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Operator algebra

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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 $\mathbb {Z}$ •Terminal ring $0=\mathbb {Z} /1\mathbb {Z}$ 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 modulon •Ring of integers •p-adic integers $\mathbb {Z} _{p}$ •p-adic numbers $\mathbb {Q} _{p}$ •Prüferp-ring $\mathbb {Z} (p^{\infty })$
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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Infunctional analysis, a branch ofmathematics, anoperator algebra is analgebra ofcontinuous linear operators on atopological vector space, with the multiplication given by thecomposition of mappings.

The results obtained in the study of operator algebras are often phrased inalgebraic terms, while the techniques used are often highlyanalytic.^[1] Although the study of operator algebras is usually classified as a branch of functional analysis, it has direct applications torepresentation theory,differential geometry,quantum statistical mechanics,quantum information, andquantum field theory.

Overview

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

An operator algebra is typically required to beclosed in a specified operatortopology 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 areaxiomatized and algebras with certain topological structure become the subject of the research.

Though algebras of operators are studied in various contexts (for example, algebras ofpseudo-differential operators acting on spaces ofdistributions), the termoperator algebra is usually used in reference to algebras ofbounded operators on aBanach space or, even more specifically in reference to algebras of operators on aseparable Hilbert space, endowed with theoperator norm topology.

In the case of operators on a Hilbert space, theHermitian adjoint map on operators gives a naturalinvolution, which provides an additional algebraic structure that can be imposed on the algebra. In this context, the best studied examples areself-adjoint operator algebras, meaning that they are closed under taking adjoints. These includeC*-algebras,von Neumann algebras, andAW*-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 closedsubalgebra 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 ofcomplex-valued continuous functions on alocally compact space, or that ofmeasurable functions on astandard measurable space. Thus, general operator algebras are often regarded as a noncommutative generalizations of these algebras, or the structure of thebase space on which the functions are defined. This point of view is elaborated as the philosophy ofnoncommutative 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,
manycommutative subspace lattice algebras,
manylimit algebras.

References

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^Theory of Operator Algebras I ByMasamichi Takesaki, Springer 2012, p vi

v t e 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 anApproximate 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

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

Operators

Algebras

Open problems

Applications

Advanced topics

Category

v t e 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 continuityAC $ba(\Sigma )$ c space Banach coordinateBK Besov $B_{p,q}^{s}(\mathbb {R} )$ Birnbaum–Orlicz Bounded variationBV Bs space ContinuousC(K) withK compact Hausdorff Hardy H^p HilbertH Morrey–Campanato $L^{\lambda ,p}(\Omega )$ ℓ^p $\ell ^{\infty }$ L^p $L^{\infty }$ weighted Schwartz $S\left(\mathbb {R} ^{n}\right)$ Segal–BargmannF Sequence space Sobolev W^k,p Sobolev inequality Triebel–Lizorkin Wiener amalgam $W(X,L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

Industrial and applied mathematics

Computational

Mathematical software	Arbitrary-precision arithmetic Finite element analysis Tensor software Interactive geometry software Optimization software Statistical software Numerical-analysis software Numerical libraries Solvers

Algebraic structures	Algebra of physical space Particle physics and representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra