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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

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Banach algebra techniques in operator theory

Operator theory --- Banach algebras --- 517.986 --- 512 --- Hilbert space --- Functional analysis --- Banach spaces --- Hyperspace --- Inner product spaces --- Algebras, Banach --- Banach rings --- Metric rings --- Normed rings --- Topological algebras --- Topological algebras. Theory of infinite-dimensional representations --- Algebra --- Banach algebras. --- Hilbert space. --- Operator theory. --- 512 Algebra --- 517.986 Topological algebras. Theory of infinite-dimensional representations --- Commutative algebra --- Banach, Algèbres de --- Algèbres commutatives --- Banach, Algèbres de. --- Algèbres commutatives. --- Opérateurs linéaires --- Linear operators --- Banach, Algèbres de. --- Algèbres commutatives. --- Opérateurs linéaires. --- Linear operators. --- Algèbres commutatives --- Banach, Algèbres de --- C etoile-algebres

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Operator theory

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