Verfügbarkeit: The structure of compact groups

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

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Bibliographische Detailangaben
Hauptverfasser:	Hofmann, Karl H. 1932- (VerfasserIn), Morris, Sidney A. 1947- (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Berlin ; Boston De Gruyter [2023]
Ausgabe:	5th edition
Schriftenreihe:	De Gruyter studies in mathematics Volume 25
Schlagworte:	Kompakte Gruppe Abelsche Gruppen Kategorientheorie Kompakte Gruppen Luge Algebra Topologische Gruppen Abelian groups category theory compact groups Lie algebra topological groups.
Online-Zugang:	https://www.degruyter.com/isbn/9783111171630 Inhaltsverzeichnis
Beschreibung:	xli, 1032 Seiten 24 cm x 17 cm
ISBN:	9783111171630 3111171639

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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 5th edition
264		1	\|a Berlin ; Boston \|b De Gruyter \|c [2023]
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653			\|a Kategorientheorie
653			\|a Kompakte Gruppen
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653			\|a Topologische Gruppen
653			\|a Abelian groups
653			\|a category theory
653			\|a compact groups
653			\|a Lie algebra
653			\|a topological groups.
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Datensatz im Suchindex

_version_	1805075535662940160
adam_text	HIGHLIGHTS OF THE STRUCTURE THEORY OF COMPACT GROUPS . IX 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 . 50 REFERENCES FOR THIS CHAPTER - ADDITIONAL READING . 50 CHAPTER 3. THE IDEAS OF PETER & WEYL, TANNAKA, HOPF, & HOCHSCHILD 51 PART 1: THE CLASSICAL THEOREM OF PETER AND WEYL . 52 AN EXCURSION INTO LINEAR ALGEBRA . 57 THE G-MODULES E' 8 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 K[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] . 118 POSTSCRIPT . 123 REFERENCES FOR THIS CHAPTER - ADDITIONAL READING . 125 CHAPTER 4. CHARACTERS . 126 PART 1: CHARACTERS OF FINITE DIMENSIONAL REPRESENTATIONS . 126 PART 2: THE STRUCTURE THEOREM OF EFIN . 133 THE STRUCTURE OF HILBERT G-MODULES . 142 THE STRUCTURE OF WEAKLY COMPLETE G-MODULES . 143 CYCLIC MODULES . 144 POSTSCRIPT . 146 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 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 TOPOLOGY - COHOMOLOGY COHOMOLOGY OF COMPACT ABELIAN GROUPS . 473 PART 6: ASPECTS OF SET THEORY ARC COMPONENTS AND BOREL SUBSETS . 479 PART 7: THE BOHR COMPACTIFICATION: THE CASE OF ABELIAN TOPOLOGICAL GROUPS . 482 POSTSCRIPT . 487 REFERENCES FOR THIS CHAPTER - ADDITIONAL READING . 490 CHAPTER 9. THE STRUCTURE OF COMPACT GROUPS . 491 PART 1: THE FUNDAMENTAL STRUCTURE THEOREMS OF COMPACT GROUPS APPROXIMATING COMPACT GROUPS BY COMPACT LIE GROUPS . 492 THE CLOSEDNESS OF COMMUTATOR SUBGROUPS . 493 SEMISIMPLE COMPACT CONNECTED GROUPS . 494 THE LEVI-MAL ' CEV STRUCTURE THEOREM FOR COMPACT GROUPS . 508 MAXIMAL CONNECTED ABELIAN SUBGROUPS . 516 THE SPLITTING STRUCTURE THEOREM . 522 SUPPLEMENTING THE IDENTITY COMPONENT . 523 PART 2: THE STRUCTURE THEOREMS FOR THE EXPONENTIAL FUNCTION THE EXPONENTIAL FUNCTION OF COMPACT GROUPS 528 THE DIMENSION OF COMPACT GROUPS . 535 LOCALLY EUCLIDEAN COMPACT GROUPS ARE COMPACT LIE GROUPS . 541 PART 3: THE CONNECTIVITY STRUCTURE OF COMPACT GROUPS ARC CONNECTIVITY . 545 LOCAL CONNECTIVITY . 550 COMPACT GROUPS AND INDECOMPOSABLE CONTINUA . 553 PART 4: SOME HOMOLOGICAL ALGEBRA FOR COMPACT GROUPS THE PROJECTIVE COVER OF CONNECTED COMPACT GROUPS . 555 PART 5: THE AUTOMORPHISM GROUP OF COMPACT GROUPS . 564 THE IWASAWA THEORY OF AUTOMORPHISM GROUPS . 566 SIMPLE COMPACT GROUPS AND THE COUNTABLE LAYER THEOREM . 574 THE STRUCTURE OF COMPACT FC-GROUPS . 576 THE COMMUTATIVITY DEGREE OF A COMPACT GROUP . 581 POSTSCRIPT . 583 REFERENCES FOR THIS CHAPTER - ADDITIONAL READING . 586 CHAPTER 10. COMPACT GROUP ACTIONS . 587 A PREPARATION INVOLVING COMPACT SEMIGROUPS . 588 ORBITS, ORBIT SPACE, AND ISOTROPY . 588 EQUIVARIANCE AND CROSS SECTIONS . 592 TRIVIALITY OF AN ACTION . 597 QUOTIENT ACTIONS, TOTALLY DISCONNECTED G-SPACES . 607 COMPACT LIE GROUP ACTIONS ON LOCALLY COMPACT SPACES . 608 TRIVIALITY THEOREMS FOR COMPACT GROUP ACTIONS . 610 SPLIT MORPHISMS . 615 ACTIONS OF COMPACT GROUPS AND ACYCLICITY . 628 FIXED POINTS OF COMPACT ABELIAN GROUP ACTIONS . 630 TRANSITIVE ACTIONS OF COMPACT GROUPS . 631 SZENTHE ' S THEORY OF TRANSITIVE ACTIONS OF COMPACT GROUPS . 633 POSTSCRIPT . 644 REFERENCES FOR THIS CHAPTER - ADDITIONAL READING . 648 CHAPTER 11. THE STRUCTURE OF FREE COMPACT GROUPS . 649 THE CATEGORY THEORETICAL BACKGROUND . 649 SPLITTING THE IDENTITY COMPONENT . 654 THE CENTER OF A FREE COMPACT GROUP . 655 THE COMMUTATOR SUBGROUP OF A FREE COMPACT GROUP . 665 FREENESS VERSUS PROJECTIVITY . 682 POSTSCRIPT . 685 REFERENCES FOR THIS CHAPTER - ADDITIONAL READING . 686 CHAPTER 12. CARDINAL INVARIANTS OF COMPACT GROUPS . 687 SUITABLE SETS . 687 GENERATING RANK AND DENSITY . 692 THE CARDINAL INVARIANTS OF CONNECTED COMPACT GROUPS . 697 CARDINAL INVARIANTS IN THE ABSENCE OF CONNECTIVITY . 701 ON THE LOCATION OF SPECIAL GENERATING SETS . 704 POSTSCRIPT . 709 REFERENCES FOR THIS CHAPTER - ADDITIONAL READING . 710 APPENDIX 1. ABELIAN GROUPS . 711 EXAMPLES . 711 FREE ABELIAN GROUPS . 715 PROJECTIVE GROUPS . 722 TORSION SUBGROUPS . 723 PURE SUBGROUPS . 724 FREE QUOTIENTS . 726 DIVISIBILITY . 727 SOME HOMOLOGICAL ALGEBRA . 738 EXACT SEQUENCES . 741 WHITEHEAD ' S PROBLEM . 752 POSTSCRIPT . 765 REFERENCES FOR THIS APPENDIX - ADDITIONAL READING . 765 APPENDIX 2. COVERING SPACES AND GROUPS . 766 COVERING SPACES AND SIMPLE CONNECTIVITY . 766 THE GROUP OF COVERING TRANSFORMATIONS . 779 UNIVERSAL COVERING GROUPS . 780 GROUPS GENERATED BY LOCAL GROUPS . 782 POSTSCRIPT . 786 REFERENCES FOR THIS APPENDIX - ADDITIONAL READING . 787 APPENDIX 3. A PRIMER OF CATEGORY THEORY . 788 PART 1: GENERAL CATEGORY THEORY . 788 CATEGORIES, MORPHISMS . 788 POINTED CATEGORIES . 796 TYPES OF MORPHISMS . 797 FUNCTORS . 805 NATURAL TRANSFORMATIONS . 817 EQUIVALENCE OF CATEGORIES . 821 LIMITS . 822 THE CONTINUITY OF ADJOINTS . 826 THE LEFT ADJOINT EXISTENCE THEOREM . 826 THE BOHR COMPACTIFICATION . 831 PART 2: COMMUTATIVE MONOIDAL CATEGORIES AND THEIR MONOIDS-HOPF ALGEBRAS 833 PART 2-1: THE QUINTESSENTIAL DIAGRAM CHASE . 833 PART 2-2: CONNECTED GRADED COMMUTATIVE HOPF ALGEBRAS . 843 PART 2-3: DUALITY OF GRADED HOPF ALGEBRAS . 859 PART 2-4: AN APPLICATION TO COMPACT MONOIDS . 861 PART 2-5: WEAKLY COMPLETE SYMMETRIC HOPF ALGEBRAS OVER R AND C . 863 PART 2-6: WEAKLY COMPLETE GROUP ALGEBRAS AND UNIVERSAL ENVELOPING ALGEBRAS 869 PART 2-7 ENVELOPING ALGEBRAS VERSUS GROUP ALGEBRAS . 875 POSTSCRIPT . 879 REFERENCES FOR THIS APPENDIX - ADDITIONAL READING . 880 APPENDIX 4. SELECTED RESULTS ON TOPOLOGY AND TOPOLOGICAL GROUPS 881 THE ARC COMPONENT TOPOLOGY . 881 THE WEIGHT OF A TOPOLOGICAL SPACE . 884 METRIZABILITY OF TOPOLOGICAL GROUPS . 889 DUALITY OF VECTOR SPACES . 897 SUBGROUPS OF TOPOLOGICAL GROUPS . 898 WALLACE ' S LEMMA . 900 CANTOR CUBES AND DYADIC SPACES . 900 SOME BASIC FACTS ON COMPACT MONOIDS . 902 POSTSCRIPT . 905 REFERENCES FOR THIS APPENDIX - ADDITIONAL READING . 906 APPENDIX 5. MEASURES ON COMPACT GROUPS . 907 THE DEFINITION OF HAAR MEASURE . 907 THE REQUIRED BACKGROUND OF RADON MEASURE THEORY . 907 PRODUCT MEASURES . 908 THE SUPPORT OF A MEASURE . 909 MEASURES ON COMPACT GROUPS: CONVOLUTION . 911 SEMIGROUP THEORETICAL CHARACTERIZATION OF HAAR MEASURE . 914 IDEMPOTENT PROBABILITY MEASURES ON A COMPACT GROUP . 915 ACTIONS AND PRODUCT MEASURES . 916 NONMEASURABLE SUBGROUPS OF COMPACT GROUPS . 921 POSTSCRIPT . 924 REFERENCES FOR THIS APPENDIX - ADDITIONAL READING . 925 APPENDIX 6. WELL-ORDERED PROJECTIVE LIMITS, SUPERCOMPACTNESS, AND COMPACT HOMEOMORPHISM GROUPS . 926 WELL-ORDERED LIE CHAINS . 926 SUPERCOMPACTNESS . 929 COMPACT HOMEOMORPHISM GROUPS . 930 POSTSCRIPT . 931 REFERENCES FOR THIS APPENDIX - ADDITIONAL READING . 931 APPENDIX 7. WEAKLY COMPLETE VECTOR SPACES AND ALGEBRAS . 932 CHARACTER GROUPS OF TOPOLOGICAL VECTOR SPACES . 932 FINITE DIMENSIONAL TOPOLOGICAL VECTOR SPACES . 934 THE FINEST LOCALLY CONVEX TOPOLOGY . 936 WEAKLY COMPLETE TOPOLOGICAL VECTOR SPACES . 940 DUALITY AT WORK FOR WEAKLY COMPLETE TOPOLOGICAL VECTOR SPACES . 944 TOPOLOGICAL PROPERTIES OF WEAKLY COMPLETE TOPOLOGICAL VECTOR SPACES . . . 948 TENSOR PRODUCTS . 950 PRO-LIE GROUPS . 951 WEAKLY COMPLETE UNITAL ALGEBRAS . 953 THE EXPONENTIAL FUNCTION . 963 POSTSCRIPT . 965 REFERENCES FOR THIS APPENDIX - ADDITIONAL READING . 967 REFERENCES . 969 INDEX OF SYMBOLS . 991 INDEX . 993
adam_txt	HIGHLIGHTS OF THE STRUCTURE THEORY OF COMPACT GROUPS . IX 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 . 50 REFERENCES FOR THIS CHAPTER - ADDITIONAL READING . 50 CHAPTER 3. THE IDEAS OF PETER & WEYL, TANNAKA, HOPF, & HOCHSCHILD 51 PART 1: THE CLASSICAL THEOREM OF PETER AND WEYL . 52 AN EXCURSION INTO LINEAR ALGEBRA . 57 THE G-MODULES E' 8 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 K[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] . 118 POSTSCRIPT . 123 REFERENCES FOR THIS CHAPTER - ADDITIONAL READING . 125 CHAPTER 4. CHARACTERS . 126 PART 1: CHARACTERS OF FINITE DIMENSIONAL REPRESENTATIONS . 126 PART 2: THE STRUCTURE THEOREM OF EFIN . 133 THE STRUCTURE OF HILBERT G-MODULES . 142 THE STRUCTURE OF WEAKLY COMPLETE G-MODULES . 143 CYCLIC MODULES . 144 POSTSCRIPT . 146 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 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 TOPOLOGY - COHOMOLOGY COHOMOLOGY OF COMPACT ABELIAN GROUPS . 473 PART 6: ASPECTS OF SET THEORY ARC COMPONENTS AND BOREL SUBSETS . 479 PART 7: THE BOHR COMPACTIFICATION: THE CASE OF ABELIAN TOPOLOGICAL GROUPS . 482 POSTSCRIPT . 487 REFERENCES FOR THIS CHAPTER - ADDITIONAL READING . 490 CHAPTER 9. THE STRUCTURE OF COMPACT GROUPS . 491 PART 1: THE FUNDAMENTAL STRUCTURE THEOREMS OF COMPACT GROUPS APPROXIMATING COMPACT GROUPS BY COMPACT LIE GROUPS . 492 THE CLOSEDNESS OF COMMUTATOR SUBGROUPS . 493 SEMISIMPLE COMPACT CONNECTED GROUPS . 494 THE LEVI-MAL ' CEV STRUCTURE THEOREM FOR COMPACT GROUPS . 508 MAXIMAL CONNECTED ABELIAN SUBGROUPS . 516 THE SPLITTING STRUCTURE THEOREM . 522 SUPPLEMENTING THE IDENTITY COMPONENT . 523 PART 2: THE STRUCTURE THEOREMS FOR THE EXPONENTIAL FUNCTION THE EXPONENTIAL FUNCTION OF COMPACT GROUPS 528 THE DIMENSION OF COMPACT GROUPS . 535 LOCALLY EUCLIDEAN COMPACT GROUPS ARE COMPACT LIE GROUPS . 541 PART 3: THE CONNECTIVITY STRUCTURE OF COMPACT GROUPS ARC CONNECTIVITY . 545 LOCAL CONNECTIVITY . 550 COMPACT GROUPS AND INDECOMPOSABLE CONTINUA . 553 PART 4: SOME HOMOLOGICAL ALGEBRA FOR COMPACT GROUPS THE PROJECTIVE COVER OF CONNECTED COMPACT GROUPS . 555 PART 5: THE AUTOMORPHISM GROUP OF COMPACT GROUPS . 564 THE IWASAWA THEORY OF AUTOMORPHISM GROUPS . 566 SIMPLE COMPACT GROUPS AND THE COUNTABLE LAYER THEOREM . 574 THE STRUCTURE OF COMPACT FC-GROUPS . 576 THE COMMUTATIVITY DEGREE OF A COMPACT GROUP . 581 POSTSCRIPT . 583 REFERENCES FOR THIS CHAPTER - ADDITIONAL READING . 586 CHAPTER 10. COMPACT GROUP ACTIONS . 587 A PREPARATION INVOLVING COMPACT SEMIGROUPS . 588 ORBITS, ORBIT SPACE, AND ISOTROPY . 588 EQUIVARIANCE AND CROSS SECTIONS . 592 TRIVIALITY OF AN ACTION . 597 QUOTIENT ACTIONS, TOTALLY DISCONNECTED G-SPACES . 607 COMPACT LIE GROUP ACTIONS ON LOCALLY COMPACT SPACES . 608 TRIVIALITY THEOREMS FOR COMPACT GROUP ACTIONS . 610 SPLIT MORPHISMS . 615 ACTIONS OF COMPACT GROUPS AND ACYCLICITY . 628 FIXED POINTS OF COMPACT ABELIAN GROUP ACTIONS . 630 TRANSITIVE ACTIONS OF COMPACT GROUPS . 631 SZENTHE ' S THEORY OF TRANSITIVE ACTIONS OF COMPACT GROUPS . 633 POSTSCRIPT . 644 REFERENCES FOR THIS CHAPTER - ADDITIONAL READING . 648 CHAPTER 11. THE STRUCTURE OF FREE COMPACT GROUPS . 649 THE CATEGORY THEORETICAL BACKGROUND . 649 SPLITTING THE IDENTITY COMPONENT . 654 THE CENTER OF A FREE COMPACT GROUP . 655 THE COMMUTATOR SUBGROUP OF A FREE COMPACT GROUP . 665 FREENESS VERSUS PROJECTIVITY . 682 POSTSCRIPT . 685 REFERENCES FOR THIS CHAPTER - ADDITIONAL READING . 686 CHAPTER 12. CARDINAL INVARIANTS OF COMPACT GROUPS . 687 SUITABLE SETS . 687 GENERATING RANK AND DENSITY . 692 THE CARDINAL INVARIANTS OF CONNECTED COMPACT GROUPS . 697 CARDINAL INVARIANTS IN THE ABSENCE OF CONNECTIVITY . 701 ON THE LOCATION OF SPECIAL GENERATING SETS . 704 POSTSCRIPT . 709 REFERENCES FOR THIS CHAPTER - ADDITIONAL READING . 710 APPENDIX 1. ABELIAN GROUPS . 711 EXAMPLES . 711 FREE ABELIAN GROUPS . 715 PROJECTIVE GROUPS . 722 TORSION SUBGROUPS . 723 PURE SUBGROUPS . 724 FREE QUOTIENTS . 726 DIVISIBILITY . 727 SOME HOMOLOGICAL ALGEBRA . 738 EXACT SEQUENCES . 741 WHITEHEAD ' S PROBLEM . 752 POSTSCRIPT . 765 REFERENCES FOR THIS APPENDIX - ADDITIONAL READING . 765 APPENDIX 2. COVERING SPACES AND GROUPS . 766 COVERING SPACES AND SIMPLE CONNECTIVITY . 766 THE GROUP OF COVERING TRANSFORMATIONS . 779 UNIVERSAL COVERING GROUPS . 780 GROUPS GENERATED BY LOCAL GROUPS . 782 POSTSCRIPT . 786 REFERENCES FOR THIS APPENDIX - ADDITIONAL READING . 787 APPENDIX 3. A PRIMER OF CATEGORY THEORY . 788 PART 1: GENERAL CATEGORY THEORY . 788 CATEGORIES, MORPHISMS . 788 POINTED CATEGORIES . 796 TYPES OF MORPHISMS . 797 FUNCTORS . 805 NATURAL TRANSFORMATIONS . 817 EQUIVALENCE OF CATEGORIES . 821 LIMITS . 822 THE CONTINUITY OF ADJOINTS . 826 THE LEFT ADJOINT EXISTENCE THEOREM . 826 THE BOHR COMPACTIFICATION . 831 PART 2: COMMUTATIVE MONOIDAL CATEGORIES AND THEIR MONOIDS-HOPF ALGEBRAS 833 PART 2-1: THE QUINTESSENTIAL DIAGRAM CHASE . 833 PART 2-2: CONNECTED GRADED COMMUTATIVE HOPF ALGEBRAS . 843 PART 2-3: DUALITY OF GRADED HOPF ALGEBRAS . 859 PART 2-4: AN APPLICATION TO COMPACT MONOIDS . 861 PART 2-5: WEAKLY COMPLETE SYMMETRIC HOPF ALGEBRAS OVER R AND C . 863 PART 2-6: WEAKLY COMPLETE GROUP ALGEBRAS AND UNIVERSAL ENVELOPING ALGEBRAS 869 PART 2-7 ENVELOPING ALGEBRAS VERSUS GROUP ALGEBRAS . 875 POSTSCRIPT . 879 REFERENCES FOR THIS APPENDIX - ADDITIONAL READING . 880 APPENDIX 4. SELECTED RESULTS ON TOPOLOGY AND TOPOLOGICAL GROUPS 881 THE ARC COMPONENT TOPOLOGY . 881 THE WEIGHT OF A TOPOLOGICAL SPACE . 884 METRIZABILITY OF TOPOLOGICAL GROUPS . 889 DUALITY OF VECTOR SPACES . 897 SUBGROUPS OF TOPOLOGICAL GROUPS . 898 WALLACE ' S LEMMA . 900 CANTOR CUBES AND DYADIC SPACES . 900 SOME BASIC FACTS ON COMPACT MONOIDS . 902 POSTSCRIPT . 905 REFERENCES FOR THIS APPENDIX - ADDITIONAL READING . 906 APPENDIX 5. MEASURES ON COMPACT GROUPS . 907 THE DEFINITION OF HAAR MEASURE . 907 THE REQUIRED BACKGROUND OF RADON MEASURE THEORY . 907 PRODUCT MEASURES . 908 THE SUPPORT OF A MEASURE . 909 MEASURES ON COMPACT GROUPS: CONVOLUTION . 911 SEMIGROUP THEORETICAL CHARACTERIZATION OF HAAR MEASURE . 914 IDEMPOTENT PROBABILITY MEASURES ON A COMPACT GROUP . 915 ACTIONS AND PRODUCT MEASURES . 916 NONMEASURABLE SUBGROUPS OF COMPACT GROUPS . 921 POSTSCRIPT . 924 REFERENCES FOR THIS APPENDIX - ADDITIONAL READING . 925 APPENDIX 6. WELL-ORDERED PROJECTIVE LIMITS, SUPERCOMPACTNESS, AND COMPACT HOMEOMORPHISM GROUPS . 926 WELL-ORDERED LIE CHAINS . 926 SUPERCOMPACTNESS . 929 COMPACT HOMEOMORPHISM GROUPS . 930 POSTSCRIPT . 931 REFERENCES FOR THIS APPENDIX - ADDITIONAL READING . 931 APPENDIX 7. WEAKLY COMPLETE VECTOR SPACES AND ALGEBRAS . 932 CHARACTER GROUPS OF TOPOLOGICAL VECTOR SPACES . 932 FINITE DIMENSIONAL TOPOLOGICAL VECTOR SPACES . 934 THE FINEST LOCALLY CONVEX TOPOLOGY . 936 WEAKLY COMPLETE TOPOLOGICAL VECTOR SPACES . 940 DUALITY AT WORK FOR WEAKLY COMPLETE TOPOLOGICAL VECTOR SPACES . 944 TOPOLOGICAL PROPERTIES OF WEAKLY COMPLETE TOPOLOGICAL VECTOR SPACES . . . 948 TENSOR PRODUCTS . 950 PRO-LIE GROUPS . 951 WEAKLY COMPLETE UNITAL ALGEBRAS . 953 THE EXPONENTIAL FUNCTION . 963 POSTSCRIPT . 965 REFERENCES FOR THIS APPENDIX - ADDITIONAL READING . 967 REFERENCES . 969 INDEX OF SYMBOLS . 991 INDEX . 993
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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 5th edition Berlin ; Boston De Gruyter [2023] xli, 1032 Seiten 24 cm x 17 cm txt rdacontent n rdamedia nc rdacarrier De Gruyter studies in mathematics Volume 25 Kompakte Gruppe (DE-588)4164840-7 gnd rswk-swf Abelsche Gruppen Kategorientheorie Kompakte Gruppen Luge Algebra Topologische Gruppen Abelian groups category theory compact groups Lie algebra topological groups. Kompakte Gruppe (DE-588)4164840-7 s DE-604 Morris, Sidney A. 1947- Verfasser (DE-588)132019515 aut Walter de Gruyter GmbH & Co. KG (DE-588)10095502-2 pbl Erscheint auch als Online-Ausgabe, PDF 978-3-11-117260-6 Erscheint auch als Online-Ausgabe, EPUB 978-3-11-117405-1 De Gruyter studies in mathematics Volume 25 (DE-604)BV000005407 25 X:MVB https://www.degruyter.com/isbn/9783111171630 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034746170&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p vlb 20230812 DE-101 https://d-nb.info/provenance/plan#vlb
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://www.degruyter.com/isbn/9783111171630 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034746170&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV000005407
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