vertex operator algebra
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Von Neumann began investigating operator algebras in the 1930s, as rings of operators on a Hilbert space.
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In general operator algebras are non-commutative rings.
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Operator algebras can be used to study arbitrary sets of operators with little algebraic relation simultaneously.
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These lead naturally to the definition of C*-algebras and other operator algebras.
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Arne Meurman (born April 6, 1956) is a Swedish mathematician working on finite groups and vertex operator algebras.
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From this point of view, operator algebras can be regarded as a generalization of spectral theory of a single operator.
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An operator algebra is typically required to be closed in a specified operator topology inside the algebra of the whole continuous linear operators.
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In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings.
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Its influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations.
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Although the study of operator algebras is usually classified as a branch of functional analysis, it has direct applications to representation theory, differential geometry, quantum statistical mechanics, quantum information, and quantum field theory.
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The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics and topology.
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In particular, it is a set of operators with both algebraic and topological closure properties.
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All this is subsumed by the *-operations of C*-algebras.
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Leonhard Euler considered algebraic operations on numbers modulo an integer, modular arithmetic, in his generalization of Fermat's little theorem.
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In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication.
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A variant of this approach replaces relations with operations, the basis of universal algebra.
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He devised a calculating machine or abacus, by which he could perform arithmetical and algebraic operations by the sense of touch; it was known as his "palpable arithmetic", and was described in his Elements of Algebra.
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The algebra of observables in quantum mechanics is naturally an algebra of operators defined on a Hilbert space, according to Werner Heisenberg's matrix mechanics formulation of quantum theory.
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In mathematics, the category of magmas, denoted Mag, has as objects sets with a binary operation, and morphisms given by homomorphisms of operations (in the universal algebra sense).
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By gaining familiarity with the way sets operate, children learn elements common to arithmetic, algebra and geometry.
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In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
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Formal definition through primitive operations and axioms were proposed for many basic algebraic structures, such as groups, rings, and fields.
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The n-th Weyl algebra, An, is the ring of differential operators with polynomial coefficients in n variables.
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The set of all equivalence classes of central simple algebras over a given field F, under this equivalence relation, can be equipped with a group operation given by the tensor product of algebras.
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Stefaan Vaes started to work in the operator algebra research group lead by Georges Skandalis.
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