Operator algebra

Short description: Branch of functional analysis

Algebraic structure → Ring theory
Ring theory

Basic concepts

Rings

• Subrings
• Ideal
• Quotient ring

• Fractional ideal
• Total quotient ring
• Product ring
• Free product of associative algebras
• Tensor product of rings

Ring homomorphisms

• Kernel
• Inner automorphism
• Frobenius endomorphism

Algebraic structures

• Module
• Associative algebra
• Graded ring
• Involutive ring
• Category of rings

• Initial ring ℤ
• Terminal ring 0=ℤ1

Related structures

• Field

• Finite field
• Non-associative ring

• Lie ring
• Jordan ring
• Semiring

• Semifield

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Commutative algebra

Commutative rings

• Integral domain

• Integrally closed domain
• GCD domain
• Unique factorization domain
• Principal ideal domain
• Euclidean domain
• Field

• Finite field
• Composition ring

• Polynomial ring
• Formal power series ring

Algebraic number theory

• Algebraic number field
• Ring of integers
• Algebraic independence
• Transcendental number theory
• Transcendence degree

p-adic number theory and decimals

• Direct limit/Inverse limit
• Zero ring ℤ1
• Integers modulo pn ℤ/pnℤ
• Prüfer p-ring ℤ(p∞)
• Base-p circle ring 𝕋
• Base-p integers ℤ
• p-adic rationals ℤ[1/p]
• Base-p real numbers ℝ
• p-adic integers ℤp
• p-adic numbers ℚp
• p-adic solenoid 𝕋p

Algebraic geometry

• Affine variety

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Noncommutative algebra

Noncommutative rings

• Division ring
• Semiprimitive ring
• Simple ring
• Commutator

Noncommutative algebraic geometry

Free algebra

Clifford algebra

• Geometric algebra
Operator algebra

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In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings.

The results obtained in the study of operator algebras are often phrased in algebraic terms, while the techniques used are often highly analytic.[1] 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.

## Contents

* 1 Overview
* 2 See also
* 3 References
* 4 Further reading

## Overview

Operator algebras can be used to study arbitrary sets of operators with little algebraic relation simultaneously. From this point of view, operator algebras can be regarded as a generalization of spectral theory of a single operator. In general operator algebras are non-commutative rings.

An operator algebra is typically required to be closed in a specified operator topology inside the whole algebra of continuous linear operators. In particular, it is a set of operators with both algebraic and topological closure properties. In some disciplines such properties are axiomized and algebras with certain topological structure become the subject of the research.

Though algebras of operators are studied in various contexts (for example, algebras of pseudo-differential operators acting on spaces of distributions), the term operator algebra is usually used in reference to algebras of bounded operators on a Banach space or, even more specially in reference to algebras of operators on a separable Hilbert space, endowed with the operator norm topology.

In the case of operators on a Hilbert space, the Hermitian adjoint map on operators gives a natural involution, which provides an additional algebraic structure that can be imposed on the algebra. In this context, the best studied examples are self-adjoint operator algebras, meaning that they are closed under taking adjoints. These include C*-algebras, von Neumann algebras, and AW*-algebra. C*-algebras can be easily characterized abstractly by a condition relating the norm, involution and multiplication. Such abstractly defined C*-algebras can be identified to a certain closed subalgebra of the algebra of the continuous linear operators on a suitable Hilbert space. A similar result holds for von Neumann algebras.

Commutative self-adjoint operator algebras can be regarded as the algebra of complex-valued continuous functions on a locally compact space, or that of measurable functions on a standard measurable space. Thus, general operator algebras are often regarded as a noncommutative generalizations of these algebras, or the structure of the base space on which the functions are defined. This point of view is elaborated as the philosophy of noncommutative geometry, which tries to study various non-classical and/or pathological objects by noncommutative operator algebras.

Examples of operator algebras that are not self-adjoint include:

* nest algebras,
* many commutative subspace lattice algebras,
* many limit algebras.

## See also

* Banach algebra - Particular kind of algebraic structure
* Physics:Matrix mechanics - Formulation of quantum mechanics
* Vertex operator algebra - Algebra used in 2D conformal field theories and string theory

## References

1. ↑ Theory of Operator Algebras I By Masamichi Takesaki, Springer 2012, p vi

## Further reading

* Blackadar, Bruce (2005). Operator Algebras: Theory of C*-Algebras and von Neumann Algebras. Encyclopaedia of Mathematical Sciences. Springer-Verlag. ISBN 3-540-28486-9.
* M. Takesaki, Theory of Operator Algebras I, Springer, 2001.

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Functional analysis (topics)

Topological vector spaces|

* Asplund
* Banach (list)
* Banach lattice
* Barrelled
* Bornological
* Brauner
* F-space
* Fréchet (tame)
* Hilbert (Inner product space
* Polarization identity)
* LF-space
* Locally convex (Seminorms/Minkowski functionals)
* Mackey
* Montel
* Nuclear
* Normed (norm)
* Quasinormed
* Reflexive
* Riesz
* Smith
* Stereotype
* Strictly convex
* Webbed
* Topological tensor product (of Hilbert spaces)

Topologies of function spaces|

* Dual
* Dual space (Dual norm)
* Operator
* Ultraweak
* Weak (polar
* operator)
* Mackey
* Strong (polar
* operator)
* Ultrastrong
* Uniform convergence

Linear operators|

* Adjoint
* Bilinear (form
* operator
* sesquilinear)
* (Un)Bounded
* Closed
* Compact (on Hilbert spaces)
* (Dis)Continuous
* Densely defined
* Fredholm
* Hilbert–Schmidt
* Functionals (positive)
* Normal
* Nuclear
* Self-adjoint
* Strictly singular
* Trace class
* Transpose
* Unitary

Operator theory|

* Banach algebras
* C*-algebras
* Spectrum (C*-algebra
* radius)
* Spectral theory (of ODEs
* Spectral theorem)
* Polar decomposition
* Singular value decomposition

Theorems|

* Banach–Alaoglu
* Banach–Mazur
* Banach–Saks
* Banach–Schauder (open mapping)
* Banach–Steinhaus (Uniform boundedness)
* Bessel's inequality
* Cauchy–Schwarz inequality
* Closed graph
* Closed range
* Eberlein–Šmulian
* Freudenthal spectral
* Gelfand–Mazur
* Gelfand–Naimark
* Goldstine
* Hahn–Banach (hyperplane separation)
* Kakutani fixed-point
* Krein–Milman
* Lomonosov's invariant subspace
* Mackey–Arens
* Mazur's lemma
* M. Riesz extension
* Riesz representation
* Parseval's identity
* Schauder fixed-point

Analysis|

* Abstract Wiener space
* Bochner space
* Differentiation in Fréchet spaces
* Derivatives (Fréchet
* Gateaux
* functional
* holomorphic)
* Integrals (Bochner
* Dunford
* Gelfand–Pettis
* regulated
* Paley–Wiener
* weak)
* Functional calculus (Borel
* continuous
* holomorphic)
* Inverse function theorem (Nash–Moser theorem)
* Measures (Lebesgue
* Projection-valued
* Vector)
* Weakly measurable function

Types of sets|

* Absolutely convex
* Absorbing
* Balanced
* Bounded
* Convex
* Convex cone (subset)
* Linear cone (subset)
* Radial
* Star-shaped
* Symmetric
* Zonotope

Subsets / set operations|

* Algebraic interior (core)
* Bounding points
* Convex hull
* Extreme point
* Interior
* Minkowski addition
* Polar

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Industrial and applied mathematics

Computational|

* Computational logic
* Algorithms
* design
* analysis
* Information theory
* Coding theory
* Cryptography

Discrete|

* Computer algebra
* Computational number theory
* Combinatorics
* Graph theory
* Discrete geometry

Mathematical
physics|

| Algebraic|

* Operator algebra
* Representation theory
* Renormalization group
* Feynman integral
* M-theory

|

Analytic|

* Differential geometry
* Fourier analysis
* Harmonic analysis
* Functional analysis
* Operator theory

Analysis|

* Approximation theory
* Numerical analysis
* Differential equations
* Dynamical systems
* Control theory
* Variational calculus

Probability theory|

* Distributions (random variables)
* Stochastic processes / analysis
* Path integral
* Stochastic variational calculus

Decision sciences|

* Statistics
* Operations research
* Optimization
* Game theory
* Mathematical economics
* Mathematical finance

Other applications|

* Biology
* Chemistry
* Psychology
* Sociology
* "The Unreasonable Effectiveness of Mathematics in the Natural Sciences"

* Category
* Mathematics portal / outline / topics list

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