TOPOLOGICAL DYNAMICS: AN INTERNATIONAL SYMPOSIUM
CONTENTS I THE CUNTZ SEMIGROUP AND THE CLASSI?CATION OF C?-ALGEBRAS INTRODUCTION THE CUNTZ SEMIGROUP THE CATEGORY CU FUNCTIONALS AND CU STRUCTURE OF THE CUNTZ SEMIGROUP ELLIOTT’S PROGRAM NUCLEARITY COUNTEREXAMPLES A NEW CONJECTURE Z-STABILITY STRICT COMPARISON NUCLEAR DIMENSION THE CONJECTURE: PRINCIPLE AND PROGRESS NUTS AND BOLTS: PROOF SKETCHES PROOF SKETCH FOR THEOREM 5 4 7 PROOF SKETCH FOR (A SPECIAL CASE OF) THEOREM 5 4 8 BIBLIOGRAPHY II AN INTRODUCTION TO CROSSED PRODUCT C*-ALGEBRAS AND MINIMAL DYNAMICS INTRODUCTION AND MOTIVATION GROUP ACTIONS EXAMPLES OF GROUP ACTIONS ON LOCALLY COMPACT SPACES EXAMPLES OF GROUP ACTIONS ON NONCOMMUTATIVE C?-ALGEBRAS ADDITIONAL EXAMPLES OF GENERALIZED GAUGE ACTIONS CONTENTS GROUP C?-ALGEBRAS AND CROSSED PRODUCTS C?-ALGEBRAS OF DISCRETE GROUPS SIMPLICITY OF THE REDUCED C?-ALGEBRA OF A FREE GROUP C?-ALGEBRAS OF LOCALLY COMPACT GROUPS CROSSED PRODUCTS REDUCED CROSSED PRODUCTS COMPUTATION OF SOME EXAMPLES OF CROSSED PRODUCTS 10 SOME STRUCTURE THEORY FOR CROSSED PRODUCTS BY FINITE GROUPS 10 1 INTRODUCTORY REMARKS ON THE STRUCTURE OF C?-ALGEBRAS 10 2 CROSSED PRODUCTS BY FINITE GROUPS 10 3 THE ROKHLIN PROPERTY FOR ACTIONS OF FINITE GROUPS 10 4 THE TRACIAL ROKHLIN PROPERTY FOR ACTIONS OF FINITE GROUPS 11 CROSSED PRODUCTS BY MINIMAL HOMEOMORPHISMS 11 1 MINIMAL ACTIONS AND THEIR CROSSED PRODUCTS 11 2 CLASSI?ABILITY: INTRODUCTION AND A SPECIAL CASE 11 3 MINIMAL HOMEOMORPHISMS OF FINITE-DIMENSIONAL SPACES 12 LARGE SUBALGEBRAS AND APPLICATIONS TO CROSSED PRODUCTS 12 1 THE CUNTZ SEMIGROUP 12 2 LARGE SUBALGEBRAS 12 3 BASIC PROPERTIES OF LARGE SUBALGEBRAS 12 4 LARGE SUBALGEBRAS AND THE RADIUS OF COMPARISON 12 5 LARGE SUBALGEBRAS IN CROSSED PRODUCTS BY Z 12 6 APPLICATION TO THE RADIUS OF COMPARISON OF CROSSED PRODUCTS 12 7 OPEN PROBLEMS ON LARGE SUBALGEBRAS III C?-ALGEBRAS AND TOPOLOGICAL DYNAMICS: FINITE APPROXIMATION AND PARADOXICALITY 13 INTRODUCTION 14 INTERNAL MEASURE-THEORETIC PHENOMENA 14 1 AMENABLE GROUPS AND NUCLEARITY 14 2 AMENABLE ACTIONS, NUCLEARITY, AND EXACTNESS 14 3 THE TYPE SEMIGROUP, INVARIANT MEASURES, AND PURE IN?NITENESS 14 4 THE UNIVERSAL MINIMAL SYSTEM 14 5 MINIMAL ACTIONS, PURE IN?NITENESS, AND NUCLEARITY CONTENTS 15 EXTERNAL MEASURE-THEORETIC PHENOMENA 15 1 SO?C GROUPS, SO?C ACTIONS, AND HYPERLINEARITY 15 2 ENTROPY 15 3 COMBINATORIAL INDEPENDENCE 15 4 MEAN DIMENSION 16 INTERNAL TOPOLOGICAL PHENOMENA 16 1 LOCALLY FINITE GROUPS AND AF ALGEBRAS 16 2 DIMENSION AND K-THEORETIC CLASSI?CATION 16 3 MINIMAL HOMEOMORPHISMS OF ZERO-DIMENSIONAL SPACES 16 4 MINIMAL HOMEOMORPHISMS OF FINITE-DIMENSIONAL SPACES 16 5 MEAN DIMENSION AND COMPARISON IN THE CUNTZ SEMIGROUP 17 EXTERNAL TOPOLOGICAL PHENOMENA 17 1 GROUPS WHICH ARE LOCALLY EMBEDDABLE INTO FINITE GROUPS 17 2 CHAIN RECURRENCE, RESIDUALLY FINITE ACTIONS, AND MF ALGEBRAS IV MINIMAL TOPOLOGICAL SYSTEMS AND ORBIT EQUIVALENCE 18 INTRODUCTION 19 CANTOR DYNAMICS 19 1 DE?NITIONS OF SOME DYNAMICAL CONCEPTS 19 2 CANTOR MINIMAL SYSTEMS 19 3 DYNAMIC AND ORDERED BRATTELI DIAGRAMS 19 3 1 BRATTELI DIAGRAMS 19 3 2 ORDERED BRATTELI DIAGRAMS AND THE BRATTELI–VERSHIK MODEL 20 ´ETALE EQUIVALENCE RELATIONS 20 1 ´ETALE EQUIVALENCE RELATIONS 20 2 ISOMORPHISM AND ORBIT EQUIVALENCE 20 3 AF-RELATIONS 20 4 INVARIANTS OF ´ETALE EQUIVALENCE RELATIONS 20 4 1 PRE-ORDERED AND ORDERED GROUPS 20 4 2 DIMENSION GROUPS 20 4 3 PRE-ORDERED GROUPS ASSOCIATED TO AN ´ETALE EQUIVALENCE RELATION 20 5 THE BRATTELI–ELLIOTT–KRIEGER THEOREM 21 THE ABSORPTION THEOREM CONTENTS 22 ORBIT EQUIVALENCE OF AF-EQUIVALENCE RELATIONS 23 ORBIT EQUIVALENCE OF MINIMAL ACTIONS OF ABELIAN GROUPS 23 1 ORBIT EQUIVALENCE OF MINIMAL Z-ACTIONS ON THE CANTOR SET 23 2 ORBIT EQUIVALENCE OF MINIMAL ACTIONS OF ABELIAN GROUPS 23 3 A TOPOLOGICAL KRIEGER THEOREM: STRONG ORBIT EQUIVALENCE 24 ORBIT REALIZATION: ORMES’ RESULTS 24 1 ORBIT REALIZATION FOR MINIMAL HOMEOMORPHISMS 24 2 ORBIT EQUIVALENCE OF A CANTOR MINIMAL SYSTEM AND ITS (CONTINUOUS) SPECTRUM 24 3 STRONG ORBIT EQUIVALENCE FOR MINIMAL HOMEOMORPHISMS 25 FULL GROUPS 25 1 THE MEASURABLE CASE 25 1 1 ALGEBRAIC PROPERTIES OF [R] 25 1 2 THE FULL GROUP AS A TOPOLOGICAL GROUP 25 2 THE TOPOLOGICAL CASE 25 2 1 TOPOLOGICAL DYE’S RECONSTRUCTION THEOREMS 25 2 2 THE FULL GROUP OF A CANTOR MINIMAL SYSTEM AS A TOPOLOGICAL GROUP