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Differential geometry. Global analysis --- Nonlinear functional analysis. --- Topological algebras. --- Nonlinear functional analysis --- Topological algebras

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Topological groups. Lie groups --- 517.986 --- Topological algebras. Theory of infinite-dimensional representations --- Harmonic analysis. --- Locally compact Abelian groups. --- Segal algebras. --- 517.986 Topological algebras. Theory of infinite-dimensional representations

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to the Encyclopaedia Subseries on Operator Algebras and Non-Commutative Geometry The theory of von Neumann algebras was initiated in a series of papers by Murray and von Neumann in the 1930's and 1940's. A von Neumann algebra is a self-adjoint unital subalgebra M of the algebra of bounded operators of a Hilbert space which is closed in the weak operator topology. According to von Neumann's bicommutant theorem, M is closed in the weak operator topology if and only if it is equal to the commutant of its commutant. A factor is a von Neumann algebra with trivial centre and the work of Murray and von Neumann contained a reduction of all von Neumann algebras to factors and a classification of factors into types I, II and III. C* -algebras are self-adjoint operator algebras on Hilbert space which are closed in the norm topology. Their study was begun in the work of Gelfand and Naimark who showed that such algebras can be characterized abstractly as involutive Banach algebras, satisfying an algebraic relation connecting the norm and the involution. They also obtained the fundamental result that a commutative unital C* -algebra is isomorphic to the algebra of complex valued continuous functions on a compact space - its spectrum. Since then the subject of operator algebras has evolved into a huge mathematical endeavour interacting with almost every branch of mathematics and several areas of theoretical physics.

Operator algebras --- Algèbres d'opérateurs --- 517.986 --- banachalgebra --- dicht --- tensoren --- integralen --- duaal --- Von Neumann --- poolse ruimte --- c*-algebra --- topologie --- spoor --- borel --- commutant --- Algebras, Operator --- Operator theory --- Topological algebras --- Topological algebras. Theory of infinite-dimensional representations --- Operator algebras. --- 517.986 Topological algebras. Theory of infinite-dimensional representations --- Analytical spaces --- Algèbres d'opérateurs --- Operator theory. --- Mathematical physics. --- Operator Theory. --- Theoretical, Mathematical and Computational Physics. --- Physical mathematics --- Physics --- Functional analysis --- Mathematics --- Algèbres topologiques --- Algèbres topologiques. --- Algèbres topologiques.

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Analytical spaces --- C*-algebras --- Automorphisms --- 517.986 --- Algebras, C star --- Algebras, W star --- C star algebras --- W star algebras --- W*-algebras --- Banach algebras --- Group theory --- Symmetry (Mathematics) --- Topological algebras. Theory of infinite-dimensional representations --- C*-algebras. --- Automorphisms. --- 517.986 Topological algebras. Theory of infinite-dimensional representations --- Algebres topologiques --- C etoile-algebres --- C star algebres

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Analytical spaces --- Banach algebras --- 517.9 --- Algebras, Banach --- Banach rings --- Metric rings --- Normed rings --- Banach spaces --- Topological algebras --- Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- Banach algebras. --- 517.9 Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- Algèbres topologiques. --- Topological algebras. --- Analyse fonctionnelle --- Functional analysis --- Algèbres topologiques. --- Banach, Algèbres de --- Algèbres topologiques --- Banach, Algèbres de