Elementary Representation Theory in Hilbert Space

1973 ◽  
pp. 145-227
Author(s):  
Steven A. Gaal
Keyword(s):  
Hilbert Space ◽  
Representation Theory ◽  
Elementary Representation

scholarly journals Parabolic induction and restriction via -algebras and Hilbert -modules

2016 ◽  
Vol 152 (6) ◽  
pp. 1286-1318 ◽  
Author(s):  
Pierre Clare ◽  
Tyrone Crisp ◽  
Nigel Higson
Keyword(s):  
Hilbert Space ◽  
Representation Theory ◽  
Inner Product ◽  
Reductive Groups ◽  
Hilbert Modules ◽  
Parabolic Induction ◽  
Full Treatment ◽  
Reduced Group ◽  
The Right ◽  
Tempered Representation

This paper is about the reduced group $C^{\ast }$-algebras of real reductive groups, and about Hilbert $C^{\ast }$-modules over these $C^{\ast }$-algebras. We shall do three things. First, we shall apply theorems from the tempered representation theory of reductive groups to determine the structure of the reduced $C^{\ast }$-algebra (the result has been known for some time, but it is difficult to assemble a full treatment from the existing literature). Second, we shall use the structure of the reduced $C^{\ast }$-algebra to determine the structure of the Hilbert $C^{\ast }$-bimodule that represents the functor of parabolic induction. Third, we shall prove that the parabolic induction bimodule admits a secondary inner product, using which we can define a functor of parabolic restriction in tempered representation theory. We shall prove in a sequel to this paper that parabolic restriction is adjoint, on both the left and the right, to parabolic induction in the context of tempered unitary Hilbert space representations.

scholarly journals Self adjoint operators and matrix measures

1971 ◽  
Vol 4 (3) ◽  
pp. 289-305 ◽  
Author(s):  
Patrick J. Browne
Keyword(s):  
Hilbert Space ◽  
Representation Theory ◽  
Spectral Representation ◽  
Adjoint Operator ◽  
Multiplication Operator ◽  
Linear Operators ◽  
Positive Matrix ◽  
Adjoint Operators ◽  
Matrix Measures

Given a self adjoint operator, T, on a Hilbert space H, and given an integer n ≥ 1, we produce a collection , N ∈ L, of n × n positive matrix measures and a unitary map U: such that UTU−1, restricted to the co-ordinate space , is the multiplication operator F(t) → tF(t) in that space. This is a generalization of the spectral representation theory of Dunford and Schwartz, Linear operators, II (1966).

scholarly journals Strongly Continuous Representations in the Hilbert Space: A Far-Reaching Concept

Universe ◽  
2021 ◽  
Vol 7 (8) ◽  
pp. 285
Author(s):  
Julio Marny Hoff da Silva ◽  
Gabriel Marcondes Caires da Rocha
Keyword(s):  
Hilbert Space ◽  
Representation Theory ◽  
Quantum Physics ◽  
Fundamental Notion ◽  
Significant Aspect ◽  
Projective Representations ◽  
Continuous Representations

We revisit the fundamental notion of continuity in representation theory, with special attention to the study of quantum physics. After studying the main theorem in the context of representation theory, we draw attention to the significant aspect of continuity in the analytic foundations of Wigner’s work. We conclude the paper by reviewing the connection between continuity, the possibility of defining certain local groups, and their relation to projective representations.

scholarly journals The role of Heisenberg group in Harmonic analysis: دور زمرة هايزنبرغ في التحليل التوافقي

2019 ◽  
Vol 3 (3) ◽  
Author(s):  
Soha Ali Salamah
Keyword(s):  
Hilbert Space ◽  
Fourier Transform ◽  
Representation Theory ◽  
Heisenberg Group ◽  
Harmonic Analysis ◽  
Lie Groups ◽  
Mathematical Concepts ◽  
Group Fourier Transform ◽  
The Relationship

In this paper we talk about Heisenberg group, the most know example from the lie groups. After that we discuss the representation theory of this group, and the relationship between the representation theory of the Heisenberg group and the position and momentum operatorsو and momentum operators.ors. ielationship between the representation theory of the Heisenberg group and the position and momen, that shows how we will make the connection between the Heisenberg group and physics. we have considered only the Schr dinger picture. That is, all the representations we considered are realized on the Hilbert space . we define the group Fourier transform on the Heisenberg group as an operator valued function, and other facts and properties. The main aim of our research is having the formula of Schr dinger Representation that connect physics with the Heisenberg group. Depending on this Representation we will study new formulas for some mathematical concepts such us Fourier Transform and  .

scholarly journals Representations of normed algebras with minimal left ideals

1974 ◽  
Vol 19 (2) ◽  
pp. 173-190 ◽  
Author(s):  
Bruce A. Barnes
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Representation Theory ◽  
Banach Spaces ◽  
Banach Algebras ◽  
Normed Algebra ◽  
Bounded Operators ◽  
General Representation ◽  
Finite Dimensional ◽  
Dimensional Range

The theory of *-representations of Banach *-algebras on Hilbert space is one of the most useful and most successful parts of the theory of Banach algebras. However, there are only scattered results concerning the representations of general Banach algebras on Banach spaces. It may be that a comprehensive representation theory is impossible. Nevertheless, for some special algebras interesting and worthwhile results can be proved. This is true for (Y), the algebra of all bounded operators on a Banach space Y, and for (Y), the subalgebra of (Y) consisting of operators with finite dimensional range. The representations of (Y) are studied in a recent paper by H. Porta and E. Berkson (6), and in another recent paper (8), P. Chernoff determines the structure of the representations of (Y) (and also of some more general algebras of operators). In both these papers, (Y), which is the socle of the algebras under consideration, plays an important role in the theory. This suggests the possibility that a more general representation theory can be formulated in the case of a normed algebra with a nontrivial socle. This we attempt to do in this paper.

STRUCTURE OF IRREDUCIBLE SU(2) PARAFERMION MODULES DERIVED VIA THE FEIGIN-FUCHS CONSTRUCTION

1992 ◽  
Vol 07 (06) ◽  
pp. 1233-1265 ◽  
Author(s):  
PAUL A. GRIFFIN ◽  
OSCAR F. HERNÁNDEZ
Keyword(s):  
Hilbert Space ◽  
Representation Theory ◽  
Correlation Functions ◽  
Fock Space ◽  
Coulomb Gas ◽  
Vertex Operators ◽  
Chiral Algebra ◽  
Zero Modes ◽  
Highest Weight ◽  
Brst Cohomology

We show how the Feigin-Fuchs Coulomb-gas construction, with two free Gaussian bosons, can be used to derive the representation theory of the SU(2) parafermion models. We identify the generators of the chiral algebra within the bosonic Fock space and derive the chiral algebra of the finitely reducible models, which correspond to the SU(2) and SU(1, 1) parafermion algebras. We then focus on the SU(2) highest-weight modules in the remainder of the paper. Unitarity of the modules requires that the states of the parafermion theory be independent of the zero modes of two fermionic vertex operators of the bosonic theory. The expressions for the Virasoro highest weights of the models are doubly degenerate in the bosonic Fock space. We formulate the correlation functions of these operators in the parafermion Hilbert space, and in particular, the fusion rules for the Virasoro highest weights are derived in an elegant way. Finally, the irreducible parafermion characters are derived. We discuss the connection between our analysis and previous work on representation theory based on BRST cohomology.

Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group

2005 ◽  
Vol 57 (3) ◽  
pp. 598-615
Author(s):  
Keri A. Kornelson
Keyword(s):  
Hilbert Space ◽  
Representation Theory ◽  
Heisenberg Group ◽  
Differential Operators ◽  
Difference Operator ◽  
Local Solvability ◽  
Laplacian Operator ◽  
Difference Operators ◽  
3 Dimensional ◽  
Discrete Heisenberg Group

AbstractDifferential operators Dx, Dy , and Dz are formed using the action of the 3-dimensional discrete Heisenberg group G on a set S, and the operators will act on functions on S. The Laplacian operator is a difference operator with variable differences which can be associated to a unitary representation of G on the Hilbert space L2(S). Using techniques from harmonic analysis and representation theory, we show that the Laplacian operator is locally solvable.

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