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K-theory for group C*-algebras and semigroup C*-algebras

Author: Joachim J R Cuntz; Siegfried Echterhoff; Xin Li, (Mathematics professor); Guoliang Yu
Publisher: Cham : Birkhäuser, 2017.
Series: Oberwolfach seminars, 47.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Printed edition:
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Joachim J R Cuntz; Siegfried Echterhoff; Xin Li, (Mathematics professor); Guoliang Yu
ISBN: 9783319599151 3319599151
OCLC Number: 1017876815
Description: 1 online resource (x, 322 pages).
Contents: Machine generated contents note: 1.Introduction --
2.Crossed products and the Mackey --
Rieffel --
Green machine / Siegfried Echterhoff --
2.1.Introduction --
2.2.Some preliminaries --
2.2.1.C*-algebras --
2.2.2.Multiplier algebras --
2.2.3.Commutative C*-algebras and functional calculus --
2.2.4.Representation and ideal spaces of C*-algebras --
2.2.5.Tensor products --
2.3.Actions and their crossed products --
2.3.1.Haar measure and vector-valued integration on groups --
2.3.2.C*-dynamical systems and their crossed products --
2.4.Crossed products versus tensor products --
2.5.The correspondence categories --
2.5.1.Hilbert modules --
2.5.2.Morita equivalences --
2.5.3.The correspondence categories --
2.5.4.The equivariant correspondence categories --
2.5.5.Induced representations and ideals --
2.5.6.The Fell topologies and weak containment --
2.6.Green's imprimitivity theorem and applications --
2.6.1.The imprimitivity theorem --
2.6.2.The Takesaki --
Takai duality theorem Note continued: 3.3.2.Kasparov's bivariant K-groups --
3.3.3.The Kasparov product --
3.3.4.Higher K K-groups and Bott-periodicity --
3.3.5.Excision in K K-theory --
3.4.The Baum --
Connes conjecture --
3.4.1.The universal proper G-space --
3.4.2.The Baum --
Connes assembly map --
3.4.3.Proper G-algebras and the Dirac dual-Dirac method --
3.4.4.The Baum --
Connes conjecture for group extensions --
3.5.The going-down (or restriction) principle and applications --
3.5.1.The going-down principle --
3.5.2.Applications of the going-down principle --
3.5.3.Crossed products by actions on totally disconnected spaces --
4.Quantitative K-theory for geometric operator algebras / Guoliang Yu --
4.1.Introduction --
4.2.Geometric C*-algebras --
4.3.Quantitative K-theory for C*-algebras --
4.4.A quantitative Mayer --
Vietoris sequence --
4.5.Dynamic asymptotic dimension and K-theory of crossed product C-algebras --
4.6.Asymptotic dimension for geometric C*-algebras and the Kunneth formula Note continued: 4.7.Quantitative X-theory for Banach algebras --
5.Semigroup C*-algebras / Xin Li --
5.1.Introduction --
5.2.C*-algebras generated by left regular representations --
5.3.Examples --
5.3.1.The natural numbers --
5.3.2.Positive cones in totally ordered groups --
5.3.3.Monoids given by presentations --
5.3.4.Examples from rings in general, and number theory in particular --
5.3.5.Finitely generated abelian cancellative semigroups --
5.4.Preliminaries --
5.4.1.Embedding semigroups into groups --
5.4.2.Graph products --
5.4.3.Krull rings --
5.5.C*-algebras attached to inverse semigroups, partial dynamical systems and groupoids --
5.5.1.Inverse semigroups --
5.5.2.Partial dynamical systems --
5.5.3.Etale groupoids --
5.5.4.The universal groupoid of an inverse semigroup --
5.5.5.Inverse semigroup C*-algebras as groupoid C*-algebras --
5.5.6.C*-algebras of partial dynamical systems as C*-algebras of partial transformation groupoids Note continued: 5.5.7.The case of inverse semigroups admitting an idempotent pure partial homomorphism to a group --
5.6.Amenability and nuclearity --
5.6.1.Groups and groupoids --
5.6.2.Amenability for semigroups --
5.6.3.Comparing reduced C*-algebras for left cancellative semigroups and their left inverse hulls --
5.6.4.C*-algebras generated by semigroups of projections --
5.6.5.The independence condition --
5.6.6.Construction of full semigroup C*-algebras --
5.6.7.Crossed product and groupoid C*-algebra descriptions of reduced semigroup C*-algebras --
5.6.8.Amenability of semigroups in terms of C*-algebras --
5.6.9.Nuclearity of semigroup C*-algebras and the connection to amenability --
5.7.Topological freeness, boundary quotients, and C*-simplicity --
5.8.The Toeplitz condition --
5.9.Graph products --
5.9.1.Constructible right ideals --
5.9.2.The independence condition --
5.9.3.The Toeplitz condition --
5.10.K-theory
Series Title: Oberwolfach seminars, 47.
Responsibility: by Joachim Cuntz, Siegfried Echterhoff, Xin Li, Guoliang Yu.

Abstract:

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  Read more...

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