Category: Von Neumann algebras

Singular trace
In mathematics, a singular trace is a trace on a space of linear operators of a separable Hilbert space that vanisheson operators of finite rank. Singular traces are a feature of infinite-dimensional
Hilbert algebra
In mathematics, Hilbert algebras and left Hilbert algebras occur in the theory of von Neumann algebras in: * Commutation theorem for traces#Hilbert algebras * Tomita–Takesaki theory#Left Hilbert alg
Weak trace-class operator
In mathematics, a weak trace class operator is a compact operator on a separable Hilbert space H with singular values the same order as the harmonic sequence.When the dimension of H is infinite, the i
Ultraweak topology
In functional analysis, a branch of mathematics, the ultraweak topology, also called the weak-* topology, or weak-* operator topology or σ-weak topology, on the set B(H) of bounded operators on a Hilb
Commutation theorem for traces
In mathematics, a commutation theorem for traces explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the presence of a trace. The first such result was p
Finite von Neumann algebra
In mathematics, a finite von Neumann algebra is a von Neumann algebra in which every isometry is a unitary. In other words, for an operator V in a finite von Neumann algebra if , then . In terms of th
Abelian von Neumann algebra
In functional analysis, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute. The prototypical example of an abelian von Neumann algebr
Continuous geometry
In mathematics, continuous geometry is an analogue of complex projective geometry introduced by von Neumann , where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can
In the theory of von Neumann algebras, a subfactor of a factor is a subalgebra that is a factor and contains . The theory of subfactors led to the discovery of the Jones polynomial in knot theory.
Temperley–Lieb algebra
In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrices, invented by Neville Temperley and Elliott Lieb. It is also related to integrable mode
Affiliated operator
In mathematics, affiliated operators were introduced by Murray and von Neumann in the theory of von Neumann algebras as a technique for using unbounded operators to study modules generated by a single
Schröder–Bernstein theorems for operator algebras
The Schröder–Bernstein theorem from set theory has analogs in the context operator algebras. This article discusses such operator-algebraic results.
Sherman–Takeda theorem
In mathematics, the Sherman–Takeda theorem states that if A is a C*-algebra then its double dual is a W*-algebra, and is isomorphic to the weak closure of A in the universal representation of A. The t
Commutator subspace
In mathematics, the commutator subspace of a two-sided ideal of bounded linear operators on a separable Hilbert space is the linear subspace spanned by commutators of operators in the ideal with bound
Tomita–Takesaki theory
In the theory of von Neumann algebras, a part of the mathematical field of functional analysis, Tomita–Takesaki theory is a method for constructing modular automorphisms of von Neumann algebras from t
Central carrier
In the context of von Neumann algebras, the central carrier of a projection E is the smallest central projection, in the von Neumann algebra, that dominates E. It is also called the central support or
Crossed product
In mathematics, and more specifically in the theory of von Neumann algebras, a crossed productis a basic method of constructing a new von Neumann algebra from a von Neumann algebra acted on by a group
Hyperfinite type II factor
In mathematics, there are up to isomorphism exactly two separably acting hyperfinite type II factors; one infinite and one finite. Murray and von Neumann proved that up to isomorphism there is a uniqu
Von Neumann bicommutant theorem
In mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of t
Baer ring
In abstract algebra and functional analysis, Baer rings, Baer *-rings, Rickart rings, Rickart *-rings, and AW*-algebras are various attempts to give an algebraic analogue of von Neumann algebras, usin
Ultrastrong topology
In functional analysis, the ultrastrong topology, or σ-strong topology, or strongest topology on the set B(H) of bounded operators on a Hilbert space is the topology defined by the family of seminorms
Kaplansky density theorem
In the theory of von Neumann algebras, the Kaplansky density theorem, due to Irving Kaplansky, is a fundamental approximation theorem. The importance and ubiquity of this technical tool led to comment
Von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a spec
Group algebra of a locally compact group
In functional analysis and related areas of mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra)
Fuglede−Kadison determinant
In mathematics, the Fuglede−Kadison determinant of an invertible operator in a finite factor is a positive real number associated with it. It defines a multiplicative homomorphism from the set of inve
Approximately finite-dimensional
No description available.
Connes embedding problem
Connes' embedding problem, formulated by Alain Connes in the 1970s, is a major problem in von Neumann algebra theory. During that time, the problem was reformulated in several different areas of mathe
Gelfand representation
In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things: * a way of representing commutative Banach algebras as algebras of continuous fu
Calkin correspondence
In mathematics, the Calkin correspondence, named after mathematician John Williams Calkin, is a bijective correspondence between two-sided ideals of bounded linear operators of a separable infinite-di
Dixmier trace
In mathematics, the Dixmier trace, introduced by Jacques Dixmier, is a non-normal trace on a space of linear operators on a Hilbert space larger than the space of trace class operators. Dixmier traces
Direct integral
In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of v