scholarly journals The Plancherel formula for the horocycle space and generalizations

1996 ◽  
Vol 61 (1) ◽  
pp. 42-56 ◽  
Author(s):  
Ronald L. Lipsman
Keyword(s):  
Harmonic Analysis ◽  
Regular Representation ◽  
Plancherel Formula ◽  
Intertwining Operators ◽  
Direct Integral ◽  
Regular Representations ◽  
Spectral Distributions ◽  
Theoretic Technique ◽  
Direct Integral Decomposition ◽  
Integral Decomposition

AbstractThe Plancherel formula for the horocycle space, and several generalizations, is derived within the framework of quasi-regular representations which have monomial spectrum. The proof uses only machinery from the Penney-Fujiwara distribution-theoretic technique; no special semisimple harmonic analysis is needed. The Plancherel formulas obtained include the spectral distributions and the intertwining operators that effect the direct integral decomposition of the quasi-regular representation.

scholarly journals FULL REGULARITY FOR A C*-ALGEBRA OF THE CANONICAL COMMUTATION RELATIONS

2009 ◽  
Vol 21 (05) ◽  
pp. 587-613 ◽  
Author(s):  
HENDRIK GRUNDLING ◽  
KARL-HERMANN NEEB
Keyword(s):  
Regular Representation ◽  
Symplectic Space ◽  
Direct Integral ◽  
Canonical Commutation Relations ◽  
C Algebra ◽  
Commutation Relations ◽  
C Algebras ◽  
Regular Representations ◽  
Direct Integral Decomposition ◽  
Integral Decomposition

The Weyl algebra — the usual C*-algebra employed to model the canonical commutation relations (CCRs), has a well-known defect, in that it has a large number of representations which are not regular and these cannot model physical fields. Here, we construct explicitly a C*-algebra which can reproduce the CCRs of a countably dimensional symplectic space (S, B) and such that its representation set is exactly the full set of regular representations of the CCRs. This construction uses Blackadar's version of infinite tensor products of nonunital C*-algebras, and it produces a "host algebra" (i.e. a generalized group algebra, explained below) for the σ-representation theory of the Abelian group S where σ(·,·) ≔ eiB(·,·)/2. As an easy application, it then follows that for every regular representation of [Formula: see text] on a separable Hilbert space, there is a direct integral decomposition of it into irreducible regular representations (a known result).

scholarly journals The Plancherel formula for the horocycle spaces and generalizations, II

1998 ◽  
Vol 65 (2) ◽  
pp. 194-211
Author(s):  
Ronald L. Lipsman
Keyword(s):  
Homogeneous Spaces ◽  
Regular Representation ◽  
Plancherel Formula ◽  
New Paradigm ◽  
Direct Integral ◽  
Stability Group ◽  
Nuclear Operators ◽  
Infinite Multiplicity ◽  
Direct Integral Decomposition ◽  
Integral Decomposition

AbstractThe Plancherel formula for various semisimple homogeneous spaces with non-reductive stability group is derived within the framework of the Bonnet Plancherel formula for the direct integral decomposition of a quasi-regular representation. These formulas represent a continuation of the author's program to establish a new paradigm for concrete Plancherel analysis on homogeneous spaces wherein the distinction between finite and infinite multiplicity is de-emphasized. One interesting feature of the paper is the computation of the Bonnet nuclear operators corresponding to certain exponential representations (roughly those induced from infinite-dimensional representations of a subgroup). Another feature is a natural realization of the direct integral decomposition over a canonical set of concrete irreducible representations, rather than over the unitary dual.

The concept of a filter

1967 ◽  
Vol 63 (1) ◽  
pp. 221-227 ◽  
Author(s):  
E. J. Hannan
Keyword(s):  
Linear Operator ◽  
Symmetric Space ◽  
Random Process ◽  
Response Function ◽  
Second Order ◽  
Closed Linear Operator ◽  
Direct Integral ◽  
Group Of Isometries ◽  
Direct Integral Decomposition ◽  
Integral Decomposition

AbstractIt is proved that for a second-order, homogeneous, random process on a globally symmetric space a filter, that is a closed linear operator which is invariant under a group of isometries of the space, may be fully described through a response function, that is that it has a direct integral decomposition into components which are scalar multiples of the identity.

The Asymptotic Ratio Set and Direct Integral Decompositions of a Von Neumann Algebra

1971 ◽  
Vol 23 (4) ◽  
pp. 598-607 ◽  
Author(s):  
Ole A. Nielsen
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Direct Integral ◽  
Von Neumann ◽  
Ratio Set ◽  
Almost All ◽  
Direct Integral Decomposition ◽  
Integral Decomposition ◽  
Remarkable Progress

The fact that any von Neumann algebra on a separable Hilbert space has an essentially unique direct integral decomposition into factors means that there is a global as well as a local aspect to any partial classification of von Neumann algebras. More precisely, suppose that J is a statement about von Neumann algebras which is either true or false for any given von Neumann algebra. Then a von Neumann algebra is said to satisfy J globally if it satisfies J, and to satsify J locally if almost all the factors appearing in some (and hence in any) central decomposition of it satisfy J . In a recent paper [3], H. Araki and E. J. Woods introduced the notion of the asymptotic ratio set of a factor, and by means of this they made remarkable progress in the classification of factors.

Bloch wave homogenisation of quasiperiodic media

2020 ◽  
pp. 1-21
Author(s):  
SISTA SIVAJI GANESH ◽  
VIVEK TEWARY
Keyword(s):  
Bloch Wave ◽  
Periodic Operator ◽  
Almost Periodic ◽  
Periodic Media ◽  
Direct Integral ◽  
Bloch Waves ◽  
Direct Integral Decomposition ◽  
Quasiperiodic Media ◽  
Integral Decomposition ◽  
Periodic Operators

Quasiperiodic media is a class of almost periodic media which is generated from periodic media through a ‘cut and project’ procedure. Quasiperiodic media displays some extraordinary optical, electronic and conductivity properties which call for the development of methods to analyse their microstructures and effective behaviour. In this paper, we develop the method of Bloch wave homogenisation for quasiperiodic media. Bloch waves are typically defined through a direct integral decomposition of periodic operators. A suitable direct integral decomposition is not available for almost periodic operators. To remedy this, we lift a quasiperiodic operator to a degenerate periodic operator in higher dimensions. Approximate Bloch waves are obtained for a regularised version of the degenerate operator. Homogenised coefficients for quasiperiodic media are obtained from the first Bloch eigenvalue of the regularised operator in the limit of regularisation parameter going to zero. A notion of quasiperiodic Bloch transform is defined and employed to obtain homogenisation limit for an equation with highly oscillating quasiperiodic coefficients.

Restricting Representations of Completely Solvable Lie Groups

1990 ◽  
Vol 42 (5) ◽  
pp. 790-824 ◽  
Author(s):  
R. L. Lipsman
Keyword(s):  
Lie Groups ◽  
Dual Problem ◽  
Nilpotent Groups ◽  
Solvable Lie Groups ◽  
Direct Integral ◽  
Induced Representations ◽  
Simply Connected ◽  
Direct Integral Decomposition ◽  
Integral Decomposition ◽  
Restriction Problem

We are concerned here with the problem of describing the direct integral decomposition of a unitary representation obtained by restriction from a larger group. This is the dual problem to the more commonly investigated problem of decomposing induced representations. In this paper we work in the context of completely solvable Lie groups—more general than nilpotent, but less general than exponential solvable. Moreover, the groups involved are simply connected. The restriction problem was considered originally in [2] and in [6] for nilpotent groups.

scholarly journals THE CONTINUOUS SPECTRUM IN DISCRETE SERIES BRANCHING LAWS

2013 ◽  
Vol 24 (07) ◽  
pp. 1350049 ◽  
Author(s):  
BENJAMIN HARRIS ◽  
HONGYU HE ◽  
GESTUR ÓLAFSSON
Keyword(s):  
Continuous Spectrum ◽  
Lie Group ◽  
Discrete Series ◽  
Series Representation ◽  
Direct Integral ◽  
Branching Laws ◽  
Reductive Lie Group ◽  
A New Technique ◽  
Direct Integral Decomposition ◽  
Integral Decomposition

If G is a reductive Lie group of Harish-Chandra class, H is a symmetric subgroup, and π is a discrete series representation of G, the authors give a condition on the pair (G, H) which guarantees that the direct integral decomposition of π|H contains each irreducible representation of H with finite multiplicity. In addition, if G is a reductive Lie group of Harish-Chandra class, and H ⊂ G is a closed, reductive subgroup of Harish-Chandra class, the authors show that the multiplicity function in the direct integral decomposition of π|H is constant along "continuous parameters". In obtaining these results, the authors develop a new technique for studying multiplicities in the restriction π|H via convolution with Harish-Chandra characters. This technique has the advantage of being useful for studying the continuous spectrum as well as the discrete spectrum.

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