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C*-algebra Extensions and K-homology by Ronald G. Douglas Pdf
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.
$C^*$-Algebra Extensions of $C(X)$ by Huaxin Lin Pdf
This work shows that the Weyl-von Neumann theorem for unitaries holds for $\sigma$-unital $AF$-algebras and their multiplier algebras. Lin studies $E(X,A)$, the quotient of $\mathrm{{\mathbf{Ext}}}^{eu}_s(C(X),A)$ by a special class of trivial extension, dubbed totally trivial extensions. This leads to a BDF-type classification for extensions of $C(X)$ by a $\sigma$-unital purely infinite simple $C^*$-algebra with trivial $K_1$-group. Lin also shows that, when $X$ is a compact subset of the plane, every extension of $C(X)$ by a finite matroid $C^*$-algebra is totally trivial. Classification of these extensions for nice spaces is given, as are some other versions of the Weyl-von Neumann-Berg theorem.
C*-Algebra Extensions and K-Homology. (AM-95), Volume 95 by Ronald G. Douglas Pdf
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.
The subject of C*-algebras received a dramatic revitalization in the 1970s by the introduction of topological methods through the work of Brown, Douglas, and Fillmore on extensions of C*-algebras and Elliott's use of $K$-theory to provide a useful classification of AF algebras. These results were the beginning of a marvelous new set of tools for analyzing concrete C*-algebras. This book is an introductory graduate level text which presents the basics of the subject through a detailed analysis of several important classes of C*-algebras. The development of operator algebras in the last twenty years has been based on a careful study of these special classes. While there are many books on C*-algebras and operator algebras available, this is the first one to attempt to explain the real examples that researchers use to test their hypotheses. Topics include AF algebras, Bunce–Deddens and Cuntz algebras, the Toeplitz algebra, irrational rotation algebras, group C*-algebras, discrete crossed products, abelian C*-algebras (spectral theory and approximate unitary equivalence) and extensions. It also introduces many modern concepts and results in the subject such as real rank zero algebras, topological stable rank, quasidiagonality, and various new constructions. These notes were compiled during the author's participation in the special year on C*-algebras at The Fields Institute for Research in Mathematical Sciences during the 1994–1995 academic year. The field of C*-algebras touches upon many other areas of mathematics such as group representations, dynamical systems, physics, $K$-theory, and topology. The variety of examples offered in this text expose the student to many of these connections. Graduate students with a solid course in functional analysis should be able to read this book. This should prepare them to read much of the current literature. This book is reasonably self-contained, and the author has provided results from other areas when necessary.
An Extension of Mackey's Method to Banach *-Algebraic Bundles by James Michael Gardner Fell Pdf
The main object of the present memoir is to show that the methods and results of Mackey (1958) and Blattner (1963) on the group extension representation problem go through without any essential change in the larger context of homogeneous Banach *-algebraic bundles (with enough cross sections). In order to dispense with separability we shall follow the topological methods of Blattner rather than Mackey's more detailed measure-theoretic analysis. Except for the last section, Part II of this memoir is in fact a rewriting of much of Blattner's papers (1963), making the modifications necessary in the larger context of bundles. The last Section 17 gives an account of the 'Mackey obstruction' in the nonseparable case, leading to an analogue (Theorem 17.2) of Theorem 8.2 of Mackey's paper for homogeneous Banach *-algebraic bundles, without separability restrictions. This is the culminating point of the present memoir.
Homological Algebra (PMS-19), Volume 19 by Henry Cartan,Samuel Eilenberg Pdf
When this book was written, methods of algebraic topology had caused revolutions in the world of pure algebra. To clarify the advances that had been made, Cartan and Eilenberg tried to unify the fields and to construct the framework of a fully fledged theory. The invasion of algebra had occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. This book presents a single homology (and also cohomology) theory that embodies all three; a large number of results is thus established in a general framework. Subsequently, each of the three theories is singled out by a suitable specialization, and its specific properties are studied. The starting point is the notion of a module over a ring. The primary operations are the tensor product of two modules and the groups of all homomorphisms of one module into another. From these, "higher order" derived of operations are obtained, which enjoy all the properties usually attributed to homology theories. This leads in a natural way to the study of "functors" and of their "derived functors." This mathematical masterpiece will appeal to all mathematicians working in algebraic topology.
Commutative Algebra and Noncommutative Algebraic Geometry by David Eisenbud,Srikanth B. Iyengar,Anurag K. Singh,Michel Van den Bergh,J. Toby Stafford Pdf
This book surveys fundamental current topics in these two areas of research, emphasising the lively interaction between them. Volume 2 focuses on the most recent research.
Noncommutative Geometry and Cayley-smooth Orders by Lieven Le Bruyn Pdf
Noncommutative Geometry and Cayley-smooth Orders explains the theory of Cayley-smooth orders in central simple algebras over function fields of varieties. In particular, the book describes the etale local structure of such orders as well as their central singularities and finite dimensional representations. After an introduction to partial d
Field Theory by I. S. Luthar,Inder Bir S. Passi Pdf
Starting with the basic notions and results in algebraic extensions, in this final volume the authors give an exposition of the work of Galois on the solubility of equations by radicals, including Kummer and Artin-Schreier extensions before providing an extensive study on field theory.
Normal Surface Singularities by András Némethi Pdf
This monograph provides a comprehensive introduction to the theory of complex normal surface singularities, with a special emphasis on connections to low-dimensional topology. In this way, it unites the analytic approach with the more recent topological one, combining their tools and methods. In the first chapters, the book sets out the foundations of the theory of normal surface singularities. This includes a comprehensive presentation of the properties of the link (as an oriented 3-manifold) and of the invariants associated with a resolution, combined with the structure and special properties of the line bundles defined on a resolution. A recurring theme is the comparison of analytic and topological invariants. For example, the Poincaré series of the divisorial filtration is compared to a topological zeta function associated with the resolution graph, and the sheaf cohomologies of the line bundles are compared to the Seiberg–Witten invariants of the link. Equivariant Ehrhart theory is introduced to establish surgery-additivity formulae of these invariants, as well as for the regularization procedures of multivariable series. In addition to recent research, the book also provides expositions of more classical subjects such as the classification of plane and cuspidal curves, Milnor fibrations and smoothing invariants, the local divisor class group, and the Hilbert–Samuel function. It contains a large number of examples of key families of germs: rational, elliptic, weighted homogeneous, superisolated and splice-quotient. It provides concrete computations of the topological invariants of their links (Casson(–Walker) and Seiberg–Witten invariants, Turaev torsion) and of the analytic invariants (geometric genus, Hilbert function of the divisorial filtration, and the analytic semigroup associated with the resolution). The book culminates in a discussion of the topological and analytic lattice cohomologies (as categorifications of the Seiberg–Witten invariant and of the geometric genus respectively) and of the graded roots. Several open problems and conjectures are also formulated. Normal Surface Singularities provides researchers in algebraic and differential geometry, singularity theory, complex analysis, and low-dimensional topology with an invaluable reference on this rich topic, offering a unified presentation of the major results and approaches.
Differential Algebra, Complex Analysis and Orthogonal Polynomials by Primitivo B. Acosta Humanez,Francisco Marcellán Pdf
Presents the 2007-2008 Jairo Charris Seminar in Algebra and Analysis on Differential Algebra, Complex Analysis and Orthogonal Polynomials, which was held at the Universidad Sergio Arboleda in Bogota, Colombia.
Designed to make learning introductory algebraic geometry as easy as possible, this text is intended for advanced undergraduates and graduate students who have taken a one-year course in algebra and are familiar with complex analysis. This newly updated second edition enhances the original treatment's extensive use of concrete examples and exercises with numerous figures that have been specially redrawn in Adobe Illustrator. An introductory chapter that focuses on examples of curves is followed by a more rigorous and careful look at plane curves. Subsequent chapters explore commutative ring theory and algebraic geometry as well as varieties of arbitrary dimension and some elementary mathematics on curves. Upon finishing the text, students will have a foundation for advancing in several different directions, including toward a further study of complex algebraic or analytic varieties or to the scheme-theoretic treatments of algebraic geometry. 2015 edition.
Set Theoretical Logic-The Algebra of Models by W Felscher Pdf
This is an introduction to mathematical logic in which all the usual topics are presented: compactness and axiomatizability of semantical consequence, Löwenheim-Skolem-Tarski theorems, prenex and other normal forms, and characterizations of elementary classes with the help of ultraproducts. Logic is based exclusively on semantics: truth and satisfiability of formulas in structures are the basic notions. The methods are algebraic in the sense that notions such as homomorphisms and congruence relations are applied throughout in order to gain new insights. These concepts are developed and can be viewed as a first course on universal algebra. The approach to algorithms generating semantical consequences is algebraic as well: for equations in algebras, for propositional formulas, for open formulas of predicate logic, and for the formulas of quantifier logic. The structural description of logical consequence is a straightforward extension of that of equational consequence, as long as Boolean valued propositions and Boolean valued structures are considered; the reduction of the classical 2-valued case then depends on the Boolean prime ideal theorem.
