action on (coming from commuting action)      209—215
Crossed product, approximate identity of protections in      19
Crossed product, invariant ideals in      212—213
Crossed product, K-theory of      38 49 50 53 210—211
Crossed product, Morita equivalence with fixed point algebra      4 236—239 243 285 298
Crossed product, preserves exactness      51
Crossed product, reduced by $F_{n}$       210—211 213—215 219 221
Crossed product, reduced by locally compact croup      209
Crossed product, simplicity of      4 236 261
Cuntz algebra      236 251 261—263 268 272
Cuntz, J.      3 12 59 68 69 72 73 74 75 84 95 100 101 103 107 108 117 125 211 251 261 262 263 268 272
Cyclic group      4 5 140 143 167 177—181 189 191 200—209 213—222 227 229 232 264—265 271 274 275 279—282 285 288 292 299 302 305 307—308
Cyclic group of prime order      189 229 258 270 273 299
Cyclic group of prime power order      230 231 300
Cyclic, topologically      10
Dauns — Hofmann theorem      277
Degenerate element of E(A,B)      121—122
Degree of element      191
Degree of homomorphism of C*-algebras      121
Degree of homomorphism of groups      191
Degree of operator      121
Derivation, equivariant      72 85 87 90—102 106—107
Derivation, inner      91 98
Derivation, lifting of      72 85 87 93 95
dg (Haar measure)      18
Diagonal action on spaces      4 143 189
Diagonal action on tensor products      70
Diagonal map      129 159
Difference map      128
Differential form      112
Differential operator      112
Dimension of local ring      216
Dimension, covering      189
Dimension, global      215 216
Dimension, homological      189—190 216—216 232
Dimension, projective      215
Dimension, virtual      14
Direct limit of G-algebras      33—34
Direct limit of G-algebras, discrete KK-theory and      155—156
Direct limit of G-algebras, equivariant K-theory of      34—36 53
Direct limit of G-algebras, equivariant KK-theory of      155—156 324—325
Direct limit of G-algebras, K-freeness of      141
Direct limit of G-algebras, locally discrete K-theory and      141
Direct limit of G-algebras, total K-freeness of      141
Direct sum of elements of E(A,B)      121
Direct sum of matrices      28
Direct sum of prequasihomomorphisms      78
Direct sum of quasihomomorphisms      78
Directly finite      252
Discrete K-theory      132 153 157 163 262 270 290 293 306—309
Discrete KK-theory      151—154 181 182 233 306—308 328
Discrete KK-theory and AF actions      307 308 328
Discrete KK-theory and direct limits      155—156
Discrete KK-theory and ideals      154
Discrete KK-theory and locally representable actions      328
Discrete KK-theory and quotients      154
Discrete Sobolev chain      111
Discrete valuation ring      216
Dixmier, J.      19 39 52 149 157 164 208 252 268 275 277 278 281 311
Double centralizer      44
Douglas, R.G.      68
Dual action      50
Dual action, action on K-theory induced by      50—51
Dual bundle      110 156
Dual space of C*-algebra      238
Dual space of C*-algebra, second      73 283
Duality, Takai      60 164 213 258 264 283 316
d|n (d divides n)      200
E(A,B) (set of Kasparov bimodules)      120
Effros, E.G.      3 12 140 252 269 286 287 294 297
Eilenberg, S.      144
Elementary group      180—181
Elliott, G.A.      298
Elliptic operator      58 68 109 112 167 179
Equivalence bimodule, Morita      236—240
Equivalence, exterior      44—48 53 55 134 135 139 141 198—199 206 212 219 227—228 296
Equivalence, generalized G-homotopy      107—109
Equivalence, Murray — von Neumann      28—33 41—42
Equivalence, Murray — von Neumann, in C*-algebraic sense      31—33 46—47 127
Equivalence, unitary      31
Equivariant derivation      72 85 87 90—102 106—107
Equivariant extension      125 215 289 325
Equivariant function      13
Equivariant homomorphism      15
Equivariant K-theory      13—15 17 51 55
Equivariant K-theory and Banach algebras      17 51
Equivariant K-theory and Bott periodicity      52 56—58
Equivariant K-theory and Bott periodicity, compared to KK-theory      129
Equivariant K-theory and commutative algebras      27
Equivariant K-theory and Connes Isomorphism Theorem (Connes' Thom Isomorphism Theorem)      190 209 210 214
Equivariant K-theory and continuity      34—36 53
Equivariant K-theory and crossed products by Z, R and $F_{n}$       209 210—211 212—214 219 221 227—228
Equivariant K-theory and direct limits      34—36 53
Equivariant K-theory and equivariant KK-theory      125—129
Equivariant K-theory and exact sequences      52 55
Equivariant K-theory and exterior equivalent actions      45—48 53 55 134 139 198—199 206 219 227—228 296
Equivariant K-theory and G-spaces      13—15 55
Equivariant K-theory and hereditary subalgebras      62—63 141 242
Equivariant K-theory and homotopy invariance      38 52
Equivariant K-theory and idempotents      28—33 34—38 41 43
Equivariant K-theory and induction      167—170 177 180 231 259 296—297
Equivariant K-theory and Mayer — Vietoris sequence      67
Equivariant K-theory and order      294—295 297 305
Equivariant K-theory and pairing      191—200
Equivariant K-theory and positive elements      294—285 297
Equivariant K-theory and product type actions      293
Equivariant K-theory and projections      31—33
Equivariant K-theory and restriction      8 167 168—170 177—180 231 294—298
Equivariant K-theory and stably isomorphic G-algebras      46 53—54
Equivariant K-theory and tensor products      4 67 189—190 200 215 218—221 229—233 235
Equivariant K-theory and twisted products      64—66
Equivariant K-theory and unitaries      58—60 217
Equivariant K-theory and vector bundles      2 6—8 12 13—15 27 111 129 156—157 189
Equivariant K-theory end projective modules      12 17 21—22 27 30 39 41 45 47 60 132—133 167—170 197—198 241
Equivariant KK-theory      78—79 116
Equivariant KK-theory and (pre-)quasihomomorphisms      68 78—81 90—91 95—115 119—120 124—129 160—161 171—174 321
Equivariant KK-theory and additivity      325 326
Equivariant KK-theory and AF G-algebras      68 308 326
Equivariant KK-theory and Bott periodicity      109—115
Equivariant KK-theory and Bott periodicity, compared to K-theory      129
Equivariant KK-theory and continuity      156 325
Equivariant KK-theory and direct limits      155—156 324—325
Equivariant KK-theory and equivariant K-theory      125—129
Equivariant KK-theory and exact sequences      124
Equivariant KK-theory and extensions      325
Equivariant KK-theory and G-spaces      129

Equivariant KK-theory and homotopy invariance      79
Equivariant KK-theory and induction      167 171—177 181 321—323
Equivariant KK-theory and invertible elements      107
Equivariant KK-theory and Kasparov bimodules      68 109 117—124
Equivariant KK-theory and locally representable actions      327
Equivariant KK-theory and products      68 81 90—91 95—105 106—109 114—116 129 152 159—161 171—177 309 321 327
Equivariant KK-theory and restriction      167 171—173 176—177 181 182 320—323
Equivariant KK-theory and stably isomorphic G-algebras      109
Equivariant KK-theory and twisted products      320—323
Equivariant localized Kuenneth theorem      4 67 189—190 200 215 218—221 229—233 235
Equivariant prequasihomomorphism      73 75 see
Equivariant quasihomomorphism      75 see
Equivariant stable isomorphism      219 227
Equivariantly contractible      54 61 218
Essential ideal      69—70 73—75 97 139
Expectation, conditional      250
Expectation, conditional, generalized      42
Extended (pre)-quasihomomorphism      see "Extendible (pre)-quasihomomorphism"
Extendible prequasihomomorphism      81—85 87—90 93 95—100 105
Extendible quasihomomorphism      68 81—86 89 90 102—103
Extension of (pre-)quasihomomorphisms      see "Extendible (pre-)quasihomomorphism"
Extension of C*-algebras      125 215 270 289 325
Extension of groups      312
Extension, equivariant      125 215 289 325
Extension, K-freeness of action on      137
Extension, KK-freeness of action on      154
Extension, locally discrete K-theory and      133
Extension, total K-freeness of action on      138
Exterior equivalent actions      44—48 53 55 134 135 139 141 198—199 208 212 219 227—228 296
Exterior power      110
External tensor product      129 189
f*(E) (pullback)      14
f*([E]) (induced map on K-theory)      14
Fell, J.M.G.      240 244
Fillmore, P.A.      68
Finite algebra      252
Finite covering dimension      189
Finitely generated projective $(G,A,\alpha)$ -module      16 see
Finitely generated projective module over a C*-algebra      12 l5 16—30 39 41 45 47 50 132—133 167—170 197—198 238 241
Fixed point algebra, K-theory of      4 240—242 250 270—274 285 see
Flat module      204
Fomenko, A.T.      58 125
Fredholm index      112 125 127
Fredholm operator on Hilbert module      125—127
Fredholm operator on Hilbert space      112—113 l15
Free Abelian group      202 215 217—218
Free action      1—5 7—9 131—135 138—139 142—143 146 151 156—157 161—167 189 221 222 227 234 236 242—244 247—248 270 272 275 27B—283 285 289—290 299 309
Free compactification      164—166
Free group      209—211 213—215 219 221 229
Free module over C*-algebra      16 17 197 298
Free module over ring      216—218 220
Free resolution      202
Free unitization      131 164
Full hereditary subalgebra      62—63 141—142 242 251
Fundamental group      189
G-algebra      15
G-Banach algebra      15—22 28—34 38 69
G-homotopy equivalence, generalized      107—109
G-homotopy of homomorphisms      38
G-homotopy of prequasihomomorphisms      76
G-homotopy of quasihomomorphisms      75
G-invariant subquotient      251 254 256—261 274 279—282
G-multiplier algebra      68 69—74 81—86 90—101 105—107 119—120 124 158 325
G-simple algebra      141 252 261 270 310 313 324 326
G-space      1 6—15 22—27 55—56 62—64 110 129 131—135 138—139 142—143 147 149 151 156—167 189 208 234 243—244 247—249 254—257 270 275—282 289—290
G-space, equivariant K-theory of      13—15 55
G-space, equivariant KK-theory and      129
G-space, free (Hausdorff)      1 7—9 131—133 138—139 142—143 156—157 162—167 189 234 243—244 247—248 270 278 289—290 see
G-space, hereditary saturation and      248
G-space, K-freeness and      138
G-space, KK-freeness and      157 161
G-space, locally discrete K-theory and      132—134
G-space, saturation and      243 248
G-space, total K-freeness and      138
G-space, twisted product of      64
G-vector bundle      6—7 13—15 23 26—27 129 189
Garding's Lemma      112
Generalized conditional expectation      42
Generalized G-homotopy equivalence      107—109
Geometric realization      215 217
Glimm, J.      312
Global dimension      215 216
GNS construction      43
Graded module      191
Graded tensor product      191
Grading operator      121 122
Green, P.      38 42 66 117 211
Griffiths, P.      112
Group C*-algebra      50 56 164 243—244
Group C*-algebra, reduced of $F_{n}$       229
Group ring      50 230
Guichardet, a.      159
Gx (orbit, = $G\cdot \{ x\}$ )      10
GZ (saturation of subset of a G-space)      10
Haar measure, conventions concerning      18
Half-exact functor      108
Halmos, P.R.      165
Handelman, D.      252 286 292 294 297 298 305
Harris, J.      112
Hereditary K-saturation      270—274 281 285 299
Hereditary K-saturation up to 2-torsion      272
Hereditary saturation      236—237 246—251 270—275 278 281—285 299—301
Hereditary subalgebra      41 43 62—63 137 141—142 237 242 246—251 267 270—274 282
Hereditary subalgebra, (total) K-freeness and      141 142
Hereditary subalgebra, equivariant K-theory of      62—63 141 242
Hereditary subalgebra, full      62—63 141—142 242 251
Hermitian metric (on vector bundle)      24—27 111
Highly K-visible C*-algebra      236 251—253 256—257 259 261 264—270 272 285
Hilbert (C*-)module      117—127 see
Hilbert (C*-)module, compact operator on      117
Hilbert (C*-)module, Fredholm operator on      125—127
Hilbert space over a C*-algebra      117—127
Hilgert, J.      3 12 191 197
Hilton, P.J.      202 220
Hodgkin, L.H.      189
Hofmann, K.H.      76 146 157 158
Holomorphic vector bundle      114
Hom(E,F) (bundle of vector bundle maps)      156
Hom(E,F) (set of maps of vector bundles or modules)      22 23
Homological dimension      189—190 215—216 232
Homology of a complex      220
Homology theory for C*-algebras      2 4 13 51 52 69 184 200 324
Homomorphism      15
Homotopy of elements of E(A,B)      121
Homotopy of extended prequasihomomorphisms      82—83
Homotopy of extended quasihomomorphisms      82
Homotopy of homomorphisms      38
Homotopy of prequasihomomorphisms      76
Homotopy of quasihomomorphisms      76
Homotopy, G-      see "G-homotopy"
Hull-kernel topology      139 149
I(G) (augmentation ideal)      14
I(G)-adic topology      1 8 132
I(U) (ideal corresponding to open set $U\subset Prim(A)$ )      253
Iberkleid, W.      190
Ideal Decomposition Lemma      4 167 182 183—188 190 222—225 228
Idempotent      12 18 20 28—43
Idempotent, minimal      39—40
Idempotents, equivariant K-theory in terms of      28—33 34—38 41 43
Idempotents, increasing net of      18
Identity of $KK^{0}_{G}(A,A)$       103
Identity, quasicentral approximate      81 84 87—89 93
Imprimitivity, system of      6 66 183
Increasing net of idempotents or projections      18
Ind (= $Ind^{G}_{H}$ )      168
Index of operator      112 125 127
Induced representation      175 see "Induction "Induction "Induction
Induction in K-theory      167—170 177 180 231 259 296—297
Induction in KK-theory      167 171—177 181 321—323
Induction in representation rings      168 170 177 180 181 231 292 296

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