|
Uniformly Bounded Representation
In mathematics, a uniformly bounded representation T of a locally compact group G on a Hilbert space H is a homomorphism into the bounded invertible operators which is continuous for the strong operator topology, and such that \sup_ \, T_g\, _ is finite. In 1947 Béla Szőkefalvi-Nagy established that any uniformly bounded representation of the integers or the real numbers is unitarizable, i.e. conjugate by an invertible operator to a unitary representation. For the integers this gives a criterion for an invertible operator to be similar to a unitary operator: the operator norms of all the positive and negative powers must be uniformly bounded. The result on unitarizability of uniformly bounded representations was extended in 1950 by Dixmier, Day and Nakamura-Takeda to all locally compact amenable groups, following essentially the method of proof of Sz-Nagy. The result is known to fail for non-amenable groups such as SL(2,R) and the free group on two generators. conjectured t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Compact Group
In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the Haar measure. This allows one to define integrals of Borel measurable functions on ''G'' so that standard analysis notions such as the Fourier transform and L^p spaces can be generalized. Many of the results of finite group representation theory are proved by averaging over the group. For compact groups, modifications of these proofs yields similar results by averaging with respect to the normalized Haar integral. In the general locally compact setting, such techniques need not hold. The resulting theory is a central part of harmonic analysis. The representation theory for locally compact abelian groups is described by Pontryagin duality. Examples and counterexamples *An ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann Conjecture
In mathematics, the von Neumann conjecture stated that a group ''G'' is non- amenable if and only if ''G'' contains a subgroup that is a free group on two generators. The conjecture was disproved in 1980. In 1929, during his work on the Banach–Tarski paradox, John von Neumann defined the concept of amenable groups and showed that no amenable group contains a free subgroup of rank 2. The suggestion that the converse might hold, that is, that every non-amenable group contains a free subgroup on two generators, was made by a number of different authors in the 1950s and 1960s. Although von Neumann's name is popularly attached to the conjecture, its first written appearance seems to be due to Mahlon Marsh Day in 1957. The Tits alternative is a fundamental theorem which, in particular, establishes the conjecture within the class of linear groups. The historically first potential counterexample is Thompson group ''F''. While its amenability is a wide open problem, the general conje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory. Single operator theory Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category. Spectrum of operators The spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Functional Analysis
The ''Journal of Functional Analysis'' is a mathematics journal published by Elsevier. Founded by Paul Malliavin, Ralph S. Phillips, and Irving Segal, its editors-in-chief are Daniel W. Stroock, Stefaan Vaes, and Cedric Villani. It is covered in databases including Scopus, the Science Citation Index The Science Citation Index Expanded – previously entitled Science Citation Index – is a citation index originally produced by the Institute for Scientific Information (ISI) and created by Eugene Garfield. It was officially launched in 1964 ..., and the SCImago Journal Rank service. References {{Mathematics-journal-stub Elsevier academic journals Mathematics journals Publications established in 1967 Semi-monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tôhoku Mathematical Journal
The ''Tohoku Mathematical Journal'' is a mathematical research journal published by Tohoku University in Japan. It was founded in August 1911 by Tsuruichi Hayashi. History Due to World War II the publication of the journal stopped in 1943 with volume 49. Publication was resumed in 1949 with the volume numbering starting again at 1. In order to distinguish between the identical numbered volumes, volumes in the first publishing period are referred to as the ''first series'' whereas the later volumes are called ''second series''. Before volume 51 of the second series the journal was called ''Tôhoku Mathematical Journal'', with a circumflex over the second letter of ''Tohoku''. Selected papers *. The first publication of the Sprague–Grundy theorem, the basis for much of combinatorial game theory, later independently rediscovered by P. M. Grundy. *. This paper describes Weiszfeld's algorithm for finding the geometric median. *. This paper, often referred to as " The Tohoku pap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Mathematics Research Notices
The ''International Mathematics Research Notices'' is a peer-reviewed mathematics journal. Originally published by Duke University Press and Hindawi Publishing Corporation, it is now published by Oxford University Press. retrieved 2015-02-26. The Executive Editor is (). According to the '' |
Direct Product Of Groups
In mathematics, specifically in group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted . 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 the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted G \oplus H. Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups. Definition Given groups (with operation ) and (with operation ), the direct product is defined as follows: The resulting algebraic object satisfies the axioms for a group. Specifically: ;Associativity: The binary operation on is associative. ;Identity: The direct product has an identity element, namely , where is the identity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gilles Pisier
Gilles I. Pisier (born 18 November 1950) is a professor of mathematics at the Pierre and Marie Curie University and a distinguished professor and A.G. and M.E. Owen Chair of Mathematics at the Texas A&M University. He is known for his contributions to several fields of mathematics, including functional analysis, probability theory, harmonic analysis, and operator theory. He has also made fundamental contributions to the theory of C*-algebras. Gilles is the younger brother of French actress Marie-France Pisier. Research Pisier has obtained many fundamental results in various parts of mathematical analysis. Geometry of Banach spaces In the "local theory of Banach spaces", Pisier and Bernard Maurey developed the theory of Rademacher type, following its use in probability theory by J. Hoffman–Jorgensen and in the characterization of Hilbert spaces among Banach spaces by S. Kwapień. Using probability in vector spaces, Pisier proved that super-reflexive Banach spaces can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Burnside Group
The Burnside problem asks whether a finitely generated group in which every element has finite Order (group theory), order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory and was influential in the development of combinatorial group theory. It is known to have a negative answer in general, as Evgeny Golod and Igor Shafarevich provided a counter-example in 1964. The problem has many refinements and variants (see #Bounded Burnside problem, bounded and #Restricted Burnside problem, restricted below) that differ in the additional conditions imposed on the orders of the group elements, some of which are still open problem, open questions. Brief history Initial work pointed towards the affirmative answer. For example, if a group ''G'' is finitely generated and the order of each element of ''G'' is a divisor of 4, then ''G'' is finite. Moreover, A. I. Kostrikin was able to prove in 1958 that among the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicolas Monod
Nicolas Monod is a professor at École Polytechnique Fédérale de Lausanne (EPFL) and known for work on bounded cohomology, ergodic theory, geometry (CAT(0) spaces), locally compact groups and amenability. He was born in Montreux, Switzerland. He obtained his PhD from ETH Zurich in 2001 with thesis "Continuous Bounded Cohomology of Locally Compact Groups" written under the direction of Marc Burger. Career Monod is a Fellow of the American Mathematical Society. He has been awarded the Gauss Lectureship and the Berwick Prize, and was an invited speaker at the International Congress of Mathematicians in 2006. He was one of the youngest Advanced Investigator awardees in the history of the European Research Council. Monod was the president of the Swiss Mathematical Society The Swiss Mathematical Society (german: Schweizerische Mathematische Gesellschaft; french: Société Mathématique Suisse), founded in Basel on September 4, 1910, is the national mathematical society of Switze ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tits Alternative
In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups. Statement The theorem, proven by Tits, is stated as follows. Consequences A linear group is not amenable if and only if it contains a non-abelian free group (thus the von Neumann conjecture, while not true in general, holds for linear groups). The Tits alternative is an important ingredient in the proof of Gromov's theorem on groups of polynomial growth. In fact the alternative essentially establishes the result for linear groups (it reduces it to the case of solvable groups, which can be dealt with by elementary means). Generalizations In geometric group theory, a group ''G'' is said to satisfy the Tits alternative if for every subgroup ''H'' of ''G'' either ''H'' is virtually solvable or ''H'' contains a nonabelian free subgroup (in some versions of the definition this condition is only required to be satisfied for all finit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
