Holdings: The structure of compact groups

The structure of compact groups: a primer for the student - a handbook for the expert

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Bibliographic Details
Main Authors:	Hofmann, Karl H. 1932- (Author), Morris, Sidney A. 1947- (Author)
Format:	Book
Language:	English
Published:	Berlin/Boston De Gruyter [2020]
Edition:	4th edition
Series:	De Gruyter studies in mathematics Volume 25
Subjects:	Kompakte Gruppe Topological groups Lie algebra Abelian Group; Category Theory; Compact Group; Lie Algebra; Topological Group Category Theory compact groups Abelian Groups
Online Access:	Inhaltsverzeichnis Inhaltsverzeichnis
Physical Description:	xxvii, 1006 Seiten
ISBN:	9783110695953

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245	1	0	\|a The structure of compact groups \|b a primer for the student - a handbook for the expert \|c Karl H. Hofmann, Sidney A. Morris
250			\|a 4th edition
264		1	\|a Berlin/Boston \|b De Gruyter \|c [2020]
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490	1		\|a De Gruyter studies in mathematics \|v Volume 25
650	0	7	\|a Kompakte Gruppe \|0 (DE-588)4164840-7 \|2 gnd \|9 rswk-swf
653			\|a Topological groups
653			\|a Lie algebra
653			\|a Abelian Group; Category Theory; Compact Group; Lie Algebra; Topological Group
653			\|a Category Theory
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653			\|a Abelian Groups
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adam_text	CONTENTS CHAPTER 1. BASIC TOPICS AND EXAMPLES ...................................................... 1 DEFINITIONS AND ELEMENTARY EXAMPLES ....................................................................... 2 ACTIONS, SUBGROUPS, QUOTIENT SPACES ........................................................................ 5 PRODUCTS OF COMPACT GROUPS ................................................................................... 10 APPLICATIONS TO ABELIAN GROUPS .............................................................................. 11 PROJECTIVE LIMITS ........................................................................................................ 17 TOTALLY DISCONNECTED COMPACT GROUPS .................................................................... 22 SOME DUALITY THEORY ................................................................................................ 24 POSTSCRIPT ..................................................................................................................... 29 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING ................................................. 30 CHAPTER 2. THE BASIC REPRESENTATION THEORY OF COMPACT GROUPS 31 SOME BASIC REPRESENTATION THEORY FOR COMPACT GROUPS ...................................... 31 THE HAAR INTEGRAL ........................................................................................................ 34 CONSEQUENCES OF HAAR MEASURE ................................................................................ 35 THE MAIN THEOREM ON HILBERT MODULES FOR COMPACT GROUPS ............................. 37 POSTSCRIPT ..................................................................................................................... 49 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING ................................................. 50 CHAPTER 3. THE IDEAS OF PETER AND WEYL ............................................. 51 PART 1: THE CLASSICAL THEOREM OF PETER AND WEYL ............................................... 52 AN EXCURSION INTO LINEAR ALGEBRA ........................................................................... 57 THE G-MODULES E 0 E, HOM(E,E) AND HOM(E, E) .......................................... 59 THE FINE STRUCTURE OF R(G, K) ................................................................................ 61 PART 2: THE GENERAL THEORY OF G-MODULES ............................................................ 68 VECTOR VALUED INTEGRATION ........................................................................................ 69 THE FIRST APPLICATION: THE AVERAGING OPERATOR .................................................... 75 COMPACT GROUPS ACTING ON CONVEX CONES ............................................................ 79 MORE MODULE ACTIONS, CONVOLUTIONS ...................................................................... 80 COMPLEXIFICATION OF REAL REPRESENTATIONS .............................................................. 85 PART 3: THE WEAKLY COMPLETE GROUP ALGEBRA ...................................................... 90 THE HOPF ASPECT OF WEAKLY COMPLETE ALGEBRAS .................................................... 94 THE DUAL OF A WEAKLY COMPLETE GROUP HOPF ALGEBRA ........................................ 101 A PRINCIPAL STRUCTURE THEOREM OF 1K[G] FOR COMPACT G .................................. 104 THE SPECTRUM OF THE K-ALGEBRA R(G, K) ............................................................. 110 THE TANNAKA-HOCHSCHILD DUALITY ............................................................................ 113 COMPACT ABELIAN GROUPS ......................................................................................... 114 THE PROBABILITY SEMIGROUP OF A COMPACT G INSIDE R[G] .................................. 117 POSTSCRIPT ................................................................................................................... 123 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING ............................................... 125 XXII CONTENTS CHAPTER 4. CHARACTERS ............................................................................... 126 PART 1: CHARACTERS OF FINITE DIMENSIONAL REPRESENTATIONS ................................ 126 PART 2: THE STRUCTURE THEOREM OF E& N ................................................................. 133 CYCLIC MODULES ........................................................................................................... 144 POSTSCRIPT ................................................................................................................... 145 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING ............................................... 146 CHAPTER 5. LINEAR LIE GROUPS ................................................................. 147 PRELIMINARIES .............................................................................................................. 147 THE EXPONENTIAL FUNCTION AND THE LOGARITHM .................................................... 149 DIFFERENTIATING THE EXPONENTIAL FUNCTION IN A BANACH ALGEBRA ........................ 161 LOCAL GROUPS FOR THE CAMPBELL -HAUSDORFF MULTIPLICATION ................................. 167 SUBGROUPS OF A -1 AND LINEAR LIE GROUPS ............................................................ 170 ANALYTIC GROUPS ........................................................................................................ 174 THE INTRINSIC EXPONENTIAL FUNCTION OF A LINEAR LIE GROUP ................................ 175 THE ADJOINT REPRESENTATION OF A LINEAR LIE GROUP ............................................. 182 SUBALGEBRAS, IDEALS, LIE SUBGROUPS, NORMAL LIE SUBGROUPS ............................. 187 NORMALIZERS, CENTRALIZERS, CENTERS ......................................................................... 193 THE COMMUTATOR SUBGROUP .................................................................................... 198 FORCED CONTINUITY OF MORPHISMS BETWEEN LIE GROUPS ....................................... 202 QUOTIENTS OF LINEAR LIE GROUPS .............................................................................. 205 THE TOPOLOGICAL SPLITTING THEOREM FOR NORMAL VECTOR SUBGROUPS ................... 209 POSTSCRIPT ................................................................................................................... 220 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING ............................................... 222 CHAPTER 6. COMPACT LIE GROUPS ............................................................ 223 COMPACT LIE ALGEBRAS .............................................................................................. 224 THE COMMUTATOR SUBGROUP OF A COMPACT LIE GROUP ....................................... 235 THE STRUCTURE THEOREM FOR COMPACT LIE GROUPS ............................................... 243 MAXIMAL TORI ............................................................................................................. 246 THE SECOND STRUCTURE THEOREM FOR CONNECTED COMPACT LIE GROUPS .............. 256 COMPACT ABELIAN LIE GROUPS AND THEIR LINEAR ACTIONS ..................................... 260 ACTION OF A MAXIMAL TORUS ON THE LIE ALGEBRA .................................................. 264 THE WEYL GROUP REVISITED ...................................................................................... 276 THE COMMUTATOR SUBGROUP OF CONNECTED COMPACT LIE GROUPS ..................... 287 ON THE AUTOMORPHISM GROUP OF A COMPACT LIE GROUP ..................................... 289 COVERING GROUPS OF DISCONNECTED COMPACT LIE GROUPS ..................................... 313 AUERBACH * S GENERATION THEOREM ............................................................................ 315 THE TOPOLOGY OF CONNECTED COMPACT LIE GROUPS ............................................. 323 POSTSCRIPT ................................................................................................................... 333 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING ............................................... 335 CHAPTER 7. DUALITY OF ABELIAN TOPOLOGICAL GROUPS ............................ 336 THE COMPACT OPEN TOPOLOGY AND HOM-GROUPS .................................................. 337 LOCAL COMPACTNESS AND DUALITY OF ABELIAN TOPOLOGICAL GROUPS ..................... 341 CONTENTS XXIII BASIC FUNCTORIAL ASPECTS OF DUALITY ...................................................................... 348 THE ANNIHILATOR MECHANISM ................................................................................... 351 CHARACTER GROUPS OF TOPOLOGICAL VECTOR SPACES .................................................. 362 THE EXPONENTIAL FUNCTION ..................................................................................... 372 WEIL * S LEMMA AND COMPACTLY GENERATED ABELIAN GROUPS ............................... 381 REDUCING LOCALLY COMPACT ABELIAN GROUPS TO COMPACT ABELIAN GROUPS . . 385 A MAJOR STRUCTURE THEOREM ................................................................................... 387 THE DUALITY THEOREM ............................................................................................. 390 THE IDENTITY COMPONENT ........................................................................................ 398 THE WEIGHT OF LOCALLY COMPACT ABELIAN GROUPS ............................................... 401 POSTSCRIPT .................................................................................................................. 403 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING ............................................... 406 CHAPTER 8. COMPACT ABELIAN GROUPS .................................................... 407 PART 1: ASPECTS OF THE ALGEBRAIC STRUCTURE DIVISIBILITY, TORSION, CONNECTIVITY ........................................................................ 408 COMPACT ABELIAN GROUPS AS FACTOR GROUPS .................................... 414 PART 2: ASPECTS OF THE POINT SET TOPOLOGICAL STRUCTURE TOPOLOGICAL DIMENSION OF COMPACT ABELIAN GROUPS .......................................... 422 ARC CONNECTIVITY ..................................................................................................... 429 LOCAL CONNECTIVITY .................................................................................................. 434 COMPACT METRIC ABELIAN GROUPS ........................................................................... 447 PART 3: ASPECTS OF ALGEBRAIC TOPOLOGY* HOMOTOPY FREE COMPACT ABELIAN GROUPS .............................................................................. 450 HOMOTOPY OF COMPACT ABELIAN GROUPS .............................................................. 455 EXPONENTIAL FUNCTION AND HOMOTOPY ................................................................... 461 THE FINE STRUCTURE OF FREE COMPACT ABELIAN GROUPS ....................................... 462 PART 4: ASPECTS OF HOMOLOGICAL ALGEBRA INJECTIVE, PROJECTIVE, AND FREE COMPACT ABELIAN GROUPS .................................. 469 PART 5: ASPECTS OF ALGEBRAIC TOPOLOGYCOHOMOLOGY COHOMOLOGY OF COMPACT ABELIAN GROUPS ............................................................ 473 PART 6: ASPECTS OF SET THEORY ARC COMPONENTS AND BOREL SUBSETS ...................................................................... 479 POSTSCRIPT .................................................................................................................. 482 REFERENCES FOR THIS CHAPTER ADDITIONAL READING ............................................... 485 CHAPTER 9. THE STRUCTURE OF COMPACT GROUPS ................................... 486 PART 1: THE FUNDAMENTAL STRUCTURE THEOREMS OF COMPACT GROUPS APPROXIMATING COMPACT GROUPS BY COMPACT LIE GROUPS ............................... 487 THE CLOSEDNESS OF COMMUTATOR SUBGROUPS ......................................................... 488 SEMISIMPLE COMPACT CONNECTED GROUPS .............................................................. 489 THE LEVI-MAL * CEV STRUCTURE THEOREM FOR COMPACT GROUPS ............................. 503 MAXIMAL CONNECTED ABELIAN SUBGROUPS .............................................................. 511 THE SPLITTING STRUCTURE THEOREM ........................................................................... 517 SUPPLEMENTING THE IDENTITY COMPONENT .............................................................. 518 XXIV CONTENTS PART 2: THE STRUCTURE THEOREMS FOR THE EXPONENTIAL FUNCTION THE EXPONENTIAL FUNCTION OF COMPACT GROUPS .................................................. 523 THE DIMENSION OF COMPACT GROUPS ....................................................................... 530 LOCALLY EUCLIDEAN COMPACT GROUPS ARE COMPACT LIE GROUPS ........................ 536 PART 3: THE CONNECTIVITY STRUCTURE OF COMPACT GROUPS ARC CONNECTIVITY ...................................................................................................... 540 LOCAL CONNECTIVITY ................................................................................................... 545 COMPACT GROUPS AND INDECOMPOSABLE CONTINUA ............................................... 548 PART 4: SOME HOMOLOGICAL ALGEBRA FOR COMPACT GROUPS THE PROJECTIVE COVER OF CONNECTED COMPACT GROUPS ....................................... 550 PART 5: THE AUTOMORPHISM GROUP OF COMPACT GROUPS ..................................... 559 THE IWASAWA THEORY OF AUTOMORPHISM GROUPS .................................................. 561 SIMPLE COMPACT GROUPS AND THE COUNTABLE LAYER THEOREM .......................... 569 THE STRUCTURE OF COMPACT FC-GROUPS ................................................................. 571 THE COMMUTATIVITY DEGREE OF A COMPACT GROUP ............................................... 576 POSTSCRIPT ................................................................................................................... 578 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING ............................................... 581 CHAPTER 10. COMPACT GROUP ACTIONS .................................................... 582 A PREPARATION INVOLVING COMPACT SEMIGROUPS .................................................... 583 ORBITS, ORBIT SPACE, AND ISOTROPY ......................................................................... 583 EQUIVARIANCE AND CROSS SECTIONS ............................................................................ 587 TRIVIALITY OF AN ACTION .............................................................................................. 592 QUOTIENT ACTIONS, TOTALLY DISCONNECTED G-SPACES ............................................ 602 COMPACT LIE GROUP ACTIONS ON LOCALLY COMPACT SPACES .................................. 603 TRIVIALITY THEOREMS FOR COMPACT GROUP ACTIONS ............................................... 605 SPLIT MORPHISMS ........................................................................................................ 610 ACTIONS OF COMPACT GROUPS AND ACYCLICITY .......................................................... 623 FIXED POINTS OF COMPACT ABELIAN GROUP ACTIONS ............................................... 625 TRANSITIVE ACTIONS OF COMPACT GROUPS ................................................................. 626 SZENTHE * S THEORY OF TRANSITIVE ACTIONS OF COMPACT GROUPS ............................. 628 POSTSCRIPT ................................................................................................................... 639 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING ............................................... 643 CHAPTER 11. THE STRUCTURE OF FREE COMPACT GROUPS ....................... 644 THE CATEGORY THEORETICAL BACKGROUND ................................................................. 643 SPLITTING THE IDENTITY COMPONENT ......................................................................... 649 THE CENTER OF A FREE COMPACT GROUP ................................................................. 650 THE COMMUTATOR SUBGROUP OF A FREE COMPACT GROUP ..................................... 660 FREENESS VERSUS PROJECTIVITY ................................................................................... 677 POSTSCRIPT ................................................................................................................... 680 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING .............................................. 681 CHAPTER 12. CARDINAL INVARIANTS OF COMPACT GROUPS ........................ 682 SUITABLE SETS ............................................................................................................. 682 CONTENTS XXV GENERATING RANK AND DENSITY ................................................................................ 687 THE CARDINAL INVARIANTS OF CONNECTED COMPACT GROUPS .................................. 692 CARDINAL INVARIANTS IN THE ABSENCE OF CONNECTIVITY .......................................... 696 ON THE LOCATION OF SPECIAL GENERATING SETS ....................................................... 699 POSTSCRIPT .................................................................................................................. 704 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING ............................................... 705 APPENDIX 1. ABELIAN GROUPS ................................................................... 706 EXAMPLES .................................................................................................................. 705 FREE ABELIAN GROUPS ................................................................................................ 710 PROJECTIVE GROUPS ..................................................................................................... 717 TORSION SUBGROUPS .................................................................................................. 718 PURE SUBGROUPS ....................................................................................................... 719 FREE QUOTIENTS .......................................................................................................... 721 DIVISIBILITY .................................................................................................................. 722 SOME HOMOLOGICAL ALGEBRA ..................................................................................... 733 EXACT SEQUENCES .................................................................................... 736 WHITEHEAD * S PROBLEM ............................................................................................. 747 POSTSCRIPT .................................................................................................................. 760 REFERENCES FOR THIS APPENDIX * ADDITIONAL READING ............................................ 760 APPENDIX 2. COVERING SPACES AND GROUPS .......................................... 761 COVERING SPACES AND SIMPLE CONNECTIVITY ............................................................ 760 THE GROUP OF COVERING TRANSFORMATIONS .............................................................. 774 UNIVERSAL COVERING GROUPS ..................................................................................... 775 GROUPS GENERATED BY LOCAL GROUPS ...................................................................... 777 POSTSCRIPT .................................................................................................................. 781 REFERENCES FOR THIS APPENDIX * ADDITIONAL READING ............................................ 782 APPENDIX 3. A PRIMER OF CATEGORY THEORY ........................................ 783 CATEGORIES, MORPHISMS ............................................................................................. 783 POINTED CATEGORIES .................................................................................................. 791 TYPES OF MORPHISMS ................................................................................................ 792 FUNCTORS .................................................................................................................... 799 NATURAL TRANSFORMATIONS ........................................................................................ 811 EQUIVALENCE OF CATEGORIES ........................................................................................ 815 LIMITS ......................................................................................................................... 816 THE CONTINUITY OF ADJOINTS ..................................................................................... 820 THE LEFT ADJOINT EXISTENCE THEOREM ................................................................... 820 COMMUTATIVE MONOIDAL CATEGORIES AND THEIR MONOIDS ..................................... 826 PART 1: THE QUINTESSENTIAL DIAGRAM CHASE ......................................................... 826 PART 2: CONNECTED GRADED COMMUTATIVE HOPF ALGEBRAS .................................. 836 PART 3: DUALITY OF GRADED HOPF ALGEBRAS ............................................................ 852 PART 4: AN APPLICATION TO COMPACT MONOIDS ....................................................... 854 PART 5: WEAKLY COMPLETE SYMMETRIC HOPF ALGEBRAS OVER R AND C ................ 856 XXVI CONTENTS POSTSCRIPT ................................................................................................................... 862 REFERENCES FOR THIS APPENDIX * ADDITIONAL READING ............................................. 863 APPENDIX 4. SELECTED RESULTS ON TOPOLOGY AND TOPOLOGICAL GROUPS 864 THE ARC COMPONENT TOPOLOGY .............................................................................. 864 THE WEIGHT OF A TOPOLOGICAL SPACE ...................................................................... 867 METRIZABILITY OF TOPOLOGICAL GROUPS ....................................................................... 872 DUALITY OF VECTOR SPACES ......................................................................................... 880 SUBGROUPS OF TOPOLOGICAL GROUPS ......................................................................... 881 WALLACE * S LEMMA ...................................................................................................... 883 CANTOR CUBES AND DYADIC SPACES ......................................................................... 883 SOME BASIC FACTS ON COMPACT MONOIDS ............................................................... 885 POSTSCRIPT ................................................................................................................... 888 REFERENCES FOR THIS APPENDIX * ADDITIONAL READING ............................................. 889 APPENDIX 5. MEASURES ON COMPACT GROUPS ........................................ 890 THE DEFINITION OF HAAR MEASURE ............................................................................ 890 THE REQUIRED BACKGROUND OF RADON MEASURE THEORY ....................................... 889 PRODUCT MEASURES ...................................................................................................... 891 THE SUPPORT OF A MEASURE ...................................................................................... 892 MEASURES ON COMPACT GROUPS: CONVOLUTION ....................................................... 894 SEMIGROUP THEORETICAL CHARACTERIZATION OF HAAR MEASURE ................................ 897 IDEMPOTENT PROBABILITY MEASURES ON A COMPACT GROUP .................................. 898 ACTIONS AND PRODUCT MEASURES .............................................................................. 899 NONMEASURABLE SUBGROUPS OF COMPACT GROUPS .................................................. 904 POSTSCRIPT ................................................................................................................... 907 REFERENCES FOR THIS APPENDIX * ADDITIONAL READING ............................................. 908 APPENDIX 6. WELL-ORDERED PROJECTIVE LIMITS, SUPERCOMPACTNESS, AND COMPACT HOMEOMORPHISM GROUPS . 909 WELL-ORDERED LIE CHAINS ........................................................................................... 909 SUPERCOMPACTNESS ...................................................................................................... 912 COMPACT HOMEOMORPHISM GROUPS ......................................................................... 913 POSTSCRIPT ................................................................................................................... 914 REFERENCES FOR THIS APPENDIX * ADDITIONAL READING ............................................. 914 APPENDIX 7. WEAKLY COMPLETE VECTOR SPACES AND ALGEBRAS ........... 915 CHARACTER GROUPS OF TOPOLOGICAL VECTOR SPACES .................................................. 915 FINITE DIMENSIONAL TOPOLOGICAL VECTOR SPACES ....................................................... 917 DUALS OF VECTOR SPACES ........................................................................................... 919 WEAKLY COMPLETE TOPOLOGICAL VECTOR SPACES ....................................................... 922 DUALITY AT WORK FOR WEAKLY COMPLETE TOPOLOGICAL VECTOR SPACES ................... 927 TOPOLOGICAL PROPERTIES OF WEAKLY COMPLETE TOPOLOGICAL VECTOR SPACES .... 930 TENSOR PRODUCTS ........................................................................................................ 933 PRO-LIE GROUPS ........................................................................................................... 934 CONTENTS XXVII WEAKLY COMPLETE UNITAL ALGEBRAS ......................................................................... 936 THE EXPONENTIAL FUNCTION ..................................................................................... 940 POSTSCRIPT .................................................................................................................. 942 REFERENCES FOR THIS APPENDIX * ADDITIONAL READING ............................................ 944 REFERENCES .................................................................................................................. 945 INDEX OF SYMBOLS ..................................................................................................... 965 INDEX ......................................................................................................................... 968
adam_txt	CONTENTS CHAPTER 1. BASIC TOPICS AND EXAMPLES . 1 DEFINITIONS AND ELEMENTARY EXAMPLES . 2 ACTIONS, SUBGROUPS, QUOTIENT SPACES . 5 PRODUCTS OF COMPACT GROUPS . 10 APPLICATIONS TO ABELIAN GROUPS . 11 PROJECTIVE LIMITS . 17 TOTALLY DISCONNECTED COMPACT GROUPS . 22 SOME DUALITY THEORY . 24 POSTSCRIPT . 29 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING . 30 CHAPTER 2. THE BASIC REPRESENTATION THEORY OF COMPACT GROUPS 31 SOME BASIC REPRESENTATION THEORY FOR COMPACT GROUPS . 31 THE HAAR INTEGRAL . 34 CONSEQUENCES OF HAAR MEASURE . 35 THE MAIN THEOREM ON HILBERT MODULES FOR COMPACT GROUPS . 37 POSTSCRIPT . 49 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING . 50 CHAPTER 3. THE IDEAS OF PETER AND WEYL . 51 PART 1: THE CLASSICAL THEOREM OF PETER AND WEYL . 52 AN EXCURSION INTO LINEAR ALGEBRA . 57 THE G-MODULES E' 0 E, HOM(E,E) AND HOM(E, E)' . 59 THE FINE STRUCTURE OF R(G, K) . 61 PART 2: THE GENERAL THEORY OF G-MODULES . 68 VECTOR VALUED INTEGRATION . 69 THE FIRST APPLICATION: THE AVERAGING OPERATOR . 75 COMPACT GROUPS ACTING ON CONVEX CONES . 79 MORE MODULE ACTIONS, CONVOLUTIONS . 80 COMPLEXIFICATION OF REAL REPRESENTATIONS . 85 PART 3: THE WEAKLY COMPLETE GROUP ALGEBRA . 90 THE HOPF ASPECT OF WEAKLY COMPLETE ALGEBRAS . 94 THE DUAL OF A WEAKLY COMPLETE GROUP HOPF ALGEBRA . 101 A PRINCIPAL STRUCTURE THEOREM OF 1K[G] FOR COMPACT G . 104 THE SPECTRUM OF THE K-ALGEBRA R(G, K) . 110 THE TANNAKA-HOCHSCHILD DUALITY . 113 COMPACT ABELIAN GROUPS . 114 THE PROBABILITY SEMIGROUP OF A COMPACT G INSIDE R[G] . 117 POSTSCRIPT . 123 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING . 125 XXII CONTENTS CHAPTER 4. CHARACTERS . 126 PART 1: CHARACTERS OF FINITE DIMENSIONAL REPRESENTATIONS . 126 PART 2: THE STRUCTURE THEOREM OF E& N . 133 CYCLIC MODULES . 144 POSTSCRIPT . 145 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING . 146 CHAPTER 5. LINEAR LIE GROUPS . 147 PRELIMINARIES . 147 THE EXPONENTIAL FUNCTION AND THE LOGARITHM . 149 DIFFERENTIATING THE EXPONENTIAL FUNCTION IN A BANACH ALGEBRA . 161 LOCAL GROUPS FOR THE CAMPBELL -HAUSDORFF MULTIPLICATION . 167 SUBGROUPS OF A -1 AND LINEAR LIE GROUPS . 170 ANALYTIC GROUPS . 174 THE INTRINSIC EXPONENTIAL FUNCTION OF A LINEAR LIE GROUP . 175 THE ADJOINT REPRESENTATION OF A LINEAR LIE GROUP . 182 SUBALGEBRAS, IDEALS, LIE SUBGROUPS, NORMAL LIE SUBGROUPS . 187 NORMALIZERS, CENTRALIZERS, CENTERS . 193 THE COMMUTATOR SUBGROUP . 198 FORCED CONTINUITY OF MORPHISMS BETWEEN LIE GROUPS . 202 QUOTIENTS OF LINEAR LIE GROUPS . 205 THE TOPOLOGICAL SPLITTING THEOREM FOR NORMAL VECTOR SUBGROUPS . 209 POSTSCRIPT . 220 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING . 222 CHAPTER 6. COMPACT LIE GROUPS . 223 COMPACT LIE ALGEBRAS . 224 THE COMMUTATOR SUBGROUP OF A COMPACT LIE GROUP . 235 THE STRUCTURE THEOREM FOR COMPACT LIE GROUPS . 243 MAXIMAL TORI . 246 THE SECOND STRUCTURE THEOREM FOR CONNECTED COMPACT LIE GROUPS . 256 COMPACT ABELIAN LIE GROUPS AND THEIR LINEAR ACTIONS . 260 ACTION OF A MAXIMAL TORUS ON THE LIE ALGEBRA . 264 THE WEYL GROUP REVISITED . 276 THE COMMUTATOR SUBGROUP OF CONNECTED COMPACT LIE GROUPS . 287 ON THE AUTOMORPHISM GROUP OF A COMPACT LIE GROUP . 289 COVERING GROUPS OF DISCONNECTED COMPACT LIE GROUPS . 313 AUERBACH * S GENERATION THEOREM . 315 THE TOPOLOGY OF CONNECTED COMPACT LIE GROUPS . 323 POSTSCRIPT . 333 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING . 335 CHAPTER 7. DUALITY OF ABELIAN TOPOLOGICAL GROUPS . 336 THE COMPACT OPEN TOPOLOGY AND HOM-GROUPS . 337 LOCAL COMPACTNESS AND DUALITY OF ABELIAN TOPOLOGICAL GROUPS . 341 CONTENTS XXIII BASIC FUNCTORIAL ASPECTS OF DUALITY . 348 THE ANNIHILATOR MECHANISM . 351 CHARACTER GROUPS OF TOPOLOGICAL VECTOR SPACES . 362 THE EXPONENTIAL FUNCTION . 372 WEIL * S LEMMA AND COMPACTLY GENERATED ABELIAN GROUPS . 381 REDUCING LOCALLY COMPACT ABELIAN GROUPS TO COMPACT ABELIAN GROUPS . . 385 A MAJOR STRUCTURE THEOREM . 387 THE DUALITY THEOREM . 390 THE IDENTITY COMPONENT . 398 THE WEIGHT OF LOCALLY COMPACT ABELIAN GROUPS . 401 POSTSCRIPT . 403 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING . 406 CHAPTER 8. COMPACT ABELIAN GROUPS . 407 PART 1: ASPECTS OF THE ALGEBRAIC STRUCTURE DIVISIBILITY, TORSION, CONNECTIVITY . 408 COMPACT ABELIAN GROUPS AS FACTOR GROUPS . 414 PART 2: ASPECTS OF THE POINT SET TOPOLOGICAL STRUCTURE TOPOLOGICAL DIMENSION OF COMPACT ABELIAN GROUPS . 422 ARC CONNECTIVITY . 429 LOCAL CONNECTIVITY . 434 COMPACT METRIC ABELIAN GROUPS . 447 PART 3: ASPECTS OF ALGEBRAIC TOPOLOGY* HOMOTOPY FREE COMPACT ABELIAN GROUPS . 450 HOMOTOPY OF COMPACT ABELIAN GROUPS . 455 EXPONENTIAL FUNCTION AND HOMOTOPY . 461 THE FINE STRUCTURE OF FREE COMPACT ABELIAN GROUPS . 462 PART 4: ASPECTS OF HOMOLOGICAL ALGEBRA INJECTIVE, PROJECTIVE, AND FREE COMPACT ABELIAN GROUPS . 469 PART 5: ASPECTS OF ALGEBRAIC TOPOLOGYCOHOMOLOGY COHOMOLOGY OF COMPACT ABELIAN GROUPS . 473 PART 6: ASPECTS OF SET THEORY ARC COMPONENTS AND BOREL SUBSETS . 479 POSTSCRIPT . 482 REFERENCES FOR THIS CHAPTER ADDITIONAL READING . 485 CHAPTER 9. THE STRUCTURE OF COMPACT GROUPS . 486 PART 1: THE FUNDAMENTAL STRUCTURE THEOREMS OF COMPACT GROUPS APPROXIMATING COMPACT GROUPS BY COMPACT LIE GROUPS . 487 THE CLOSEDNESS OF COMMUTATOR SUBGROUPS . 488 SEMISIMPLE COMPACT CONNECTED GROUPS . 489 THE LEVI-MAL * CEV STRUCTURE THEOREM FOR COMPACT GROUPS . 503 MAXIMAL CONNECTED ABELIAN SUBGROUPS . 511 THE SPLITTING STRUCTURE THEOREM . 517 SUPPLEMENTING THE IDENTITY COMPONENT . 518 XXIV CONTENTS PART 2: THE STRUCTURE THEOREMS FOR THE EXPONENTIAL FUNCTION THE EXPONENTIAL FUNCTION OF COMPACT GROUPS . 523 THE DIMENSION OF COMPACT GROUPS . 530 LOCALLY EUCLIDEAN COMPACT GROUPS ARE COMPACT LIE GROUPS . 536 PART 3: THE CONNECTIVITY STRUCTURE OF COMPACT GROUPS ARC CONNECTIVITY . 540 LOCAL CONNECTIVITY . 545 COMPACT GROUPS AND INDECOMPOSABLE CONTINUA . 548 PART 4: SOME HOMOLOGICAL ALGEBRA FOR COMPACT GROUPS THE PROJECTIVE COVER OF CONNECTED COMPACT GROUPS . 550 PART 5: THE AUTOMORPHISM GROUP OF COMPACT GROUPS . 559 THE IWASAWA THEORY OF AUTOMORPHISM GROUPS . 561 SIMPLE COMPACT GROUPS AND THE COUNTABLE LAYER THEOREM . 569 THE STRUCTURE OF COMPACT FC-GROUPS . 571 THE COMMUTATIVITY DEGREE OF A COMPACT GROUP . 576 POSTSCRIPT . 578 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING . 581 CHAPTER 10. COMPACT GROUP ACTIONS . 582 A PREPARATION INVOLVING COMPACT SEMIGROUPS . 583 ORBITS, ORBIT SPACE, AND ISOTROPY . 583 EQUIVARIANCE AND CROSS SECTIONS . 587 TRIVIALITY OF AN ACTION . 592 QUOTIENT ACTIONS, TOTALLY DISCONNECTED G-SPACES . 602 COMPACT LIE GROUP ACTIONS ON LOCALLY COMPACT SPACES . 603 TRIVIALITY THEOREMS FOR COMPACT GROUP ACTIONS . 605 SPLIT MORPHISMS . 610 ACTIONS OF COMPACT GROUPS AND ACYCLICITY . 623 FIXED POINTS OF COMPACT ABELIAN GROUP ACTIONS . 625 TRANSITIVE ACTIONS OF COMPACT GROUPS . 626 SZENTHE * S THEORY OF TRANSITIVE ACTIONS OF COMPACT GROUPS . 628 POSTSCRIPT . 639 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING . 643 CHAPTER 11. THE STRUCTURE OF FREE COMPACT GROUPS . 644 THE CATEGORY THEORETICAL BACKGROUND . 643 SPLITTING THE IDENTITY COMPONENT . 649 THE CENTER OF A FREE COMPACT GROUP . 650 THE COMMUTATOR SUBGROUP OF A FREE COMPACT GROUP . 660 FREENESS VERSUS PROJECTIVITY . 677 POSTSCRIPT . 680 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING . 681 CHAPTER 12. CARDINAL INVARIANTS OF COMPACT GROUPS . 682 SUITABLE SETS . 682 CONTENTS XXV GENERATING RANK AND DENSITY . 687 THE CARDINAL INVARIANTS OF CONNECTED COMPACT GROUPS . 692 CARDINAL INVARIANTS IN THE ABSENCE OF CONNECTIVITY . 696 ON THE LOCATION OF SPECIAL GENERATING SETS . 699 POSTSCRIPT . 704 REFERENCES FOR THIS CHAPTER * ADDITIONAL READING . 705 APPENDIX 1. ABELIAN GROUPS . 706 EXAMPLES . 705 FREE ABELIAN GROUPS . 710 PROJECTIVE GROUPS . 717 TORSION SUBGROUPS . 718 PURE SUBGROUPS . 719 FREE QUOTIENTS . 721 DIVISIBILITY . 722 SOME HOMOLOGICAL ALGEBRA . 733 EXACT SEQUENCES . 736 WHITEHEAD * S PROBLEM . 747 POSTSCRIPT . 760 REFERENCES FOR THIS APPENDIX * ADDITIONAL READING . 760 APPENDIX 2. COVERING SPACES AND GROUPS . 761 COVERING SPACES AND SIMPLE CONNECTIVITY . 760 THE GROUP OF COVERING TRANSFORMATIONS . 774 UNIVERSAL COVERING GROUPS . 775 GROUPS GENERATED BY LOCAL GROUPS . 777 POSTSCRIPT . 781 REFERENCES FOR THIS APPENDIX * ADDITIONAL READING . 782 APPENDIX 3. A PRIMER OF CATEGORY THEORY . 783 CATEGORIES, MORPHISMS . 783 POINTED CATEGORIES . 791 TYPES OF MORPHISMS . 792 FUNCTORS . 799 NATURAL TRANSFORMATIONS . 811 EQUIVALENCE OF CATEGORIES . 815 LIMITS . 816 THE CONTINUITY OF ADJOINTS . 820 THE LEFT ADJOINT EXISTENCE THEOREM . 820 COMMUTATIVE MONOIDAL CATEGORIES AND THEIR MONOIDS . 826 PART 1: THE QUINTESSENTIAL DIAGRAM CHASE . 826 PART 2: CONNECTED GRADED COMMUTATIVE HOPF ALGEBRAS . 836 PART 3: DUALITY OF GRADED HOPF ALGEBRAS . 852 PART 4: AN APPLICATION TO COMPACT MONOIDS . 854 PART 5: WEAKLY COMPLETE SYMMETRIC HOPF ALGEBRAS OVER R AND C . 856 XXVI CONTENTS POSTSCRIPT . 862 REFERENCES FOR THIS APPENDIX * ADDITIONAL READING . 863 APPENDIX 4. SELECTED RESULTS ON TOPOLOGY AND TOPOLOGICAL GROUPS 864 THE ARC COMPONENT TOPOLOGY . 864 THE WEIGHT OF A TOPOLOGICAL SPACE . 867 METRIZABILITY OF TOPOLOGICAL GROUPS . 872 DUALITY OF VECTOR SPACES . 880 SUBGROUPS OF TOPOLOGICAL GROUPS . 881 WALLACE * S LEMMA . 883 CANTOR CUBES AND DYADIC SPACES . 883 SOME BASIC FACTS ON COMPACT MONOIDS . 885 POSTSCRIPT . 888 REFERENCES FOR THIS APPENDIX * ADDITIONAL READING . 889 APPENDIX 5. MEASURES ON COMPACT GROUPS . 890 THE DEFINITION OF HAAR MEASURE . 890 THE REQUIRED BACKGROUND OF RADON MEASURE THEORY . 889 PRODUCT MEASURES . 891 THE SUPPORT OF A MEASURE . 892 MEASURES ON COMPACT GROUPS: CONVOLUTION . 894 SEMIGROUP THEORETICAL CHARACTERIZATION OF HAAR MEASURE . 897 IDEMPOTENT PROBABILITY MEASURES ON A COMPACT GROUP . 898 ACTIONS AND PRODUCT MEASURES . 899 NONMEASURABLE SUBGROUPS OF COMPACT GROUPS . 904 POSTSCRIPT . 907 REFERENCES FOR THIS APPENDIX * ADDITIONAL READING . 908 APPENDIX 6. WELL-ORDERED PROJECTIVE LIMITS, SUPERCOMPACTNESS, AND COMPACT HOMEOMORPHISM GROUPS . 909 WELL-ORDERED LIE CHAINS . 909 SUPERCOMPACTNESS . 912 COMPACT HOMEOMORPHISM GROUPS . 913 POSTSCRIPT . 914 REFERENCES FOR THIS APPENDIX * ADDITIONAL READING . 914 APPENDIX 7. WEAKLY COMPLETE VECTOR SPACES AND ALGEBRAS . 915 CHARACTER GROUPS OF TOPOLOGICAL VECTOR SPACES . 915 FINITE DIMENSIONAL TOPOLOGICAL VECTOR SPACES . 917 DUALS OF VECTOR SPACES . 919 WEAKLY COMPLETE TOPOLOGICAL VECTOR SPACES . 922 DUALITY AT WORK FOR WEAKLY COMPLETE TOPOLOGICAL VECTOR SPACES . 927 TOPOLOGICAL PROPERTIES OF WEAKLY COMPLETE TOPOLOGICAL VECTOR SPACES . 930 TENSOR PRODUCTS . 933 PRO-LIE GROUPS . 934 CONTENTS XXVII WEAKLY COMPLETE UNITAL ALGEBRAS . 936 THE EXPONENTIAL FUNCTION . 940 POSTSCRIPT . 942 REFERENCES FOR THIS APPENDIX * ADDITIONAL READING . 944 REFERENCES . 945 INDEX OF SYMBOLS . 965 INDEX . 968
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author	Hofmann, Karl H. 1932- Morris, Sidney A. 1947-
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dewey-sort	3512.55
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discipline_str_mv	Mathematik
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spelling	Hofmann, Karl H. 1932- Verfasser (DE-588)115780734 aut The structure of compact groups a primer for the student - a handbook for the expert Karl H. Hofmann, Sidney A. Morris 4th edition Berlin/Boston De Gruyter [2020] © 2020 xxvii, 1006 Seiten txt rdacontent n rdamedia nc rdacarrier De Gruyter studies in mathematics Volume 25 Kompakte Gruppe (DE-588)4164840-7 gnd rswk-swf Topological groups Lie algebra Abelian Group; Category Theory; Compact Group; Lie Algebra; Topological Group Category Theory compact groups Abelian Groups Kompakte Gruppe (DE-588)4164840-7 s DE-604 Morris, Sidney A. 1947- Verfasser (DE-588)132019515 aut Erscheint auch als Online-Ausgabe, PDF 978-3-11-069599-1 Erscheint auch als Online-Ausgabe, EPUB 978-3-11-069601-1 Vorangegangen ist 978-3-11-029655-6 De Gruyter studies in mathematics Volume 25 (DE-604)BV000005407 25 B:DE-101 application/pdf https://d-nb.info/1211862887/04 Inhaltsverzeichnis DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032294008&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Hofmann, Karl H. 1932- Morris, Sidney A. 1947- The structure of compact groups a primer for the student - a handbook for the expert De Gruyter studies in mathematics Kompakte Gruppe (DE-588)4164840-7 gnd
subject_GND	(DE-588)4164840-7
title	The structure of compact groups a primer for the student - a handbook for the expert
title_auth	The structure of compact groups a primer for the student - a handbook for the expert
title_exact_search	The structure of compact groups a primer for the student - a handbook for the expert
title_exact_search_txtP	The structure of compact groups a primer for the student - a handbook for the expert
title_full	The structure of compact groups a primer for the student - a handbook for the expert Karl H. Hofmann, Sidney A. Morris
title_fullStr	The structure of compact groups a primer for the student - a handbook for the expert Karl H. Hofmann, Sidney A. Morris
title_full_unstemmed	The structure of compact groups a primer for the student - a handbook for the expert Karl H. Hofmann, Sidney A. Morris
title_short	The structure of compact groups
title_sort	the structure of compact groups a primer for the student a handbook for the expert
title_sub	a primer for the student - a handbook for the expert
topic	Kompakte Gruppe (DE-588)4164840-7 gnd
topic_facet	Kompakte Gruppe
url	https://d-nb.info/1211862887/04 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032294008&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV000005407
work_keys_str_mv	AT hofmannkarlh thestructureofcompactgroupsaprimerforthestudentahandbookfortheexpert AT morrissidneya thestructureofcompactgroupsaprimerforthestudentahandbookfortheexpert

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