Related Research Articles
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.
In mathematics, specifically in functional analysis, a C∗-algebra is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties:
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = g–1ag. This is an equivalence relation whose equivalence classes are called conjugacy classes.
In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group.
In mathematics, the free groupFS over a given set S consists of all words that can be built from members of S, considering two words to be different unless their equality follows from the group axioms. The members of S are called generators of FS, and the number of generators is the rank of the free group. An arbitrary group G is called free if it is isomorphic to FS for some subset S of G, that is, if there is a subset S of G such that every element of G can be written in exactly one way as a product of finitely many elements of S and their inverses.
In mathematics, specifically group theory, a nilpotent groupG is a group that has an upper central series that terminates with G. Equivalently, its central series is of finite length or its lower central series terminates with {1}.
In mathematics, a free abelian group or free Z-module is an abelian group with a basis, or, equivalently, a free module over the integers. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as a linear combination of basis elements with integer coefficients. For instance, the integers with addition form a free abelian group with basis {1}. Free abelian groups have properties which make them similar to vector spaces. They have applications in algebraic topology, where they are used to define chain groups, and in algebraic geometry, where they are used to define divisors. Integer lattices also form examples of free abelian groups, and lattice theory studies free abelian subgroups of real vector spaces.
A group is a set together with an associative operation which admits an identity element and such that every element has an inverse.
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 special type of C*-algebra.
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, such that representations of the algebra are related to representations of the group. As such, they are similar to the group ring associated to a discrete group.
In group theory, a branch of mathematics, the order of a group is its cardinality, that is, the number of elements in its set. If the group is seen multiplicatively, the order of an elementa of a group, sometimes also called the period length or period of a, is the smallest positive integer m such that am = e, where e denotes the identity element of the group, and am denotes the product of m copies of a. If no such m exists, a is said to have infinite order.
Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of K. There are only finitely many patterns of splitting that may occur. Although the full description of the splitting of every prime p in a general Galois extension is a major unsolved problem, the Chebotarev density theorem says that the frequency of the occurrence of a given pattern, for all primes p less than a large integer N, tends to a certain limit as N goes to infinity. It was proved by Nikolai Chebotaryov in his thesis in 1922, published in.
In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive invariant measure on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun on "mean".
In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination of finitely many elements of the finite set S and of inverses of such elements.
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 unique von Neumann algebra that is a factor of type II1 and also hyperfinite; it is called the hyperfinite type II1 factor. There are an uncountable number of other factors of type II1. Connes proved that the infinite one is also unique.
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 von Neumann algebras. The concept was introduced in 1949 by John von Neumann in one of the papers in the series On Rings of Operators. One of von Neumann's goals in this paper was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of so-called factors. Factors are analogous to full matrix algebras over a field, and von Neumann wanted to prove a continuous analogue of the Artin–Wedderburn theorem classifying semi-simple rings.
In mathematics, and more specifically in the theory of von Neumann algebras, a crossed product is a basic method of constructing a new von Neumann algebra from a von Neumann algebra acted on by a group. It is related to the semidirect product construction for groups.
In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.
In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative. Since locally compact abelian groups have a well-understood theory, Pontryagin duality, which includes the basic structures of Fourier series and Fourier transforms, the major business of non-commutative harmonic analysis is usually taken to be the extension of the theory to all groups G that are locally compact. The case of compact groups is understood, qualitatively and after the Peter–Weyl theorem from the 1920s, as being generally analogous to that of finite groups and their character theory.
In mathematics, Jordan operator algebras are real or complex Jordan algebras with the compatible structure of a Banach space. When the coefficients are real numbers, the algebras are called Jordan Banach algebras. The theory has been extensively developed only for the subclass of JB algebras. The axioms for these algebras were devised by Alfsen, Schultz & Størmer (1978). Those that can be realised concretely as subalgebras of self-adjoint operators on a real or complex Hilbert space with the operator Jordan product and the operator norm are called JC algebras. The axioms for complex Jordan operator algebras, first suggested by Irving Kaplansky in 1976, require an involution and are called JB* algebras or Jordan C* algebras. By analogy with the abstract characterisation of von Neumann algebras as C* algebras for which the underlying Banach space is the dual of another, there is a corresponding definition of JBW algebras. Those that can be realised using ultraweakly closed Jordan algebras of self-adjoint operators with the operator Jordan product are called JW algebras. The JBW algebras with trivial center, so-called JBW factors, are classified in terms of von Neumann factors: apart from the exceptional 27 dimensional Albert algebra and the spin factors, all other JBW factors are isomorphic either to the self-adjoint part of a von Neumann factor or to its fixed point algebra under a period two *-anti-automorphism. Jordan operator algebras have been applied in quantum mechanics and in complex geometry, where Koecher's description of bounded symmetric domains using Jordan algebras has been extended to infinite dimensions.
References
- ↑ Palmer, Theodore W. (2001), Banach Algebras and the General Theory of *-Algebras, Volume 2, Encyclopedia of mathematics and its applications, 79, Cambridge University Press, p. 907, ISBN 9780521366380 .
- ↑ Popa, Sorin (2007), "Deformation and rigidity for group actions and von Neumann algebras", International Congress of Mathematicians. Vol. I (PDF), Eur. Math. Soc., Zürich, pp. 445–477, doi:10.4171/022-1/18, ISBN 978-3-03719-022-7, MR 2334200 . See in particular p. 450: "LΓ is a II1 factor iff Γ is ICC".
- 1 2 Palmer (2001), p. 908.
 | This algebra-related article is a stub. You can help Wikipedia by expanding it. |