Operator algebra

Branch of functional analysis

Algebraic structure → Ring theory
Ring theory

Basic concepts

Rings

• Subrings
• Ideal
• Quotient ring

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

Ring homomorphisms

• Kernel
• Inner automorphism
• Frobenius endomorphism

Algebraic structures

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

• Initial ring Z {\displaystyle \mathbb {Z} }
• Terminal ring 0 = Z / 1 Z {\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }

Related structures

• Field

• Finite field
• Non-associative ring

• Lie ring
• Jordan ring
• Semiring

• Semifield

Commutative algebra

Commutative rings

• Integral domain

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

• Finite field
• Polynomial ring
• Formal power series ring

Algebraic number theory

• Algebraic number field
• Integers modulo n
• Ring of integers
• p-adic integers Z p {\displaystyle \mathbb {Z} _{p}}
• p-adic numbers Q p {\displaystyle \mathbb {Q} _{p}}
• Prüfer p-ring Z ( p ∞ ) {\displaystyle \mathbb {Z} (p^{\infty })}

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.

## Overview

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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 axiomatized 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*-algebras. 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

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* Banach algebra – Particular kind of algebraic structure
* Matrix mechanics – Formulation of quantum mechanics
* Topologies on the set of operators on a Hilbert space
* Vertex operator algebra – Algebra used in 2D conformal field theories and string theory

## References

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

## Further reading

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* 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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Spectral theory and *-algebras

Basic concepts|

* Involution/*-algebra
* Banach algebra
* B*-algebra
* C*-algebra
* Noncommutative topology
* Projection-valued measure
* Spectrum
* Spectrum of a C*-algebra
* Spectral radius
* Operator space

Main results|

* Gelfand–Mazur theorem
* Gelfand–Naimark theorem
* Gelfand representation
* Polar decomposition
* Singular value decomposition
* Spectral theorem
* Spectral theory of normal C*-algebras

Special Elements/Operators|

* Isospectral
* Normal operator
* Hermitian/Self-adjoint operator
* Unitary operator
* Unit

Spectrum|

* Krein–Rutman theorem
* Normal eigenvalue
* Spectrum of a C*-algebra
* Spectral radius
* Spectral asymmetry
* Spectral gap

Decomposition|

* Decomposition of a spectrum
* Continuous
* Point
* Residual
* Approximate point
* Compression
* Direct integral
* Discrete
* Spectral abscissa

Spectral Theorem|

* Borel functional calculus
* Min-max theorem
* Positive operator-valued measure
* Projection-valued measure
* Riesz projector
* Rigged Hilbert space
* Spectral theorem
* Spectral theory of compact operators
* Spectral theory of normal C*-algebras

Special algebras|

* Amenable Banach algebra
* With an Approximate identity
* Banach function algebra
* Disk algebra
* Nuclear C*-algebra
* Uniform algebra
* Von Neumann algebra
* Tomita–Takesaki theory

Finite-Dimensional|

* Alon–Boppana bound
* Bauer–Fike theorem
* Numerical range
* Schur–Horn theorem

Generalizations|

* Dirac spectrum
* Essential spectrum
* Pseudospectrum
* Structure space (Shilov boundary)

Miscellaneous|

* Abstract index group
* Banach algebra cohomology
* Cohen–Hewitt factorization theorem
* Extensions of symmetric operators
* Fredholm theory
* Limiting absorption principle
* Schröder–Bernstein theorems for operator algebras
* Sherman–Takeda theorem
* Unbounded operator

Examples|

* Wiener algebra

Applications|

* Almost Mathieu operator
* Corona theorem
* Hearing the shape of a drum (Dirichlet eigenvalue)
* Heat kernel
* Kuznetsov trace formula
* Lax pair
* Proto-value function
* Ramanujan graph
* Rayleigh–Faber–Krahn inequality
* Spectral geometry
* Spectral method
* Spectral theory of ordinary differential equations
* Sturm–Liouville theory
* Superstrong approximation
* Transfer operator
* Transform theory
* Weyl law
* Wiener–Khinchin theorem

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Functional analysis (topics – glossary)

Spaces|

* Banach
* Besov
* Fréchet
* Hilbert
* Hölder
* Nuclear
* Orlicz
* Schwartz
* Sobolev
* Topological vector

Properties|

* Barrelled
* Complete
* Dual (Algebraic / Topological)
* Locally convex
* Reflexive
* Separable

Theorems|

* Hahn–Banach
* Riesz representation
* Closed graph
* Uniform boundedness principle
* Kakutani fixed-point
* Krein–Milman
* Min–max
* Gelfand–Naimark
* Banach–Alaoglu

Operators|

* Adjoint
* Bounded
* Compact
* Hilbert–Schmidt
* Normal
* Nuclear
* Trace class
* Transpose
* Unbounded
* Unitary

Algebras|

* Banach algebra
* C*-algebra
* Spectrum of a C*-algebra
* Operator algebra
* Group algebra of a locally compact group
* Von Neumann algebra

Open problems|

* Invariant subspace problem
* Mahler's conjecture

Applications|

* Hardy space
* Spectral theory of ordinary differential equations
* Heat kernel
* Index theorem
* Calculus of variations
* Functional calculus
* Integral linear operator
* Jones polynomial
* Topological quantum field theory
* Noncommutative geometry
* Riemann hypothesis
* Distribution (or Generalized functions)

Advanced topics|

* Approximation property
* Balanced set
* Choquet theory
* Weak topology
* Banach–Mazur distance
* Tomita–Takesaki theory

* Category

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Banach space topics

Types of Banach spaces|

* Asplund
* Banach
* list
* Banach lattice
* Grothendieck
* Hilbert
* Inner product space
* Polarization identity
* (Polynomially) Reflexive
* Riesz
* L-semi-inner product
* (B
* Strictly
* Uniformly) convex
* Uniformly smooth
* (Injective
* Projective) Tensor product (of Hilbert spaces)

Banach spaces are:|

* Barrelled
* Complete
* F-space
* Fréchet
* tame
* Locally convex
* Seminorms/Minkowski functionals
* Mackey
* Metrizable
* Normed
* norm
* Quasinormed
* Stereotype

Function space Topologies|

* Banach–Mazur compactum
* Dual
* Dual space
* Dual norm
* Operator
* Ultraweak
* Weak
* polar
* operator
* 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
* kernel
* operator
* Hilbert–Schmidt
* Functionals
* positive
* Pseudo-monotone
* Normal
* Nuclear
* Self-adjoint
* Strictly singular
* Trace class
* Transpose
* Unitary

Operator theory|

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

Theorems|

* Anderson–Kadec
* 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
* Parseval's identity
* Riesz's lemma
* Riesz representation
* Robinson-Ursescu
* Schauder fixed-point
* Sobczyk's theorem

Analysis|

* Abstract Wiener space
* Banach manifold
* bundle
* Bochner space
* Convex series
* Differentiation in Fréchet spaces
* Derivatives
* Fréchet
* Gateaux
* functional
* holomorphic
* quasi
* Integrals
* Bochner
* Dunford
* Gelfand–Pettis
* regulated
* Paley–Wiener
* weak
* Functional calculus
* Borel
* continuous
* holomorphic
* Measures
* Lebesgue
* Projection-valued
* Vector
* Weakly / Strongly measurable function

Types of sets|

* Absolutely convex
* Absorbing
* Affine
* Balanced/Circled
* Bounded
* Convex
* Convex cone (subset)
* Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx))
* Linear cone (subset)
* Radial
* Radially convex/Star-shaped
* Symmetric
* Zonotope

Subsets / set operations|

* Affine hull
* (Relative) Algebraic interior (core)
* Bounding points
* Convex hull
* Extreme point
* Interior
* Linear span
* Minkowski addition
* Polar
* (Quasi) Relative interior

Examples|

* Absolute continuity AC
* b a ( Σ ) {\displaystyle ba(\Sigma )}
* c space
* Banach coordinate BK
* Besov  B p , q s ( R ) {\displaystyle B_{p,q}^{s}(\mathbb {R} )}
* Birnbaum–Orlicz
* Bounded variation BV
* Bs space
* Continuous C(K) with K compact Hausdorff
* Hardy Hp
* Hilbert H
* Morrey–Campanato  L λ , p ( Ω ) {\displaystyle L^{\lambda ,p}(\Omega )}
* ℓp
* ℓ ∞ {\displaystyle \ell ^{\infty }}
* Lp
* L ∞ {\displaystyle L^{\infty }}
* weighted
* Schwartz  S ( R n ) {\displaystyle S\left(\mathbb {R} ^{n}\right)}
* Segal–Bargmann F
* Sequence space
* Sobolev Wk,p
* Sobolev inequality
* Triebel–Lizorkin
* Wiener amalgam  W ( X , L p ) {\displaystyle W(X,L^{p})}

Applications|

* Differential operator
* Finite element method
* Mathematical formulation of quantum mechanics
* Ordinary Differential Equations (ODEs)
* Validated numerics

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* Supersymmetry
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* Renormalization group
* Representation theory
* Spacetime algebra
* Superalgebra
* Supersymmetry algebra

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* Mathematics

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