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Gelfand–Raikov Theorem
The Gel'fand–Raikov (Гельфанд–Райков) theorem is a theorem in the theory of locally compact topological groups. It states that a locally compact group is completely determined by its (possibly infinite dimensional) unitary representations. The theorem was first published in 1943. A unitary representation of a locally compact group on a Hilbert space |
Locally Compact Topological 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 *Any c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Representation
In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G'' is a locally compact ( Hausdorff) topological group and the representations are strongly continuous. The theory has been widely applied in quantum mechanics since the 1920s, particularly influenced by Hermann Weyl's 1928 book ''Gruppentheorie und Quantenmechanik''. One of the pioneers in constructing a general theory of unitary representations, for any group ''G'' rather than just for particular groups useful in applications, was George Mackey. Context in harmonic analysis The theory of unitary representations of topological groups is closely connected with harmonic analysis. In the case of an abelian group ''G'', a fairly complete picture of the representation theory of ''G'' is given by Pontryagin duality. In general, the unitary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Coefficient
In mathematics, a matrix coefficient (or matrix element) is a function on a group of a special form, which depends on a linear representation of the group and additional data. Precisely, it is a function on a compact topological group ''G'' obtained by composing a representation of ''G'' on a vector space ''V'' with a linear map from the endomorphisms of ''V'' into ''V'' 's underlying field. It is also called a representative function. They arise naturally from finite-dimensional representations of ''G'' as the matrix-entry functions of the corresponding matrix representations. The Peter–Weyl theorem says that the matrix coefficients on ''G'' are dense in the Hilbert space of square-integrable functions on ''G''. Matrix coefficients of representations of Lie groups turned out to be intimately related with the theory of special functions, providing a unifying approach to large parts of this theory. Growth properties of matrix coefficients play a key role in the classification o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more elementary kind of structure, the abelian group. The direct sum of two abelian groups A and B is another abelian group A\oplus B consisting of the ordered pairs (a,b) where a \in A and b \in B. To add ordered pairs, we define the sum (a, b) + (c, d) to be (a + c, b + d); in other words addition is defined coordinate-wise. For example, the direct sum \Reals \oplus \Reals , where \Reals is real coordinate space, is the Cartesian plane, \R ^2 . A similar process can be used to form the direct sum of two vector spaces or two modules. We can also form direct sums with any finite number of summands, for example A \oplus B \oplus C, provided A, B, and C are the same kinds of algebraic structures (e.g., all abelian groups, or all vector sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Representation
In mathematics, the tensor representations of the general linear group are those that are obtained by taking finitely many tensor products of the fundamental representation and its dual. The irreducible factors of such a representation are also called tensor representations, and can be obtained by applying Schur functors (associated to Young tableaux). These coincide with the rational representations of the general linear group. More generally, a matrix group is any subgroup of the general linear group. A tensor representation of a matrix group is any representation that is contained in a tensor representation of the general linear group. For example, the orthogonal group O(''n'') admits a tensor representation on the space of all trace-free symmetric tensors of order two. For orthogonal groups, the tensor representations are contrasted with the spin representations. The classical group In mathematics, the classical groups are defined as the special linear groups over t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Conjugate Representation
In mathematics, if is a group and is a representation of it over the complex vector space , then the complex conjugate representation is defined over the complex conjugate vector space as follows: : is the conjugate of for all in . is also a representation, as one may check explicitly. If is a real Lie algebra and is a representation of it over the vector space , then the conjugate representation is defined over the conjugate vector space as follows: : is the conjugate of for all in .This is the mathematicians' convention. Physicists use a different convention where the Lie bracket of two real vectors is an imaginary vector. In the physicist's convention, insert a minus in the definition. is also a representation, as one may check explicitly. If two real Lie algebras have the same complexification, and we have a complex representation of the complexified Lie algebra, their conjugate representations are still going to be different. See spinor for some examples ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone–Weierstrass Theorem
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass in 1885 using the Weierstrass transform. Marshall H. Stone considerably generalized the theorem and simplified the proof . His result is known as the Stone–Weierstrass theorem. The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval , an arbitrary compact Hausdorff space is considered, and instead of the algebra of polynomial functions, a variety of other families of continuous functions on X are shown to suffice, as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfand–Naimark Theorem
In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra ''A'' is isometrically *-isomorphic to a C*-subalgebra of bounded operators on a Hilbert space. This result was proven by Israel Gelfand and Mark Naimark in 1943 and was a significant point in the development of the theory of C*-algebras since it established the possibility of considering a C*-algebra as an abstract algebraic entity without reference to particular realizations as an operator algebra. Details The Gelfand–Naimark representation π is the direct sum of representations π''f'' of ''A'' where ''f'' ranges over the set of pure states of A and π''f'' is the irreducible representation associated to ''f'' by the GNS construction. Thus the Gelfand–Naimark representation acts on the Hilbert direct sum of the Hilbert spaces ''H''''f'' by : \pi(x) bigoplus_ H_f= \bigoplus_ \pi_f(x)H_f. π(''x'') is a bounded linear operator since it is the direct sum of a family of operators, each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
