49 articles
Murray–von Neumann classification of II₁ factors
The Murray–von Neumann classification of II₁ factors is the program begun by Murray and von Neumann in 1936 and 1943, which introduced the fundamental group invariant of a II₁ factor.
Nest algebra
A nest algebra is the algebra of bounded operators on a Hilbert space leaving invariant every subspace in a nest, a totally ordered collection of closed subspaces.
Noncommutative integration
Noncommutative integration is a branch of operator algebra theory extending measure and integration theory to von Neumann algebras, reducing to classical integration when the algebra is abelian.
Normed algebra
A normed algebra is an associative algebra equipped with a submultiplicative norm, meaning |ab| ≤ |a|·|b|; when the norm is complete it is called a Banach algebra.
Nuclear dimension
Nuclear dimension is a numerical invariant of C-algebras, introduced by Wilhelm Winter and Joachim Zacharias in 2009 as a noncommutative version of topological covering dimension.
Operator space theory
Operator space theory studies Banach spaces embedded in C-algebras, where matrix-level norms distinguish spaces that look identical as Banach spaces, merging C-algebra and Banach space theory.
Planar algebra
A planar algebra is an algebraic structure of vector spaces acted on by planar diagrams, introduced by Vaughan Jones in 1999 to axiomatize subfactor invariants.
Predual and ultraweak topology of von Neumann algebras
The predual of a von Neumann algebra is the Banach space whose dual it is, unique up to isometric isomorphism; it determines the ultraweak topology and normal maps.
Projection lattice and comparison theory
The projection lattice of a von Neumann algebra is its complete lattice of projections, ordered by positivity, with Murray–von Neumann equivalence and comparison theory deciding when projections are equivalent.
Reflexive operator algebra
A reflexive operator algebra is an algebra of bounded operators equal to the algebra of all operators sharing its invariant subspaces, a notion tied to the invariant subspace problem.
Spectral triple
A spectral triple is data (A, H, D) encoding a geometric space, the basic object of Alain Connes' noncommutative geometry, where it generalizes Riemannian manifolds with the Dirac operator.
Spectrum (functional analysis)
The spectrum, also called the operator spectrum, is the set of scalars λ for which T − λI has no bounded inverse, generalizing a matrix's eigenvalues.
Stable rank
Stable rank, or topological stable rank, is a numerical invariant of a C-algebra, introduced by Marc Rieffel in 1983 as the first noncommutative generalization of covering dimension.
Standard form of a von Neumann algebra
The standard form of a von Neumann algebra is a canonical Hilbert-space representation with a self-dual cone, proved to exist and be unique by Haagerup in 1975.
Subfactor
A subfactor is an inclusion of factors within von Neumann algebras, a theory initiated by Vaughan Jones in the early 1980s, whose Jones index takes quantized values.
Tomita–Takesaki theory
Tomita–Takesaki theory, also known as Tomita's theorem, associates to a von Neumann algebra a modular operator and conjugation, and is central to operator algebra theory and quantum field theory.
Uniform algebra
A uniform algebra is a closed subalgebra of continuous functions on a compact Hausdorff space, containing the constants and separating points, studied for uniform approximation.
Von Neumann algebra
A von Neumann algebra, also called a W-algebra, is a weakly closed self-adjoint algebra of bounded operators on a Hilbert space, introduced by John von Neumann in 1929.
Wiener algebra
The Wiener algebra A(𝕋), or W(𝕋), is a commutative Banach algebra of continuous periodic functions with absolutely summable Fourier coefficients, introduced by Norbert Wiener and known for Wiener's lemma.