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E-grāmata: K-Theory for Group C*-Algebras and Semigroup C*-Algebras

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  • Formāts: PDF+DRM
  • Sērija : Oberwolfach Seminars 47
  • Izdošanas datums: 24-Oct-2017
  • Izdevniecība: Birkhauser Verlag AG
  • Valoda: eng
  • ISBN-13: 9783319599151
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K-Theory for Group C*-Algebras and Semigroup C*-AlgebrasPalielināt
  • Formāts: PDF+DRM
  • Sērija : Oberwolfach Seminars 47
  • Izdošanas datums: 24-Oct-2017
  • Izdevniecība: Birkhauser Verlag AG
  • Valoda: eng
  • ISBN-13: 9783319599151
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This book gives an account of the necessary background for group algebras and crossed products for actions of a group or a semigroup on a space and reports on some very recently developed techniques with applications to particular examples. Much of the material is available here for the first time in book form. The topics discussed are among the most classical and intensely studied C*-algebras. They are important for applications in fields as diverse as the theory of unitary group representations, index theory, the topology of manifolds or ergodic theory of group actions.

Part of the most basic structural information for such a C*-algebra is contained in its K-theory. The determination of the K-groups of C*-algebras constructed from group or semigroup actions is a particularly challenging problem. Paul Baum and Alain Connes proposed a formula for the K-theory of the reduced crossed product for a group action that would permit, in principle, its computation. By work of many hands, the formula has by now been verified for very large classes of groups and this work has led to the development of a host of new techniques. An important ingredient is Kasparov's bivariant K-theory.

More recently, also the C*-algebras generated by the regular representation of a semigroup as well as the crossed products for actions of semigroups by endomorphisms have been studied in more detail.





Intriguing examples of actions of such semigroups come from ergodic theory as well as from algebraic number theory. The computation of the K-theory of the corresponding crossed products needs new techniques. In cases of interest the K-theory of the algebras reflects ergodic theoretic or number theoretic properties of the action.
Preface ix
1 Introduction
1(4)
2 Crossed products and the Mackey--Rieffel--Green machine
5(76)
Siegfried Echterhoff
2.1 Introduction
5(1)
2.2 Some preliminaries
6(5)
2.2.1 C*-algebras
6(1)
2.2.2 Multiplier algebras
7(1)
2.2.3 Commutative C*-algebras and functional calculus
7(2)
2.2.4 Representation and ideal spaces of C*-algebras
9(1)
2.2.5 Tensor products
10(1)
2.3 Actions and their crossed products
11(5)
2.3.1 Haar measure and vector-valued integration on groups
11(1)
2.3.2 C*-dynamical systems and their crossed products
12(4)
2.4 Crossed products versus tensor products
16(3)
2.5 The correspondence categories
19(12)
2.5.1 Hilbert modules
20(1)
2.5.2 Morita equivalences
21(3)
2.5.3 The correspondence categories
24(1)
2.5.4 The equivariant correspondence categories
25(1)
2.5.5 Induced representations and ideals
26(4)
2.5.6 The Fell topologies and weak containment
30(1)
2.6 Green's imprimitivity theorem and applications
31(12)
2.6.1 The imprimitivity theorem
31(8)
2.6.2 The Takesaki--Takai duality theorem
39(1)
2.6.3 Permanence properties of exact groups
40(3)
2.7 Induced representations and the ideal structure of crossed products
43(21)
2.7.1 Induced representations of groups and crossed products
43(9)
2.7.2 The ideal structure of crossed products
52(9)
2.7.3 The Mackey machine for transformation groups
61(3)
2.8 The Mackey--Rieffel--Green machine for twisted crossed products
64(17)
2.8.1 Twisted actions and twisted crossed products
64(4)
2.8.2 The twisted equivariant correspondence category and the stabilisation trick
68(3)
2.8.3 Twisted Takesaki-Takai duality
71(1)
2.8.4 Stability of exactness under group extensions
71(2)
2.8.5 Induced representations of twisted crossed products
73(1)
2.8.6 Twisted group algebras, actions on κ and Mackey's little group method
74(7)
3 Bivariant K K-Theory and the Baum--Connes conjecure
81(68)
Siegfried Echterhoff
3.1 Introduction
81(2)
3.2 Operator X-Theory
83(5)
3.3 Kasparov's equivariant K K-theory
88(19)
3.3.1 Graded C*-algebras and Hilbert modules
88(2)
3.3.2 Kasparov's bivariant K-groups
90(3)
3.3.3 The Kasparov product
93(5)
3.3.4 Higher K K-groups and Bott-periodicity
98(8)
3.3.5 Excision in K K-theory
106(1)
3.4 The Baum--Connes conjecture
107(22)
3.4.1 The universal proper G-space
107(3)
3.4.2 The Baum--Connes assembly map
110(5)
3.4.3 Proper G-algebras and the Dirac dual-Dirac method
115(13)
3.4.4 The Baum--Connes conjecture for group extensions
128(1)
3.5 The going-down (or restriction) principle and applications
129(20)
3.5.1 The going-down principle
129(7)
3.5.2 Applications of the going-down principle
136(3)
3.5.3 Crossed products by actions on totally disconnected spaces
139(10)
4 Quantitative K-theory for geometric operator algebras
149(18)
Guoliang Yu
4.1 Introduction
149(2)
4.2 Geometric C*-algebras
151(2)
4.3 Quantitative K-theory for C*-algebras
153(1)
4.4 A quantitative Mayer--Vietoris sequence
154(3)
4.5 Dynamic asymptotic dimension and K-theory of crossed product C-algebras
157(2)
4.6 Asymptotic dimension for geometric C*-algebras and the Kunneth formula
159(2)
4.7 Quantitative X-theory for Banach algebras
161(6)
5 Semigroup C*-algebras
167(106)
Xin Li
5.1 Introduction
167(1)
5.2 C*-algebras generated by left regular representations
168(1)
5.3 Examples
169(6)
5.3.1 The natural numbers
169(1)
5.3.2 Positive cones in totally ordered groups
170(1)
5.3.3 Monoids given by presentations
170(3)
5.3.4 Examples from rings in general, and number theory in particular
173(1)
5.3.5 Finitely generated abelian cancellative semigroups
174(1)
5.4 Preliminaries
175(7)
5.4.1 Embedding semigroups into groups
175(1)
5.4.2 Graph products
176(4)
5.4.3 Krull rings
180(2)
5.5 C*-algebras attached to inverse semigroups, partial dynamical systems and groupoids
182(22)
5.5.1 Inverse semigroups
182(6)
5.5.2 Partial dynamical systems
188(4)
5.5.3 Etale groupoids
192(3)
5.5.4 The universal groupoid of an inverse semigroup
195(2)
5.5.5 Inverse semigroup C*-algebras as groupoid C*-algebras
197(3)
5.5.6 C*-algebras of partial dynamical systems as C*-algebras of partial transformation groupoids
200(3)
5.5.7 The case of inverse semigroups admitting an idempotent pure partial homomorphism to a group
203(1)
5.6 Amenability and nuclearity
204(35)
5.6.1 Groups and groupoids
205(3)
5.6.2 Amenability for semigroups
208(1)
5.6.3 Comparing reduced C*-algebras for left cancellative semigroups and their left inverse hulls
209(7)
5.6.4 C*-algebras generated by semigroups of projections
216(6)
5.6.5 The independence condition
222(8)
5.6.6 Construction of full semigroup C*-algebras
230(2)
5.6.7 Crossed product and groupoid C*-algebra descriptions of reduced semigroup C*-algebras
232(3)
5.6.8 Amenability of semigroups in terms of C*-algebras
235(3)
5.6.9 Nuclearity of semigroup C*-algebras and the connection to amenability
238(1)
5.7 Topological freeness, boundary quotients, and C*-simplicity
239(10)
5.8 The Toeplitz condition
249(7)
5.9 Graph products
256(12)
5.9.1 Constructible right ideals
257(4)
5.9.2 The independence condition
261(4)
5.9.3 The Toeplitz condition
265(3)
5.10 K-theory
268(2)
5.11 Further developments, outlook, and open questions
270(3)
6 Algebraic actions and their C*-algebras
273(24)
Joachim Cuntz
6.1 Introduction
273(2)
6.2 Single algebraic endomorphisms
275(8)
6.2.1 The K-theory of u[ φ]
278(3)
6.2.2 Examples
281(2)
6.3 Actions by a family of endomorphisms, ring C*-algebras
283(3)
6.4 Regular C*-algebras for ax + b - semigroups
286(3)
6.5 The K-theory for C*λ(R × R×)
289(3)
6.6 KMS-states
292(5)
7 Semigroup C*-algebras and toric varieties
297(10)
Joachim Cuntz
7.1 Introduction
297(1)
7.2 Toric varieties
298(3)
7.3 The regular C*-algebra for a toric semigroup
301(6)
Bibliography 307
Joachim Cuntz is a full Professor at the Westfälische Wilhelms-Universität in Münster, Germany.

Siegfried Echterhoff is a full Professor at the Westfälische Wilhelms-Universität in Münster, Germany.

Xin Li is a Senior Lecturer in Pure Mathematics at Queen Mary University of London, United Kingdom.

Guoliang Yu is Powell Chair in Mathematics and Professor at the Texas A&M University, USA.
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