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Branch of functional analysis
functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication
Operator_algebra
Algebra used in 2D conformal field theories and string theory
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string
Vertex_operator_algebra
*-algebra of bounded operators on a Hilbert space
mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains
Von_Neumann_algebra
Topological complex vector space
of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties:
C*-algebra
In mathematics, Jordan operator algebras are real or complex Jordan algebras with the compatible structure of a Banach space. When the coefficients are
Jordan_operator_algebra
Idempotent linear transformation from a vector space to itself
In linear algebra and functional analysis, a projection is a linear transformation P {\displaystyle P} from a vector space to itself (an endomorphism)
Projection_(linear_algebra)
Fundamentals of the Theory of Operator Algebras is a four-volume textbook on the classical theory of operator algebras written by Richard Kadison and John
Fundamentals of the Theory of Operator Algebras
Fundamentals_of_the_Theory_of_Operator_Algebras
Mathematical study of linear operators
collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory. Single
Operator_theory
Algebraic manipulation of "true" and "false"
and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction (and) denoted
Boolean_algebra
Given a unital C*-algebra A {\displaystyle {\mathcal {A}}} , a *-closed subspace S containing 1 is called an operator system. One can associate to each
Operator_system
Mathematical structure in abstract algebra
mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra; read as "star-algebra") is a mathematical structure consisting of
*-algebra
In functional analysis, a reflexive operator algebra A is an operator algebra that has enough invariant subspaces to characterize it. Formally, A is reflexive
Reflexive_operator_algebra
Monster and modular connection
now known to be underlain by a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky
Monstrous_moonshine
Branch of mathematics
formalism. It includes operator-algebraic methods based on C*-algebras, von Neumann algebras, and spectral triples; algebraic approaches to noncommutative
Noncommutative_geometry
Linear operator acting on modular forms
"Hecke algebras", although sometimes the link to Hecke operators is not entirely obvious. These algebras include certain quotients of the group algebras of
Hecke_operator
Concept in mathematics
universal enveloping algebra. In addition, the enveloping algebra gives a precise definition for the Casimir operators. Because Casimir operators commute with
Universal_enveloping_algebra
Non-perturbative approach to quantum field theory
non-perturbative approach to quantum field theory. One example is the vertex operator algebra, which has been used to construct two-dimensional conformal field theories
Operator_product_expansion
Not-necessarily-associative commutative algebra satisfying (xy)(xx) = x(y(xx))
In abstract algebra, a Jordan algebra is a nonassociative algebra (with unit) over a field whose multiplication satisfies the following axioms: x y =
Jordan_algebra
Function acting on function spaces
spaces. Operators on these spaces are known as sequence transformations. Bounded linear operators over a Banach space form a Banach algebra in respect
Operator_(mathematics)
Vector space equipped with a bilinear product
mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure
Algebra_over_a_field
construct a generalized ★-product operator algebra from a given arbitrary finite-dimensional Poisson manifold. This operator algebra amounts to the deformation
Kontsevich quantization formula
Kontsevich_quantization_formula
Set with operations obeying given axioms
Vertex operator algebra Von Neumann algebra: a *-algebra of operators on a Hilbert space equipped with the weak operator topology. Algebraic structures
Algebraic_structure
Particular kind of algebraic structure
is a closed *-subalgebra of the algebra of bounded operators on some Hilbert space. Measure algebra: A Banach algebra consisting of all Radon measures
Banach_algebra
Algebra based on a vector space with a quadratic form
mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure
Clifford_algebra
Associative algebra together with a Lie bracket that satisfies Leibniz's law
different Poisson algebra, one that would be much larger. For a vertex operator algebra (V, Y, ω, 1), the space V/C2(V) is a Poisson algebra with {a, b} =
Poisson_algebra
Generalization of the concept of a direct sum in mathematics
of von Neumann algebras. The concept was introduced in 1949 by John von Neumann in one of the papers in the series On Rings of Operators. One of von Neumann's
Direct_integral
Theory of relational databases
of relational algebra is to define operators that transform one or more input relations to an output relation. Given that these operators accept relations
Relational_algebra
Algebra associated to any vector space
In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle
Exterior_algebra
ring). *-algebra Affine Lie algebra Akivis algebra Algebra for a monad Albert algebra Alternative algebra AW*-algebra Azumaya algebra Banach algebra Birman–Wenzl
List_of_algebras
Invariant of vertex algebra
vertex operator algebra. Many important representation theoretic properties of the vertex algebra are logically related to properties of its Zhu algebra or
Zhu_algebra
Algebraic structure
where ⟨S, ·, +, ′, 0, 1⟩ is a Boolean algebra and postfix I designates a unary operator, the interior operator, satisfying the identities: xI ≤ x xII
Interior_algebra
Canonical commutation or anticommutation relations
(\cdot ,\cdot )} . In the theory of operator algebras, the CCR algebra over H {\displaystyle H} is the unital C*-algebra generated by elements { W ( f ) :
CCR_and_CAR_algebras
Topological algebra associated to continuous groups
the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra), such that
Group algebra of a locally compact group
Group_algebra_of_a_locally_compact_group
Operators useful in quantum mechanics
creation operator. In general, the CCR algebra is infinite dimensional. If we take a Banach space completion, it becomes a C*-algebra. The CCR algebra over
Creation and annihilation operators
Creation_and_annihilation_operators
Mathematical operator
as an algebraic lattice in this context. Conversely, if C is an algebraic poset, then the closure operator is finitary. Each closure operator on a finite
Closure_operator
analogs in the context of operator algebras. This article discusses such operator-algebraic results. Suppose M is a von Neumann algebra and E, F are projections
Schröder–Bernstein theorems for operator algebras
Schröder–Bernstein_theorems_for_operator_algebras
Mathematical method in functional analysis
Hilbert algebras. The modular operator is trivial and the corresponding von Neumann algebra is a direct sum of type I and type II von Neumann algebras. Examples:
Tomita–Takesaki_theory
Algebraic structure with addition and multiplication
group rings in representation theory, operator algebras in functional analysis, rings of differential operators, and cohomology rings in topology. The
Ring_(mathematics)
Banach space of a dual
is the Banach space L1(R) of integrable functions. In operator algebra, if a dual Banach/operator space A {\displaystyle A} is realized as the dual of
Predual
Sporadic simple group
group acts on a vertex operator algebra over the field with 3 elements. This vertex operator algebra contains the E8 Lie algebra over F3, giving the embedding
Thompson_sporadic_group
analysis, a branch of mathematics, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute. The
Abelian_von_Neumann_algebra
Theorem
Stinespring,[when?] is a result from operator theory that represents any completely positive map on a C*-algebra A as a composition of two completely
Stinespring_dilation_theorem
Type of order at absolute zero
Chiral operator product algebra and edge excitations of a FQH droplet (pdf),Nucl. Phys. B422, 476 (1994): Used chiral operator product algebra to construct
Topological_order
Mathematics theorem in functional analysis
theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-subalgebra of bounded operators on a Hilbert space. This result was
Gelfand–Naimark_theorem
Raising and lowering operators in quantum mechanics
linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases
Ladder_operator
Algebraic structure used in analysis
In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket
Lie_algebra
Formulation of quantum mechanics on a Hilbert Space
in the state ω {\displaystyle \omega } . If the C*-algebra is the algebra of all bounded operators on a Hilbert space H {\displaystyle \mathbb {H} }
Dirac–von_Neumann_axioms
Algebra of possibly unbounded operators
In mathematics, an O*-algebra is an algebra of possibly unbounded operators defined on a dense subspace of a Hilbert space. The original examples were
O*-algebra
Sporadic simple group
over the finite field of order 2. Höhn (1996) constructed a vertex operator algebra acted on by the baby monster. Conway and Norton suggested in their
Baby_monster_group
Mathematical function, in linear algebra
mathematics, and more specifically in linear algebra, a linear map, linear mapping, or linear operator is a particular kind of function between vector
Linear_map
American mathematician (1939–2024)
analysis. He also wrote texts on operator algebras, including a general text on the subject titled A Course in Operator Theory (American Mathematical Society)
John_B._Conway
space is a subspace of a C*-algebra. The category of operator spaces includes operator systems and operator algebras. For operator systems, in addition to
Operator_space
AW*-algebra is an algebraic generalization of a W*-algebra. They were introduced by Irving Kaplansky in 1951. As operator algebras, von Neumann algebras,
AW*-algebra
Topics referred to by the same term
Look up algebra in Wiktionary, the free dictionary. Algebra may refer to: Elementary algebra Universal algebra Abstract algebra Linear algebra Relational
Algebra_(disambiguation)
Theory in mathematics
of C*-algebras by Lawrence G. Brown, Ronald G. Douglas, and Peter Arthur Fillmore in 1977. In turn, it has had great success in operator algebraic formalism
KK-theory
Mathematical theory on random variables
theory of operator algebras. Given a free group on some number of generators, we can consider the von Neumann algebra generated by the group algebra, which
Free_probability
German mathematician
mathematician, working on geometric group theory, algebraic topology, ergodic theory of group actions, and operator algebras. Thom received in 2000 his Certificate
Andreas_Thom_(mathematician)
Japanese mathematician
Sciences, the University of Tokyo. His primary area of expertise is operator algebra theory. Born in Ōta, Tokyo, Kawahigashi was raised in a family where
Yasuyuki_Kawahigashi
Ring that is also a vector space or a module
In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center
Associative_algebra
Branch of applied mathematics
some parts of the mathematical fields of linear algebra, the spectral theory of operators, operator algebras and, more broadly, functional analysis. Nonrelativistic
Mathematical_physics
Reasoning about equations with free variables
focus on the process of algebraization itself, like classifying various forms of algebraizability using the Leibniz operator (Czelakowski 2003). A homogeneous
Algebraic_logic
Measure of the "size" of linear operators
targets Operator algebra – Branch of functional analysis Operator theory – Mathematical study of linear operators Topologies on the set of operators on a
Operator_norm
English mathematician
December 1932) is an English mathematician working on operator algebras who introduced nest algebras. He was elected a Fellow of the Royal Society in 1977
John_Robert_Ringrose
Sporadic simple group
Duncan (2006) used the 28-dimensional lattice to construct a vertex operator algebra acted on by the double cover. Alternatively, the double cover can be
Rudvalis_group
French mathematician
24 May 1924) is a French mathematician. He worked on operator algebras, especially C*-algebras, and wrote several of the standard reference books on
Jacques_Dixmier
Branch of algebra that studies commutative rings
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both
Commutative_algebra
Topics referred to by the same term
relation where elements of a set are self-related Reflexive operator algebra, an operator algebra that has enough invariant subspaces to characterize it Reflexive
Reflexive
Kind of linear transformation
space of bounded linear operators L ( H ) {\displaystyle L(H)} on a Hilbert space H becomes a C*-algebra and especially an operator space. It is possible
Bounded_operator
Representation of a Lie algebra as a set of linear transformations
momentum operators. The notion is closely related to that of a representation of a Lie group. Roughly speaking, the representations of Lie algebras are the
Lie_algebra_representation
theory. In the 1980s, the Rota-Baxter operator of weight 0 in the context of Lie algebras was rediscovered as the operator form of the classical Yang–Baxter
Rota–Baxter_algebra
Japanese mathematician
in Sendai) is a Japanese mathematician working in the theory of operator algebras. Takesaki studied at Tohoku University, earning a bachelor's degree
Masamichi_Takesaki
Mathematical inequality relating inner products and norms
theory, e.g. for operator-convex functions and operator algebras, where the domain and/or range are replaced by a C*-algebra or W*-algebra. An inner product
Cauchy–Schwarz_inequality
Infinite dimensional Lie algebra occurring in quantum field theory
density operators in quantum field theories define an infinite-dimensional Lie algebra called a current algebra. Mathematically these are Lie algebras consisting
Current_algebra
Free object in the category of associative algebras
In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since
Free_algebra
American mathematician
American mathematician known for his contributions to the study of operator algebras. Born in New York City in 1925, Kadison was a Gustave C. Kuemmerle
Richard_Kadison
Area of mathematics
algorithm design, computational complexity, numerical methods and computer algebra. Computational mathematics refers also to the use of computers for mathematics
Computational_mathematics
Study of abstract algebraic structures
Concretely, this is just an action of i , as this generates the algebra, and the operator representing i (the image of i in End(V)) is denoted J to avoid
Algebra_representation
Correspondence in functional analysis
{\displaystyle A} to the identity operator on H {\displaystyle H} . A state on a C ∗ {\displaystyle C^{*}} -algebra A {\displaystyle A} is a positive
Gelfand–Naimark–Segal construction
Gelfand–Naimark–Segal_construction
Theory of subatomic structure
1007/BF01232032. Frenkel, Igor; Lepowsky, James; Meurman, Arne (1988). Vertex Operator Algebras and the Monster. Pure and Applied Mathematics. Vol. 134. Academic
String_theory
Branch of mathematics
algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations
Homological_algebra
French mathematician (born 1947)
French mathematician, known for his contributions to the study of operator algebras and noncommutative geometry. He was a professor at the Collège de
Alain_Connes
Positive element Positive linear functional operator algebra nest algebra reflexive operator algebra Calkin algebra Gelfand representation Gelfand–Naimark
List of functional analysis topics
List_of_functional_analysis_topics
248-dimensional exceptional simple Lie group
several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding
E8_(mathematics)
Type of monoidal category
of vertex operator algebras. There is a well-established theory that associates to every conformal field theory a vertex operator algebra. When this
Modular_tensor_category
Annual mathematics conference series
Numerical analysis The other major branch of operator theory, Operator algebras (C* and von Neumann Algebras), is not heavily represented at IWOTA and has
International Workshop on Operator Theory and its Applications
International_Workshop_on_Operator_Theory_and_its_Applications
Distinguished element of a Lie algebra's center
Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example
Casimir_element
Algebraic study of differential equations
differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators as algebraic objects
Differential_algebra
branch of mathematics, nest algebras are a class of operator algebras that generalise the upper-triangular matrix algebras to a Hilbert space context.
Nest_algebra
Unique extension of pure states in Hilbert spaces
ℓ2 and two related C*-algebras: the algebra B {\displaystyle B} of all continuous linear operators from ℓ2 to ℓ2, and the algebra D {\displaystyle D} of
Kadison–Singer_problem
American mathematician (1934–2011)
November 1934 – 15 November 2011) was a mathematician specializing in operator algebras who worked as a professor of Mathematics at the University of California
William_Arveson
Scientific area at the interface between computer science and mathematics
evaluation is fundamental in computer algebra. For example, the operator "=" of equation is also, in most computer algebra systems, the name of the program
Computer_algebra
Study of discrete mathematical structures
function fields. Algebraic structures occur as both discrete examples and continuous examples. Discrete algebras include: Boolean algebra used in logic gates
Discrete_mathematics
Topics referred to by the same term
incidence structure consisting of points and parallel classes of lines An operator algebra in local quantum field theory ε-net (computational geometry), a concept
Net
useful in operator theory, due to the von Neumann double commutant theorem, which relates the algebraic and analytic structures of operator algebras. Specifically
Bicommutant
Differential algebra
In abstract algebra, the Weyl algebras are abstracted from the ring of differential operators with polynomial coefficients. They are named after Hermann
Weyl_algebra
Riemannian manifold with SU(n) holonomy
In algebraic and differential geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has certain properties
Calabi–Yau_manifold
Sum of elements on the main diagonal
In linear algebra, the trace of a square matrix A, denoted tr(A), is defined as a sum of the elements on its main diagonal, a 11 + a 22 + ⋯ + a n n {\displaystyle
Trace_(linear_algebra)
extends uniquely preserving norm, to a weak-operator continuous linear functional ρ on the von Neumann algebra Φ(A)−. If ρ is hermitian, or positive, the
Universal representation (C*-algebra)
Universal_representation_(C*-algebra)
Argentine physicist (1940–2021)
Virasoro–Shapiro amplitude, the Virasoro algebra, the super Virasoro algebra, the Virasoro vertex operator algebra, the Virasoro group, the Virasoro conjecture
Miguel Ángel Virasoro (physicist)
Miguel_Ángel_Virasoro_(physicist)
Transformation in quantum mechanics
directions. There is a close link to other methods of boson mapping of operator algebras: in particular, the (non-Hermitian) Dyson–Maleev technique, and to
Holstein–Primakoff transformation
Holstein–Primakoff_transformation
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OPERATOR ALGEBRA
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OPERATOR ALGEBRA
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OPERATOR ALGEBRA
OPERATOR ALGEBRA
OPERATOR ALGEBRA
OPERATOR ALGEBRA
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