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This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.

Matrices. --- Algebra, Matrix --- Cracovians (Mathematics) --- Matrix algebra --- Matrixes (Algebra) --- Algebra, Abstract --- Algebra, Universal --- Matrices --- 512.64 --- 512.64 Linear and multilinear algebra. Matrix theory --- Linear and multilinear algebra. Matrix theory --- Addition. --- Analytic continuation. --- Arithmetic mean. --- Banach space. --- Binomial theorem. --- Block matrix. --- Bochner's theorem. --- Calculation. --- Cauchy matrix. --- Cauchy–Schwarz inequality. --- Characteristic polynomial. --- Coefficient. --- Commutative property. --- Compact space. --- Completely positive map. --- Complex number. --- Computation. --- Continuous function. --- Convex combination. --- Convex function. --- Convex set. --- Corollary. --- Density matrix. --- Diagonal matrix. --- Differential geometry. --- Eigenvalues and eigenvectors. --- Equation. --- Equivalence relation. --- Existential quantification. --- Extreme point. --- Fourier transform. --- Functional analysis. --- Fundamental theorem. --- G. H. Hardy. --- Gamma function. --- Geometric mean. --- Geometry. --- Hadamard product (matrices). --- Hahn–Banach theorem. --- Harmonic analysis. --- Hermitian matrix. --- Hilbert space. --- Hyperbolic function. --- Infimum and supremum. --- Infinite divisibility (probability). --- Invertible matrix. --- Lecture. --- Linear algebra. --- Linear map. --- Logarithm. --- Logarithmic mean. --- Mathematics. --- Matrix (mathematics). --- Matrix analysis. --- Matrix unit. --- Metric space. --- Monotonic function. --- Natural number. --- Open set. --- Operator algebra. --- Operator system. --- Orthonormal basis. --- Partial trace. --- Positive definiteness. --- Positive element. --- Positive map. --- Positive semidefinite. --- Positive-definite function. --- Positive-definite matrix. --- Probability measure. --- Probability. --- Projection (linear algebra). --- Quantity. --- Quantum computing. --- Quantum information. --- Quantum statistical mechanics. --- Real number. --- Riccati equation. --- Riemannian geometry. --- Riemannian manifold. --- Riesz representation theorem. --- Right half-plane. --- Schur complement. --- Schur's theorem. --- Scientific notation. --- Self-adjoint operator. --- Sign (mathematics). --- Special case. --- Spectral theorem. --- Square root. --- Standard basis. --- Summation. --- Tensor product. --- Theorem. --- Toeplitz matrix. --- Unit vector. --- Unitary matrix. --- Unitary operator. --- Upper half-plane. --- Variable (mathematics).

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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