Section
Operator Algebras
C*-algebras, von Neumann algebras, their maps, representations, states, weights, and structure theory.
Core idea
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Operator algebras
Operator and C*-algebra foundations
- C*-algebra
- C*-algebra C_0(X)
- C*-algebra of bounded operators
- C*-algebra of compact operators
- C*-identity
- C*-subalgebra
- Canonical operator trace
- Commutative C*-algebra
- Concrete C*-algebra
- Graded C*-algebra
- Even and odd operators on a graded Hilbert space
- Generated C*-subalgebra
- Involutive algebra
- Matrix C*-algebra
- Opposite algebra
- Projection in a C*-algebra
- Unital C*-algebra
- Unitary element of a C*-algebra
Positivity and order
- Order on self-adjoint elements
- Order-unit space
- Positive and negative parts of a self-adjoint element
- Positive cone of a C*-algebra
- Positive element of a C*-algebra
- Positive square root in a C*-algebra
- Self-adjoint element of a C*-algebra
Maps and morphisms
- *-automorphism
- *-homomorphism
- *-isomorphism
- Completely bounded map
- Completely positive contraction
- Completely positive map
- Completely positive order-zero map
- Conditional expectation of C*-algebras
- Faithful *-homomorphism
- Kadison–Schwarz inequality
- n-positive map
- Nondegenerate *-homomorphism
- Positive linear map
- Stinespring dilation theorem
- Unital *-homomorphism
- Unital completely positive map
Ideals and quotients
- Closed two-sided ideal of a C*-algebra
- Corona C*-algebra
- Essential ideal of a C*-algebra
- Full hereditary C*-subalgebra
- Hereditary C*-subalgebra
- Hull–kernel topology
- Kernel of a *-homomorphism
- Multiplier algebra
- Primitive ideal of a C*-algebra
- Primitive ideal space
- Quotient C*-algebra
- Strict topology on a multiplier algebra
Approximate identities, unitizations, and multipliers
- Approximate identity in a C*-algebra
- Double centralizer of a C*-algebra
- Extension of a nondegenerate homomorphism to multiplier algebras
- Quasicentral approximate identity
- Strictly positive element of a C*-algebra
- Unitization of a C*-algebra
- σ-unital C*-algebra
Spectra and functional calculus
- Character space of a commutative C*-algebra
- Continuous functional calculus
- Gelfand duality
- Gelfand transform
- Spectral mapping theorem
- Spectral permanence
States, characters, and traces
- Cauchy–Schwarz inequality for a positive functional
- Character of a C*-algebra
- Faithful state on a C*-algebra
- Lower semicontinuous trace
- Positive linear functional
- Pure state of a C*-algebra
- Singular state on a von Neumann algebra
- State on a C*-algebra
- State space of a C*-algebra
- Trace on a C*-algebra
- Tracial state
Representations and enveloping von Neumann algebras
- Bounded star-representation
- Cyclic representation of a C*-algebra
- Direct sum of C*-representations
- Disjoint representations
- Factorial representation
- Faithful representation of a C*-algebra
- Gelfand–Naimark theorem
- GNS construction
- GNS uniqueness theorem
- Irreducible representation of a C*-algebra
- Nondegenerate representation of a C*-algebra
- Pure states and irreducible GNS representations
- Quasi-equivalent representations
- Representation of a C*-algebra
- States separate positive elements
- Unitary equivalence of C*-representations
- Universal representation of a C*-algebra
- Vector state
- Von Neumann algebra generated by a representation
Type I C*-algebras
- Elementary C*-algebra
- Glimm's type I theorem
- Liminal C*-algebra
- Liminal C*-algebras are type I
- Type I C*-algebra
- Type I composition series
- Type I factor representation
- Type I stability under ideals and quotients
Group C*-algebras
- Amenability and equality of full and reduced group C*-algebras
- Canonical quotient from full to reduced group C*-algebra
- Full group C*-algebra
- Group C*-algebras of a discrete group
- Nuclearity of reduced group C*-algebras and amenability for discrete groups
- Reduced group C*-algebra
- Unitary representations and representations of the full group C*-algebra
Crossed products
- Amenable action on a C*-algebra
- C*-dynamical system
- Canonical conditional expectation on a reduced crossed product
- Canonical map from full to reduced crossed product
- Covariant representation of a C*-dynamical system
- Crossed-product convolution *-algebra C_c(G,A)
- Dual action on a crossed product
- Fell bundle
- Full crossed product
- Integrated form of a covariant representation
- Reduced crossed product
- Regular covariant representation
- Takai duality
- Transformation-group C*-algebra
Universal C*-algebras
- C*-algebra by generators and relations
- Enveloping C*-algebra
- Maximal C*-seminorm
- Universal C*-algebra
Exactness and tensor products
- C*-tensor norm
- Completely positive approximation property
- Exact C*-algebra
- Local lifting property
- Local reflexivity for C*-algebras
- Maximal C*-tensor product
- Minimal C*-tensor product
- Nuclear C*-algebra
- Nuclear C*-algebras are exact
- Short exact sequence of C*-algebras
- Weak expectation property
Continuous fields and C*-bundles
- Continuous C_0(X)-algebra
- Continuous field of C*-algebras
- Continuous-trace C*-algebra
- C_0(X)-algebra
- C_0(X)-algebras and upper-semicontinuous C*-bundles
- Dixmier–Douady class
- Fiber of a C_0(X)-algebra
- Section algebra of a C*-bundle
- Upper-semicontinuous C*-bundle
Hilbert modules, correspondences, and Morita theory
- Adjointable operator on a Hilbert C*-module
- Brown-Green-Rieffel stabilization theorem
- C*-correspondence
- Cauchy-Schwarz inequality for Hilbert C*-modules
- Compact operator on a Hilbert C*-module
- Countably generated Hilbert C*-module
- Cuntz-Pimsner algebra
- Exterior tensor product of Hilbert C*-modules
- Fock module of a C*-correspondence
- Green imprimitivity theorem
- Hilbert C*-module
- Imprimitivity bimodule
- Internal tensor product of C*-correspondences
- Kasparov stabilization theorem
- Linking algebra
- Multiplier module
- Regular operator on a Hilbert C*-module
- Rieffel induction
- Standard Hilbert C*-module
- Strong Morita equivalence of C*-algebras
- Toeplitz representation of a C*-correspondence
- Toeplitz-Pimsner algebra
Von Neumann algebra foundations and topologies
- Bicommutant
- Commutant
- Sakai characterization theorem
- Strong operator topology
- Strong-* operator topology
- Ultrastrong operator topology
- Ultrastrong-star topology
- Ultraweak operator topology
- Von Neumann algebra
- Von Neumann bicommutant theorem
- Weak operator topology
Projections and comparison theory
- Abelian projection
- Central support of a projection
- Comparison theorem for projections
- Finite projection
- Infinite projection
- Minimal projection
- Murray–von Neumann equivalence of projections
- Murray–von Neumann subequivalence of projections
- Properly infinite projection
Maps, preduals, and normal states
- Faithful normal state
- Normal *-homomorphism
- Normal completely positive map
- Normal conditional expectation
- Normal functional
- Normal linear map
- Normal positive map
- Normal representation
- Normal state
- Normal unital completely positive map
- Positive cone of the predual
- Predual of a von Neumann algebra
- Support projection of a normal positive functional
- Unique predual theorem
- Vector functional on a von Neumann algebra
Weights, modular theory, and standard form
- Centralizer of a weight
- Connes cocycle derivative
- Cyclic vector
- Cyclic-separating duality
- Extended positive cone of a von Neumann algebra
- Entire analytic element for a one-parameter automorphism group
- Faithful normal semifinite trace
- Faithful weight
- Finite normal trace on a von Neumann algebra
- GNS construction for a weight
- Left ideal of a weight
- Modular automorphism group
- Modular conjugation
- Modular KMS condition
- Modular operator
- Natural positive cone
- Normal semifinite faithful weight
- Normal weight
- Operator-valued weight
- Semifinite weight
- Separating vector
- Spatial derivative
- Standard form of a von Neumann algebra
- Tomita operator
- Tomita–Takesaki theorem
- Tracial weight
- Uniqueness of standard form
- Weight on a von Neumann algebra
Von Neumann crossed products and continuous cores
- Von Neumann crossed product
- Dual action on a von Neumann crossed product
- Continuous core of a von Neumann algebra
Factors and type classification
- Center of a von Neumann algebra
- Central decomposition of a von Neumann algebra
- Commutative von Neumann algebra
- Connes classification of type III factors
- Dimension function on projections
- Finite von Neumann algebra
- Flow of weights
- Group von Neumann algebra
- Hyperfinite II₁ factor
- Properly infinite von Neumann algebra
- Semifinite von Neumann algebra
- Type decomposition of von Neumann algebras
- Type I factor
- Type I von Neumann algebra
- Type II von Neumann algebra
- Type III factor
- Type III von Neumann algebra
- Type II₁ factor
- Type II∞ factor
- Type Iₙ factor
- Type I∞ factor
- Von Neumann factor
Affiliated operators and noncommutative integration
- Breuer-Fredholm operator
- Dixmier trace
- Marcinkiewicz–Macaev ideal
- Measurable operator affiliated with a von Neumann algebra
- Noncommutative L^p space
- Operator affiliated with a von Neumann algebra
- Spectral characterization of affiliation
- Tau-compact operator
- Tau-measurable operator
- Tracial noncommutative L^p space
- Weak Schatten ideal