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

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

јвтор: Phillips N.Ch.

јннотаци€:

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.


язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1987

 оличество страниц: 371

ƒобавлена в каталог: 17.04.2008

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
$(G,C_{0}(X))$-algebra      158Ч159 162Ч164 219 248 277 278
$(Z/2Z)^{2}$      see "Action of $(Z/2Z)^{2}$"
$(Z/2Z)^{2}\times Z/3Z$      see "Action of $(Z/2Z)^{2}\times Z/3Z$"
$(Z/2Z)^{3}$      see "Action of $(Z/2Z)^{3}$"
$(\varphi/\psi)_{*}$ (difference map in K-theory)      128
$(^{n}_{j})$ (binomial coefficient)      258
$a\oplus b$ (direct sum of matrices)      28
$A\oplus_{max}B$ (maximal tensor product)      159
$A\otimes B$ (minimal tensor product)      104
$A^{+}$ (unitization)      22
$A^{G}$ (fixed point algebra)      58
$A_{\tau}$ (eigenspace for action)      137
$A_{\vartheta}$ (rotation algebra)      263
$C(G\times_{H}A)$ (twisted product)      64
$CP^{n}$ (projective space)      156
$C^{*}_{r}(G)$ (reduced group C*-algebra)      see "$C^{*}_{r}(G</a></span> <span class=subjpages><a href=
$C^{*}_{r}(G,A)$ (reduced crossed product)      see "$C^{*}_{r}(G</a></span> <span class=subjpages><a href=
$C^{*}_{r}(G,A,\alpha)$ (reduced crossed product)      209
$C_{0}(X)$ (continuous functions vanishing at $\infty$)      22
$C_{0}(X)$-algebra      157Ч159 162Ч164 219 248 277 278
$C_{0}(X,A)$ (continuous A-valued functions vanishing at $\infty$)      17
$C_{b}(X)$ (bounded continuous functions)      277
$C_{c}(X)$ (compactly supported continuous functions)      see "$C_{c}(X</a></span> <span class=subjpages><a href=
$C_{c}(X,A)$ (compactly supported continuous A -valued functions)      17
$C_{\varphi}$ (mapping cone)      61
$d_{A}$      174
$d_{G,H,A}$      174
$D_{\Phi}$ (map associated to a p.q.h.)      73
$Ext_{G}(A,B)$ (equivariant extensions)      125
$E\oplus F$ (Whitney sum)      13
$E\otimes F$ (tensor product of bundles)      13
$E\otimes_{A}B$ (tensor product of module and algebra)      17
$E_{x}$ (fiber of vector bundle)      13
$f^{+}$ (map of one point compactifications)      14
$F_{n}$ (free group)      210
$G\times_{H}X$ (twisted product)      64
$G_{x}$ (stabilizer subgroup)      7
$Hom_{f}(E,F)$ (set of vector bundle maps)      22
$Hom_{g}(E,F)$ (set of maps of vector bundles of modules)      22 23
$Hom_{\varphi}(E,F)$ (set of module maps)      23
$H^{m}$ (sheaf cohomology)      114
$H^{s}$ (Sobolev space)      111
$H^{\infty}_{B}$      117
$h_{*}$ (=$K^{G}_{*}(\cdot)_{P}$)      200
$h_{*}$ (equivariant homology theory)      184
$H_{B}$ (Hilbert space over B)      117
$Ind^{G}_{H}$ (induction map in K-theory)      168
$Ind^{G}_{H}$ (induction map in KK-theory)      174
$Ind^{G}_{H}$ (induction of modules)      187
$I^{-}_{S}$ (from ideal Decomposition Lemma)      183
$I^{G}_{H}$ (induction map in KK-theory)      171
$I^{G}_{H}$ (map in KK-theory)      171Ч177 321Ч323
$I_{A}$      174
$i_{G,H,A}$      174
$I_{S}$ (from ideal Decomposition Lemma)      183
$K(E_{1},E_{2})$ (compact module morphisms)      117
$KK^{*}_{G}(A,B)$ (equivariant KK-theory)      116
$KK^{0}_{G}(A,B)$ (equivariant KK-theory)      116
$KK^{1}_{G}(A,B)$ (equivariant KK-theory)      116
$KK_{G}(A,B)$ (equivariant KK-theory)      78
$K^{*}_{G}(X)$ (equivariant K-theory)      1
$K^{0}_{G}(X)$ (equivariant K-theory)      13 14
$K^{1}_{G}(X)$ (equivariant K-theory)      15
$K^{G}$-visibility      261
$K^{G}_{*}(A)$ (equivariant K-theory)      51
$K^{G}_{0}(A)$ (equivariant K-theory)      17 22
$K^{G}_{1}(A)$ (equivariant K-theory)      51
$L(E_{1},E_{2})$ (adjointable module homomorphisms)      117
$L(V,W)\otimes A$      28
$L^{1}(G,A)$ (twisted $L^{1}$-algebra)      see "$L^{1}(G</a></span> <span class=subjpages><a href=
$L^{1}(G,A,\alpha)$ (twisted $L^{1}$-algebra)      18
$M_{G}(A)$ (G-multiplier algebra)      69
$M_{n}$ ($n\times n$ complex matrices)      39
$M_{P}$ (localization of module)      6
$N^{G}$ (category related to Tensor Product Theorem)      221
$N^{G}_{0}$ (category related to equivariant localized Kuenneth formula)      215
$N^{G}_{1}$ (category related to equivariant located Kuenneth formula)      221
$N^{G}_{nuc}$ (category related to Tensor Product Theorem)      229
$O_{n}$ (Cuntz algebra)      241
$Proj_{G}(A)$ (finitely generated projective $(G,A,\alpha)$-modules)      23
$p\oplus q$ (direct sum of idempotents)      28
$P_{(as\ subscript)}$ (localization)      6
$P_{G}(A)$ (set of G-invariant idempotents)      28
$Q_{\Phi}$ (map associated to a p.q.h.)      73
$Res^{G}_{H}$ (restriction map in K-theory)      168
$Res^{G}_{H}$ (restriction map in KK-theory)      171
$Res^{G}_{H}$ (restriction of modules)      167
$R_{+}(G)$ (positive part of R(G))      58
$R_{P}$ (localization of ring)      6
$S^{1}$      see "Circle group"
$S^{1}$ (circle group)      19
$S^{1}$ (circle)      6
$S^{2}A$ (second suspension)      116
$S^{Ч1}M$ (localization of module)      6
$S^{Ч1}R$ (localization of ring)      6
$S_{G}(A)$ (semigroup of idempotents)      28
$U^{G}(V,A)$ (G-invariant unitaries in $L(V)\otimes A$)      58
$U^{G}_{0}(V,A)$ (identity component of $U^{G}(V,A)$)      58
$Vect_{G}(X)$ (G-vector bundles)      23
$X^{+}$ (one point compactification)      14
$Z/2Z\times Z/2Z$      see "Crossed product by $Z/2Z\times Z/2Z$"
$[\Phi]$ (class of q.h. or p.q.h. in KK-theory)      78
$[\Phi][\Psi]$ (product of classes of (pre-)quasihomomorphisms)      90
$[\varphi]$ (class of homomorphism in KK-theory)      80
$\alpha^{+}$ (action on unitization)      22
$\alpha_{(\cdot)}|_{I}$ (restriction of action to invariant subset)      136
$\Box$ (operator on differential forms)      112
$\eta\otimes_{D}\lambda$ (product in KK-theory)      104
$\eta_{*}$ (map in KK-theory)      60
$\Gamma(E)$ (sections of vector bundle)      23
$\Gamma(\alpha)$ (Connes Spectrum)      249
$\Gamma^{\infty}(E)$ ($C^{\infty}$ sections of vector bundle)      111
$\Gamma_{0}(A)$ (sections vanishing at infinity)      146
$\lambda\Phi$ (prequasihomomorphism)      73
$\lambda\Phi$ (quasihomomorphism)      75
$\lambda^{*}$ (map in KK-theory)      80
$\Lambda^{i}$ (exterior power)      110
$\lim\limits_{\longleftarrow}^{}^{1}$      156 325Ч326
$\lim\limits_{\longrightarrow}$ (direct limit)      33
$\omega$ (pairing in K-theory)      191
$\omega(A,B)$ (pairing in K-theory)      192
$\omega_{P}$ (pairing in localized K-theory)      196
$\omega_{P}(A,B)$ (pairing in localized K-theory)      200
$\oplus$ (direct sum)      see "$a\oplus b$" " " "
$\otimes$ (tensor product)      
see "$A\otimes B$" " " " " A$"/>" "
$\overline{KK}_{G}(A,B)$ (equivariant KK-theory)      
121
$\overline{U}^{G}(V,A)$ (= $U^{G}(V,A)/U^{G}_{0}(V,A)$)      58
$\overline{\partial}, \overline{\partial}^{*}$ (operators on differential forms)      112
$\Phi_{1}\oplus \Phi_{2}$ (direct sum of q.h. and p.q.h.)      78
$\Phi_{\eta}$ (prequasihomomorphism)      73
$\Phi_{\eta}$ (quasihomomorphism)      75
$\pi_{A}$      174
$\pi_{G,H,A}$      174
$\tau_{D}$ (map in KK-theory)      104
$\varphi^{A}_{\tau}$      306
$\varphi_{*}$ (map in K-theory)      21
$\widehat{ }$ (dual group, action)      see "$\widehat{G}$" "
$\widehat{G}$ (dual group)      
50
$\widehat{\alpha}$ (dual action)      50
$\widetilde{ }$      see "$\widetilde{V}$" "
$\widetilde{Sp}(\alpha)$ (strong spectrum)      
249
$\widetilde{V}$ (complex conjugate space)      39
$\widetilde{x}$ (element of C*(G,A,$\alpha$) gotten from $x\in A$)      238
$\widetilde{\Gamma}(\alpha)$ (strong Connes spectrum)      249
$\|$ $\|_{1}$ ($L^{1}$-norm)      18
$\|$ $\|_{2}$ (Hilbert module norm)      42
$\|$ $\|_{\infty}$ (supremum norm)      43
(ab)(g) (twisted convolution)      17
(G,A,$\alpha$)-module      16 see
(G,C(X))-algebra      159Ч162 279
* (adjoint)      see "a*(g)" "
* (subscript or superscript for induced map)      see "f*(E)" "f*([E])" " " " "
* (unspecified index in (co-)homology theory)      see "$h_{*}$" " " "
*-homomorphism      15
*-strong operator topology      71Ч72 118
+ (superscript for compactification or unitization)      see "$A^{+}$" " " "
a*(g) (adjoint in $L^{1}$ algebra)      
18
action      13 15
Action of $(Z/2Z)^{2}$      137Ч138 155 282 302 309
Action of $(Z/2Z)^{2}\times Z/3Z$      181
Action of $(Z/2Z)^{3}$      235
Action of $F_{n}$      209Ч211 213Ч215 219 281
Action of $S^{1}$      133 156 165 190 228 263Ч265 274 275 280Ч283 265 301Ч302 309
Action of R      201 209Ч211 213Ч214 219 227Ч228
Action of Z      190 201 209Ч214 219 227Ч228 263Ч265 283Ч284 290
Action of Z/2Z      4 5 134Ч136 143 224 251 271Ч274 286 299 300 302Ч305
Action of Z/3Z      261Ч263
Action of Z/4Z      5 299 302Ч305 309
Action of Z/6Z      299Ч300
Action of Z/nZ      4 5 140 143 167 177Ч181 189 200Ч209 213Ч222 227 229 232 264Ч265 271 274 275 279Ч282 285 288 299 302 305 307Ч308
Action of Z/pZ (p prime)      189 229 258 270 273 299
Action on a Banach algebra      15
Action on a C*-algebra      15
Action on a crossed product C*-algebra      209Ч215 see
Action on a projective module      16
Action on a space      13 see
Action on a tensor product      see "Tensor product"
Action on a vector bundle      13
Action, AF      287 see
Action, commuting      209Ч215 219 227Ч228
Action, exterior equivalent      44Ч40 53 55 134 135 139 141 198Ч199 206 212 219 227Ч228 296
Action, free      1Ч5 7Ч9 131Ч135 138Ч139 142Ч143 146 151 156Ч157 161Ч167 189 221Ч222 227 234 236 242Ч244 247Ч248 270 272 275 278Ч283 285 289Ч290 299 309
Action, hereditarily K-saturated      270Ч274 281 285 299
Action, hereditarily K-saturated up to 2-torsion      272
Action, hereditarily saturated      236Ч237 246Ч251 270Ч275 278 281Ч285 299Ч301
Action, inner      44
Action, K-free      136 see
Action, K-saturated      270Ч274 285
Action, KK-free      151 see
Action, locally representable      5 286 292Ч299 305 309 327Ч328
Action, outer      298 305
Action, product type      286 292Ч294 298Ч302 305 309
Action, restriction to subgroup of      3Ч5 137Ч138 143 162 177Ч182 272 288 300 302 305 307Ч309
Action, saturated      236Ч237 240Ч251 270 273Ч274 282
Action, totally K-free      138 see
Action, totally K-free up to torsion      178Ч179
Action, totally KK-free      152
Action, with discrete K-theory      132 153 157 163 262 270 290 293 306Ч309
Action, with discrete KK-theory      151Ч154 181 182 233 306Ч308 328
Action, with locally discrete K-theory      131Ч138 141 157 178 185 211 224 232 235 271 286Ч296 299 328
ad(u) (a $\longrightarrow$ u*au)      137
Additivity of KK-theory      325 326 see
AF action      267 286Ч320
AF action, discrete K-theory and      307 308 328
AF action, K-freeness of      5 206Ч294 298Ч305 309
AF action, KK-freeness of      5 286 307 308 328
AF action, locally discrete K-theory and      286Ч296 299 328
AF action, total K-freeness of      5 286 288Ч289 291 294Ч295 299 302 307 308Ч309 312
AF algebra      5 68 130 178 181 227Ч228 252 263 266 268Ч270 284 286Ч328 see
AF algebra, equivariant KK-theory of      68 308 328
AF algebra, K-visibility of      252
AF algebra, order on K-theory of      252 294 297 298 305
AF automorphism      290Ч291
AF G-algebra      287 286Ч328
Akemann, c.a.      120 122
Algebra, $(G,C_{0}(X))$-      158Ч159 162Ч164 219 248 277 278
Algebra, $C_{0}(X)$-      157Ч159 162Ч164 219 248 277 278
Algebra, (G,C(X))-      159Ч162 279
Algebra, AF G-      287 see
Algebra, C(X)-      76 159Ч162 278 279
Algebra, G-      15
Algebra, G-Banach      15Ч22 28Ч34 38 69
Amenable group      211 215
Approximate identity of projections, in crossed product      19
Approximate identity, quasicentral      81 84 87Ч89 93
Approximately inner automorphism      305
Artin's theorem      177 180
Arveson Spectrum      137Ч138 246 249 301
Atiyah, M.F.      1 6 7 8 9 10 24 26 56 58 68 108 109 110 111 112 113 114 115 141 152 156 179 196 216 296 326
Augmentation ideal      14
Aut(A) (automorphism group)      15
Automorphism, AF      290Ч291
Automorphism, approximately inner      305
Automorphism, inner      45
Automorphism, outer      236
Automorphism, universally weakly inner      283Ч284
Baire category theorem      8
Baire space      208 275Ч277
Banach *-algebraic bundle      240 244
Banach algebra      3 12 15Ч22 28Ч34 36Ч38 41 55 58 69
Baum, P.      2
Bimodule      236Ч241
Bimodule, Morita equivalence      236Ч240
Binomial coefficient      180 258 260
Blackadar, B.      33 34 305 306 313
Bott periodicity for K-theory      52 58
Bott periodicity for KK-theory      109Ч115
Bott periodicity, relation between maps for K- and KK-theory      129
Bratteli, O.      228 252 269 286 288 301 312
Brauer's theorem      180
Bredon, G.E.      7 10 147 151 163
Brown, L.G.      62 63 66 68 267
Bundle of C*-algebras      76 146 157
Bundle, Banach *-algebraic      240 244
Bundle, vector      2 6Ч8 12Ч15 23Ч27 110Ч115 129 156Ч157 189
Busby, R.C.      44
C(X) (continuous functions)      23
C(X)-algebra      76 159Ч162 278 279
C(X)-module      157 158
C(X)-module, locally convex      158
C(X,A) (continuous A-valued functions)      17
C*(G) (group C*-algebra)      49
C*(G,A) (crossed product)      see "C*(G src='/math_tex/c745b9b57c145ec5577b82542b2df54682.gif'
C*(G,A,$\alpha$) (crossed product)      18
C*-algebra valued inner product      42Ч43 117 238Ч241
C*-dynamical system      5
CA (cone)      54
Cauchy Ч Schwarz inequality      43
Change of group homomorphism      167 see "Restriction"
Choi, M.-D.      140
Circle group      19 30 133 156 165 190 212 216 228 263Ч265 274 275 280Ч283 285 289 301Ч303 309 312
Circle group, action of      133 156 165 190 228 263Ч265 274 275 280Ч283 285 301Ч302 309
Cocycle, one-      44 46 139 198 311
Cocycle, two-      155
Cohomology of sheaves      114
Cohomology theory, for C*-algebras      69 156 325
Cohomology theory, for spaces      1 55
Commuting actions      209Ч215 219 227Ч228
Compact operator on Hilbert module      117
Compactification, free      164Ч166
Compactification, one point      14
Complex conjugate representation      39 71 109 155 323Ч324
Complex projective space      110 156
Composition series      141Ч142 147 150 157 205 208 227 244 247 266 268 275 277Ч279
Conditional expectation      250
Conditional expectation, generalized      42
Cone over a C*-algebra      54 57 61 217Ч218
Cone, mapping      61 217Ч218
Conjugate representation      39 71 109 155 323Ч324
Connected group      2 59 71 113 189 262 284 309 314
Connecting homomorphism      48 53Ч55 57 60Ч61 191 196 204
Connes Isomorphism Theorem (Connes' Thom Isomorphism Theorem), in equivariant K-theory      190 209 210 214
Connes spectrum      4 131 137Ч138 236 249 250 263Ч264 282 301 305
Connes spectrum, strong      4Ч5 212 227 236Ч237 249Ч250 257Ч263 283 286 305
Connes, A.      2 190 209 210
Continuity of K-theory      34Ч36 53
Continuity of KK-theory      158 325
Continuous field of C*-algebras      164 208 281
Continuous field of C*-algebras, locally trivial      208
Continuous trace algebra      208 285
1 2 3 4
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