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E-raamat: Monotone Complete C*-algebras and Generic Dynamics

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Monotone Complete C*-algebras and Generic DynamicsSuurem pilt
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This monograph covers monotone complete C*-algebras, their properties and the new classification theory, with an introduction of spectroid invariants and a classification semigroup. The purpose is to expound the new theory of monotone complete C*-algebras and introduce some open problems.

The monograph analyses the key aspects of monotone C*-algebras published in the last half century, including the significant early work of Irving Kaplansky and his research fields of AW*-algebras and monotone complete C*-algebras. It subsequently moves on to show how these theories developed in recent times and provides new points of view on previous study; allowing one to restructure and reformulate the presentation of the entire theory. 

Written by experts in the field with over forty years’ experience, Monotone Complete C*-algebras and Generic Dynamics will be of great interest to graduate students and working mathematicians. The prerequisites are elementary functional analysis, point set topology and a basic knowledge of C*-algebras.

Arvustused

The monograph under review, written by two leading experts in the field, is an invaluable contribution to the study and dissemination of m.c. C*-algebras. It provides, in a historic perspective, an updated and thorough description of the recent research on this broad class of C*-algebras, highlighting its important connections with topology, logic, and ergodic theory. ... the present book is an excellent addition to the operator algebras literature, covering for the first time the two themes advertised in the title. (Florin P. Boca, zbMATH 1382.46003, 2018)

1 Introduction
1(6)
1.1 Monotone Complete Algebras of Operators
1(3)
1.2 Generic Dynamics
4(3)
2 Order Fundamentals
7(42)
2.1 Order Structures and Order Convergence
8(16)
2.1.1 Operator Monotone
19(1)
2.1.2 Monotone Closed Subspaces
19(2)
2.1.3 Regular Subalgebras and Subspaces
21(2)
2.1.4 On l2-Summable Sequences
23(1)
2.2 Monotone σ-Complete C*-Algebras
24(12)
2.2.1 Open Problem
30(1)
2.2.2 Polar Decomposition
30(1)
2.2.3 Sequentially Closed Subspaces
31(5)
2.3 Basics for Commutative Algebras
36(7)
2.3.1 Extension Theorems
41(2)
2.4 Matrix Algebras over a Monotone Complete C*-Algebra
43(6)
3 Classification and Invariants
49(14)
3.1 C*-Algebras of Small Size
49(2)
3.2 Classification Semigroup
51(7)
3.3 Spectroid and Representing Functions
58(5)
4 Commutative Algebras: Constructions and Classifications
63(28)
4.1 Boolean Algebra Preliminaries
63(6)
4.1.1 Basics
63(3)
4.1.2 Regular Open Sets
66(2)
4.1.3 Structure Space
68(1)
4.2 Commutative Algebras: General Constructions
69(11)
4.2.1 Measurable Functions
69(2)
4.2.2 Baire Spaces and Category
71(5)
4.2.3 Normal States and Wild Algebras
76(1)
4.2.4 Separable Spaces
77(2)
4.2.5 Baire Measurability
79(1)
4.2.6 Cantor Product Spaces
79(1)
4.3 Constructing and Classifying Wild Commutative Algebras
80(11)
5 Convexity and Representations
91(50)
5.1 Function Representations of Non-commutative Algebras
91(2)
5.2 Compact Convex Sets: Preliminaries
93(6)
5.3 Envelopes of C*-Algebras
99(10)
5.3.1 Open Problem
102(1)
5.3.2 Monotone Envelopes and Homomorphisms
102(5)
5.3.3 Borel Envelopes
107(2)
5.4 Representation Theorems
109(6)
5.4.1 Open Problem
115(1)
5.5 Compact Convex Sets: Semicontinuity and Approximations
115(3)
5.6 Applications of Convexity to Completions
118(7)
5.6.1 Open Problem
124(1)
5.7 Separable State Spaces and Embeddings in L(H)
125(1)
5.7.1 Open Problem
126(1)
5.8 Small C*-Algebras and Completely Positive Maps
126(9)
5.9 Small C*-Algebras and Regular σ-Completions
135(6)
6 Generic Dynamics
141(40)
6.1 Basics
141(3)
6.2 Extending Continuous Functions
144(4)
6.3 Ergodic Discrete Group Actions on Topological Spaces
148(3)
6.4 Induced Actions
151(4)
6.5 Ergodicity on Special Spaces
155(4)
6.5.1 Zero-Dimensional Spaces
157(2)
6.6 Orbit Equivalence: Zero-Dimensional Spaces
159(11)
6.7 Orbit Equivalence: Extremally Disconnected Spaces
170(4)
6.8 Orbit Equivalence: Perfect Polish Spaces
174(1)
6.9 Automorphisms and the Dixmier Algebra
175(4)
6.10 Summary and Preview
179(2)
7 Constructing Monotone Complete C*-Algebras
181(38)
7.1 Monotone Complete C*-Algebra of an Equivalence Relation
181(15)
7.1.1 Induced Actions
193(3)
7.2 Introduction to Cross-Product Algebras
196(6)
7.3 The Normaliser Algebra
202(3)
7.4 Free Dense Actions of the Dyadic Group
205(10)
7.5 Approximately Finite Dimensional Algebras
215(4)
7.5.1 Open Problems
218(1)
8 Envelopes, Completions and AW*-Algebras
219(28)
8.1 Injective Envelopes and Regular Completions
219(15)
8.1.1 Completely Positive Linear Maps and Choi-Effros Algebras
219(5)
8.1.2 Constructing Regular Completions of Unital C*-Algebras
224(10)
8.1.3 Open Problem
234(1)
8.2 What Are AW*-Algebras'?
234(5)
8.3 Projections and AW*-Algebras
239(6)
8.3.1 Projections in AW*-Algebras
239(5)
8.3.2 Open Problems
244(1)
8.4 Conclusions
245(2)
Bibliography 247(8)
Index 255
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