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E-recursion, Forcing And C*-algebras [Mīkstie vāki]

Edited by (Nus, S'pore), Edited by (The Univ Of California, Berkeley, Usa), Edited by (Chinese Academy Of Sciences, China), Edited by (Univ Of California, Berkeley, Usa), Edited by (Nus, S'pore)
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E-recursion, Forcing And C*-algebrasPalielināt
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This volume presents the lecture notes of short courses given by three leading experts in mathematical logic at the 2012 Asian Initiative for Infinity Logic Summer School. The major topics cover set-theoretic forcing, higher recursion theory, and applications of set theory to C*-algebra. This volume offers a wide spectrum of ideas and techniques introduced in contemporary research in the field of mathematical logic to students, researchers and mathematicians.
Introduction 3(1)
1 Operators on Hilbert Spaces
4(8)
1.1 Subspaces and subalgebras of B(H)
6(1)
1.1.1 Operator space
6(1)
1.1.2 Operator system
7(1)
1.1.3 Concrete C*-algebras
7(1)
1.1.4 Non-self-adjoint subalgebras
7(1)
1.1.5 von Neumann algebras
7(1)
1.2 Exercises
7(2)
1.3 Spectrum and spectral radius
9(1)
1.4 Exercises
9(1)
1.5 Normal operators and the spectral theorem
10(1)
1.5.1 Types of bounded operators
11(1)
1.6 Exercises
11(1)
2 Preliminaries on C*-Algebras
12(11)
2.1 Positivity, states and the GNS theorem
12(1)
2.2 Exercises
13(1)
2.3 Continuous functional calculus
14(3)
2.4 Exercises
17(1)
2.5 Constructions of C*-algebras
18(3)
2.5.1 Unitization
18(1)
2.5.2 Direct sums
18(1)
2.5.3 Direct products
18(1)
2.5.4 Direct limits (also called inductive limits)
19(1)
2.5.5 Matrix algebra over A
19(1)
2.5.6 Stabilization
19(1)
2.5.7 Minimal tensor product
20(1)
2.5.8 Continuous fields of C*-algebras
21(1)
2.5.9 Corners
21(1)
2.5.10 and so on
21(1)
2.6 Exercises
21(2)
3 Local Theory of C*-Algebras
23(9)
3.1 Polar decomposition
23(1)
3.2 Stability
24(2)
3.3 Exercises
26(1)
3.4 Murray-von Neumann equivalence of projections
26(2)
3.5 Exercises
28(1)
3.6 Traces
29(3)
3.7 Exercises
32(1)
4 UHF Algebras and AF Algebras
32(10)
4.1 UHF algebras
32(2)
4.2 Another look at the UHF algebras
34(1)
4.3 Exercises
35(1)
4.4 Bratteli diagrams
36(2)
4.5 Exercises
38(2)
4.6 AF algebras
40(1)
4.7 Exercises
40(2)
5 The Functor Ko
42(7)
5.1 Computation of Ko in some simple cases
43(1)
5.1.1 Ko of Mn(C)
43(1)
5.1.2 Ko of B{H)
43(1)
5.1.3 Ko of the Calkin algebra
43(1)
5.1.4 Ko of the CAR algebra
43(1)
5.1.5 Ko of other UHF algebras
43(1)
5.1.6 Ko of a *-homomorphism
44(1)
5.2 Exercises
44(1)
5.3 Cancellation property
44(2)
5.4 Classification of AF algebras
46(2)
5.5 Exercises
48(1)
6 Elliott's Program
49(4)
6.0.1 Failure of cancellation
51(1)
6.1 Exercises
51(2)
7 Abstract Classification
53(8)
7.1 Effros Borel space
55(3)
7.1.1 Spaces of countable structures
55(1)
7.1.2 Compact metric spaces
55(1)
7.1.3 Separable Banach spaces
56(1)
7.1.4 von Neumann algebras with a separable predual
57(1)
7.1.5 Separable C*-algebras
57(1)
7.2 Computation of the Elliott invariant is Borel
58(1)
7.3 Comparing complexities of analytic equivalence relations
59(1)
7.3.1 Relation Eo
59(1)
7.3.2 Essentially countable equivalence relations
59(1)
7.3.3 Countable structures
59(1)
7.3.4 Orbit equivalence relations
59(1)
7.3.5 Turbulence
60(1)
7.3.6 The dark side
60(1)
7.4 Exercises
60(1)
8 Estimating the Complexity of the Isomorphism of C*-Algebras
61(10)
8.1 Turbulence: A lower bound for complexity
61(6)
8.2 Below a group action: An upper bound for complexity
67(1)
8.3 Cuntz algebra O2
67(3)
8.4 Exercises
70(1)
9 Concluding Remarks
71(5)
9.1 The Borel-reducibility diagram
71(3)
9.2 Selected open problems
74(2)
References 76