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The subject matter of compact groups is frequently cited in fields like algebra, topology, functional analysis, and theoretical physics. This book serves the dual purpose of providing a textbook on it for upper level graduate courses or seminars, and of serving as a source book for research specialists who need to apply the structure and representation theory of compact groups. After a gentle introduction to compact groups and their representation theory, the book presents self-contained courses on linear Lie groups, on compact Lie groups, and on locally compact abelian groups. Separate appended chapters contain the material for courses on abelian groups and on category theory. However, the thrust of the book points in the direction of the structure theory of not necessarily finite dimensional, nor necessarily commutative, compact groups, unfettered by weight restrictions or dimensional bounds. In the process it utilizes infinite dimensional Lie algebras and the exponential function of arbitrary compact groups. The first edition of 1998 and the second edition of 2006 were well received by reviewers and have been frequently "ed in the areas of instruction and research. For the present new edition the text has been cleaned of typographical flaws and the content has been conceptually sharpened in some places and polished and improved in others. New material has been added to various sections taking into account the progress of research on compact groups both by the authors and other writers. Motivation was provided, among other things, by questions about the structure of compact groups put to the authors by readers through the years following the earlier editions. Accordingly, the authors wished to clarify some aspects of the book which they felt needed improvement. The list of references has increased as the authors included recent publications pertinent to the content of the book.

Compact groups. --- Lie groups. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Groups, Compact --- Locally compact groups --- Abelian Group. --- Category Theory. --- Compact Group. --- Lie Algebra. --- Topological Group.

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Following the tremendous reception of our first volume on topological groups called ""Topological Groups: Yesterday, Today, and Tomorrow"", we now present our second volume. Like the first volume, this collection contains articles by some of the best scholars in the world on topological groups. A feature of the first volume was surveys, and we continue that tradition in this volume with three new surveys. These surveys are of interest not only to the expert but also to those who are less experienced. Particularly exciting to active researchers, especially young researchers, is the inclusion of over three dozen open questions. This volume consists of 11 papers containing many new and interesting results and examples across the spectrum of topological group theory and related topics. Well-known researchers who contributed to this volume include Taras Banakh, Michael Megrelishvili, Sidney A. Morris, Saharon Shelah, George A. Willis, O'lga V. Sipacheva, and Stephen Wagner.

coarse structure --- descriptive set theory --- group representation --- thick set --- free topological group --- character --- selectively sequentially pseudocompact --- separable topological group --- quotient group --- Chabauty topology --- pseudo-?-bounded --- topological group --- free precompact Boolean group --- right-angled Artin groups --- Neretin’s group --- coarse space --- ultrafilter space --- endomorphism --- separable --- absolutely closed topological group --- tree --- strongly pseudocompact --- Gromov’s compactification --- dynamical system --- semigroup compactification --- tame function --- compact topological semigroup --- vast set --- space of closed subgroups --- reflexive group --- Lie group --- matrix coefficient --- maximal ideal --- topological semigroup --- ballean --- continuous inverse algebra --- extension --- subgroup --- Thompson’s group --- scale --- isomorphic embedding --- arrow ultrafilter --- H-space --- paratopological group --- pseudocompact --- Ramsey ultrafilter --- fibre bundle --- locally compact group --- product --- large set in a group --- Vietoris topology --- topological group of compact exponent --- Bourbaki uniformity --- p-compact --- mapping cylinder --- syndetic set --- p-adic Lie group --- Boolean topological group --- non-trivial convergent sequence --- fixed point algebra --- polish group topologies --- varieties of coarse spaces --- piecewise syndetic set --- maximal space

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Following the tremendous reception of our first volume on topological groups called ""Topological Groups: Yesterday, Today, and Tomorrow"", we now present our second volume. Like the first volume, this collection contains articles by some of the best scholars in the world on topological groups. A feature of the first volume was surveys, and we continue that tradition in this volume with three new surveys. These surveys are of interest not only to the expert but also to those who are less experienced. Particularly exciting to active researchers, especially young researchers, is the inclusion of over three dozen open questions. This volume consists of 11 papers containing many new and interesting results and examples across the spectrum of topological group theory and related topics. Well-known researchers who contributed to this volume include Taras Banakh, Michael Megrelishvili, Sidney A. Morris, Saharon Shelah, George A. Willis, O'lga V. Sipacheva, and Stephen Wagner.

coarse structure --- descriptive set theory --- group representation --- thick set --- free topological group --- character --- selectively sequentially pseudocompact --- separable topological group --- quotient group --- Chabauty topology --- pseudo-?-bounded --- topological group --- free precompact Boolean group --- right-angled Artin groups --- Neretin’s group --- coarse space --- ultrafilter space --- endomorphism --- separable --- absolutely closed topological group --- tree --- strongly pseudocompact --- Gromov’s compactification --- dynamical system --- semigroup compactification --- tame function --- compact topological semigroup --- vast set --- space of closed subgroups --- reflexive group --- Lie group --- matrix coefficient --- maximal ideal --- topological semigroup --- ballean --- continuous inverse algebra --- extension --- subgroup --- Thompson’s group --- scale --- isomorphic embedding --- arrow ultrafilter --- H-space --- paratopological group --- pseudocompact --- Ramsey ultrafilter --- fibre bundle --- locally compact group --- product --- large set in a group --- Vietoris topology --- topological group of compact exponent --- Bourbaki uniformity --- p-compact --- mapping cylinder --- syndetic set --- p-adic Lie group --- Boolean topological group --- non-trivial convergent sequence --- fixed point algebra --- polish group topologies --- varieties of coarse spaces --- piecewise syndetic set --- maximal space

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This book studies the interplay between the geometry and topology of locally symmetric spaces, and the arithmetic aspects of the special values of L-functions.The authors study the cohomology of locally symmetric spaces for GL(N) where the cohomology groups are with coefficients in a local system attached to a finite-dimensional algebraic representation of GL(N). The image of the global cohomology in the cohomology of the Borel-Serre boundary is called Eisenstein cohomology, since at a transcendental level the cohomology classes may be described in terms of Eisenstein series and induced representations. However, because the groups are sheaf-theoretically defined, one can control their rationality and even integrality properties. A celebrated theorem by Langlands describes the constant term of an Eisenstein series in terms of automorphic L-functions. A cohomological interpretation of this theorem in terms of maps in Eisenstein cohomology allows the authors to study the rationality properties of the special values of Rankin-Selberg L-functions for GL(n) x GL(m), where n + m = N. The authors carry through the entire program with an eye toward generalizations.This book should be of interest to advanced graduate students and researchers interested in number theory, automorphic forms, representation theory, and the cohomology of arithmetic groups.

Shimura varieties. --- Cohomology operations. --- Number theory. --- Arithmetic groups. --- L-functions. --- Functions, L --- -Number theory --- Group theory --- Number study --- Numbers, Theory of --- Algebra --- Operations (Algebraic topology) --- Algebraic topology --- Varieties, Shimura --- Arithmetical algebraic geometry --- Addition. --- Adele ring. --- Algebraic group. --- Algebraic number theory. --- Arithmetic group. --- Automorphic form. --- Base change. --- Basis (linear algebra). --- Bearing (navigation). --- Borel subgroup. --- Calculation. --- Category of groups. --- Coefficient. --- Cohomology. --- Combination. --- Commutative ring. --- Compact group. --- Computation. --- Conjecture. --- Constant term. --- Corollary. --- Covering space. --- Critical value. --- Diagram (category theory). --- Dimension. --- Dirichlet character. --- Discrete series representation. --- Discrete spectrum. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Elaboration. --- Embedding. --- Euler product. --- Field extension. --- Field of fractions. --- Free module. --- Freydoon Shahidi. --- Function field. --- Functor. --- Galois group. --- Ground field. --- Group (mathematics). --- Group scheme. --- Harish-Chandra. --- Hecke L-function. --- Hecke character. --- Hecke operator. --- Hereditary property. --- Induced representation. --- Irreducible representation. --- K0. --- L-function. --- Langlands dual group. --- Level structure. --- Lie algebra cohomology. --- Lie algebra. --- Lie group. --- Linear combination. --- Linear map. --- Local system. --- Maximal torus. --- Modular form. --- Modular symbol. --- Module (mathematics). --- Monograph. --- N0. --- National Science Foundation. --- Natural number. --- Natural transformation. --- Nilradical. --- Permutation. --- Prime number. --- Quantity. --- Rational number. --- Reductive group. --- Requirement. --- Ring of integers. --- Root of unity. --- SL2(R). --- Scalar (physics). --- Sheaf (mathematics). --- Special case. --- Spectral sequence. --- Standard L-function. --- Subgroup. --- Subset. --- Summation. --- Tensor product. --- Theorem. --- Theory. --- Triangular matrix. --- Triviality (mathematics). --- Two-dimensional space. --- Unitary group. --- Vector space. --- W0. --- Weyl group.

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Recent developments in diverse areas of mathematics suggest the study of a certain class of extensions of C*-algebras. Here, Ronald Douglas uses methods from homological algebra to study this collection of extensions. He first shows that equivalence classes of the extensions of the compact metrizable space X form an abelian group Ext (X). Second, he shows that the correspondence X ⃗ Ext (X) defines a homotopy invariant covariant functor which can then be used to define a generalized homology theory. Establishing the periodicity of order two, the author shows, following Atiyah, that a concrete realization of K-homology is obtained.

Analytical spaces --- 517.986 --- Topological algebras. Theory of infinite-dimensional representations --- Algebra, Homological. --- C*-algebras. --- K-theory. --- 517.986 Topological algebras. Theory of infinite-dimensional representations --- Algebra, Homological --- C*-algebras --- K-theory --- Algebraic topology --- Homology theory --- Algebras, C star --- Algebras, W star --- C star algebras --- W star algebras --- W*-algebras --- Banach algebras --- Homological algebra --- Algebra, Abstract --- K-théorie. --- Homologie. --- Addition. --- Affine transformation. --- Algebraic topology. --- Atiyah–Singer index theorem. --- Automorphism. --- Banach algebra. --- Bijection. --- Boundary value problem. --- Bundle map. --- C*-algebra. --- Calculation. --- Cardinal number. --- Category of abelian groups. --- Characteristic class. --- Chern class. --- Clifford algebra. --- Coefficient. --- Cohomology. --- Compact operator. --- Completely positive map. --- Contact geometry. --- Continuous function. --- Corollary. --- Diagram (category theory). --- Diffeomorphism. --- Differentiable manifold. --- Differential operator. --- Dimension (vector space). --- Dimension function. --- Dimension. --- Direct integral. --- Direct proof. --- Eigenvalues and eigenvectors. --- Equivalence class. --- Equivalence relation. --- Essential spectrum. --- Euler class. --- Exact sequence. --- Existential quantification. --- Fiber bundle. --- Finite group. --- Fredholm operator. --- Fredholm. --- Free abelian group. --- Fundamental class. --- Fundamental group. --- Hardy space. --- Hermann Weyl. --- Hilbert space. --- Homological algebra. --- Homology (mathematics). --- Homomorphism. --- Homotopy. --- Ideal (ring theory). --- Inner automorphism. --- Irreducible representation. --- K-group. --- Lebesgue space. --- Locally compact group. --- Maximal compact subgroup. --- Michael Atiyah. --- Monomorphism. --- Morphism. --- Natural number. --- Natural transformation. --- Normal operator. --- Operator algebra. --- Operator norm. --- Operator theory. --- Orthogonal group. --- Pairing. --- Piecewise linear manifold. --- Polynomial. --- Pontryagin class. --- Positive and negative parts. --- Positive map. --- Pseudo-differential operator. --- Quaternion. --- Quotient algebra. --- Self-adjoint operator. --- Self-adjoint. --- Simply connected space. --- Smooth structure. --- Special case. --- Stein manifold. --- Strong topology. --- Subalgebra. --- Subgroup. --- Subset. --- Summation. --- Tangent bundle. --- Theorem. --- Todd class. --- Topology. --- Torsion subgroup. --- Unitary operator. --- Universal coefficient theorem. --- Variable (mathematics). --- Von Neumann algebra. --- Homology theory. --- Homologie --- K-théorie --- C etoile-algebres