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Idempotent Matrices over Complex Group Algebras

Ioannis Emmanouil (Autor)

Buch | Softcover

XIV, 282 Seiten

2005 | 2006
Springer Berlin (Verlag)
978-3-540-27990-7 (ISBN)

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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

Ioannis Emmanouil was initiated to mathematics in Athens, Greece and then moved to Berkeley, where he studied homological algebra, algebraic geometry and K-theory with Mariusz Wodzicki. After receiving his Ph.D. from Berkeley in 1994, he taught for three years at the University of Michigan as a Hildebrandt Assistant Professor. He moved back to Europe in 1997 and after a year at IHES, returned to Greece in 1998. At present, he is a member of the faculty of the University of Athens.

Introduction.- Motivating Examples.- Reduction to Positive Characteristic.- A Homological Approach.- Completions of CG.- Appendices: Tools from Commutative Algebra.- Discrete Ring-valued Integrals.- Frobenius' Density Theorem.- Homological Techniques.- Comparison of Projections.

From the reviews:

"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)

Erscheint lt. Verlag	20.10.2005
Reihe/Serie	Universitext
Zusatzinfo	XIV, 282 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	458 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Schlagworte	Algebra • Functional Analysis • Group rings • Gruppe (mathemat.) • Gruppe (Mathematik) • Hardcover, Softcover / Mathematik/Arithmetik, Algebra • HC/Mathematik/Arithmetik, Algebra • Homological algebra • Idempotents • K0-Group • Matrizen • MSC(2000):16S34, 18G, 19A31, 19D55, 20C07, 20J05, 46L10, 46L
ISBN-10	3-540-27990-3 / 3540279903
ISBN-13	978-3-540-27990-7 / 9783540279907
Zustand	Neuware