Universitext

By: Ioannis Emmanouil

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Paperback | 20 October 2005

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The study of idempotent elements in group algebras (or, more generally, the study of classes in the K-theory of such algebras) originates from geometric and analytic considerations. For example, C.T.C. Wall [72] has shown that the problem of deciding whether a ?nitely dominated space with fundamental group? is homotopy equivalent to a ?nite CW-complex leads naturally to the study of a certain class in the reduced K-theoryK (Z?) of the group ringZ?. 0 As another example, consider a discrete groupG which acts freely, properly discontinuously, cocompactly and isometrically on a Riemannian manifold. Then, following A. Connes and H. Moscovici [16], the index of an invariant 0th-order elliptic pseudo-di?erential operator is de?ned as an element in the ? ? K -group of the reduced groupC -algebraCG. 0 r Theidempotentconjecture(alsoknownasthegeneralizedKadisonconjec- ? ? ture) asserts that the reduced groupC -algebraCG of a discrete torsion-free r groupG has no idempotents =0,1; this claim is known to be a consequence of a far-reaching conjecture of P. Baum and A. Connes [6]. Alternatively, one mayapproachtheidempotentconjectureasanassertionabouttheconnect- ness of a non-commutative space;ifG is a discrete torsion-free abelian group ? thenCG is the algebra of continuous complex-valued functions on the dual r

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"The study of idempotent elements in the group algebras originates from geometric and analytic considerations. ... This book provides an introduction to the study of these problems for graduate students and researchers ... . collects and presents in a systematic way basic techniques and important results that have been obtained during the past few decades. ... The book is suitable for independent study ... . Moreover, all chapters and appendices finish with a sufficient number of exercises that also increase the quality of the book." (S. V. Mihovski, Zentralblatt MATH, Vol. 1093 (19), 2006)

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You Can Find This Book In

Non-Fiction Mathematics Algebra Groups & Group Theory Calculus & Mathematical Analysis Functional Analysis & Transforms

Introduction	p. 1
Preliminaries	p. 1
Basic Notions from Algebra	p. 1
Basic Notions from Analysis	p. 8
The K[subscript 0]-group of a Ring	p. 15
Traces and the K[subscript 0]-group	p. 23
The Idempotent Conjectures	p. 30
The Hattori-Stallings Rank on K[subscript 0](kG)	p. 30
Idempotents in CG	p. 34
Some First Examples of Groups that Satisfy the Idempotent Conjecture	p. 37
Exercises	p. 42
Motivating Examples	p. 49
The Case of Abelian Groups	p. 49
The Geometric Rank Function	p. 50
K-theory and the Geometric Rank	p. 52
The Connectedness of Spec kG	p. 59
The Case of Finite Groups	p. 61
The Transfer Homomorphism	p. 61
Subgroups of Finite Index	p. 63
Swan's Theorem	p. 65
Exercises	p. 68
Reduction to Positive Characteristic	p. 73
The Rationality of the Canonical Trace	p. 73
Coefficient Fields of Positive Characteristic	p. 74
Lifting to the Field of Algebraic Numbers	p. 77
The Kaplansky Positivity Theorem	p. 80
Idempotent Matrices with Entries in the Complex Group Algebra	p. 85
The Support of the Hattori-Stallings Rank	p. 87
Iterates of the Frobenius Operator	p. 87
The Main Results	p. 91
An Application: the Case of Solvable Groups	p. 100
Exercises	p. 106
A Homological Approach	p. 111
Cyclic Homology of Algebras	p. 111
Basic Definitions and Results	p. 112
The Relation to K-theory	p. 125
The Cyclic Homology of Group Algebras	p. 129
The Nilpotency of Connes' Operator	p. 145
Idempotent Conjectures and the Nilpotency of S	p. 145
Closure Properties	p. 149
Exercises	p. 155
Completions of CG	p. 159
The Integrality of the Trace Conjecture	p. 159
Formulation of the Conjecture	p. 160
The Case of an Abelian Group	p. 161
The Case of a Free Group	p. 170
Induced Modules over NG	p. 179
The Center-Valued Trace on NG	p. 180
Matrices with Entries in NG	p. 195
Exercises	p. 202
Tools from Commutative Algebra	p. 207
Localization and Local Rings	p. 207
Integral Dependence	p. 213
Noether Normalization	p. 217
The Krull Intersection Theorem	p. 222
Exercises	p. 225
Discrete Ring-Valued Integrals	p. 227
Discrete Group-Valued Integrals	p. 227
Idempotent-Valued Premeasures	p. 230
Exercises	p. 232
Frobenius' Density Theorem	p. 235
The Density Theorem	p. 235
Exercises	p. 238
Homological Techniques	p. 239
Complexes and Homology	p. 239
Chain Complexes	p. 239
Double Complexes	p. 240
Tor and Ext	p. 242
Group Homology and Cohomology	p. 244
Basic Definitions	p. 244
H[superscript 2] and Extensions	p. 247
Products	p. 249
Duality	p. 253
The (co-)homology of an Extension	p. 255
Exercises	p. 257
Comparison of Projections	p. 263
Equivalence and Weak Ordering	p. 263
Exercises	p. 269
References	p. 271
Index	p. 275
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