Induced Representations of Locally Compact Groups I

Authors

George W. Mackey

Type: Article

Publication Date: 1952-01-01

Citations: 776

DOI: https://doi.org/10.2307/1969423

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Abstract

In the theory of representations of a finite group by linear transformations the closely related notions of imprimitivity and of play a prominent role. In [18] the author has generalized these notions to the case in which the group is a separable locally compact topological group and the linear transformations are unitary transformations in Hilbert space. It turns out (and this is the principle theorem of [18]) that the classical theorem of Frobenius according to which every imprimitive representation of a finite group is in a certain canonical fashion by a representation of a subgroup may be reformulated so as to remain true under the more general circumstances indicated above. This connection between representations of groups and representations of their subgroups has many interesting and useful properties in the finite case and it naturally occurs to one to study the extent to which these properties persist in general. The present paper is the first of a projected series in which it is planned to investigate this question in a systematic manner. Our principal results, formulated as Theorems 12.1, 12.2 and 13.1, are closely related; each being essentially a corollary of its predecessor. The first asserts that if L is a representation of the closed subgroup G1 of (M and UL is the corresponding induced representation of (M then the restriction of UL to the closed subgroup G2 is a sum over the G1: G2 double cosets of certain induced representations o G 2 . The second gives a similar decomposition of the Kronecker product UL f? UM where L and M are representations of G1 and G2 respectively. The third provides a usable formula for computing the strong intertwining numbers of the induced representations UL and UM of (M. The in question are ordinary discrete sums only when there are at most countably many non trivial double cosets. In general direct integrals as defined by von Neumann in [26] must be used and we must restrict ourselves to the case in which the relevant double coset decomposition of (M is measurable. As we have shown in detail elsewhere [20] these theorems for finite groups imply certain classical results; in particular the Frobenius reciprocity theorem and the Shoda criteria for the irreducibility and unitary equivalence of monomial representations. Our theorems yield generalizations of these results but these generalizations may be regarded as satisfactory only insofar as they deal with representations whose irreducible constituents are discrete and finite dimensional. More far reaching generalizations must be sought in other directions. We

Topics

Locations

Annals of Mathematics
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Induced Representations of Locally Compact Groups II. The Frobenius Reciprocity Theorem

1953-09-01

George W. Mackey

In the study of the relationship between the representation theory of a group and those of its various subgroups an important role is played by Frobenius's notion of induced representation. … In the study of the relationship between the representation theory of a group and those of its various subgroups an important role is played by Frobenius's notion of induced representation. To every representation L of the subgroup G of the finite group 65 there is assigned a well defined representation UL of ( called the representation of M induced by L. In I of this series [10] the author began a systematic study of a generalization of this notion in which (5 is a separable locally compact topological group and the spaces of L and UL are possibly infinite dimensional Hilbert spaces. In particular [10] contains a generalization of the Frobenius reciprocity theorem; that is the theorem asserting that UL contains the irreducible representation M of (D just as many times as the restriction of M to G contains the irreducible representation L of G. The generalization contained in [10] is unsatisfactory in that it deals only with the discrete finite dimensional irreducible components of the representations concerned and becomes vacuous when these representations decompose in a continuous fashion or have no finite dimensional irreducible components. A more satisfactory generalization has been obtained by Mautner [13, 14]. It deals in a consequent fashion with continuously decomposable representations of 5 but is restricted by the requirement that G be compact. This means in particular that only discretely decomposable representations of G need be dealt with. The principal result of the present article is a generalization of the Frobenius reciprocity theorem which deals effectively with continuously decomposable representations of both M and G. Since in compensation for the hypothesis that G be compact, we require that both G and 5 have regular representations which are of type I, our theorem does not quite include that of Mautner. However we show in addition that whenever the regular representation of G is not only of type I but also discretely decomposable then the requirement that the regular representation of (M be of type I can be eliminated. Thus our methods also yield a result which includes that of Mautner. In neither of our results is it necessary to assume that the groups concerned are unimodular. A noteworthy feature of our principal theorem is that it includes a reciprocity not only for the multiplicities but for the measures involved in the continuous decompositions as well. The basic idea in our approach is a new proof of the Frobenius reciprocity theorem in the finite case which has the advantage of generalizing significantly

A theorem on induced representations

1962-12-01

Robert J. Blattner

In [1], we proved a criterion for the disjointness of two induced representations UL and UM of a Lie group G, where L and M are finite-dimensional unitary representations of … In [1], we proved a criterion for the disjointness of two induced representations UL and UM of a Lie group G, where L and M are finite-dimensional unitary representations of compact subgroups H and K, respectively, of G. The purpose of this paper is to improve this theorem by getting a stronger conclusion, while dropping the conditions that G be a Lie group arld H be compact, and that L and M be finite-dimensional. Moreover, the restriction on K is weakened to read: G has arbitrarily small neighborhoods of the identity invariant under the adjoint action of K on G. Finally, the proof given below is fairly elementary, while the proof in [I ] is quite involved. Notations and conventions: Let Al be a topological vector space. C(G, CU) will denote the space of continuous functions from G to cA, equipped with the topology of uniform convergence on compact subsets of G while Co(G, CLL) will denote the space of those fC-C(G, CLL) with compact support. If cAl is omitted it is understood that ca=C. If cUl is another topological vector space, ?(ca, cU4) will denote the space of continuous linear maps from cA into cal equipped with the topology of bounded convergence. All integrations are with respect to right Haar measure. For any locally compact group G, G will denote its modular function. If f, g Co (G), f o g will denote the convolution of f and g, and f* is defined by f *(x) = c G(x) -f(x-1)If L and M are representations of G, R(L, M) will denote the space of intertwining operators for L and M (see [3]), while I(L, M) will denote the dimension of R(L, M). For the definition of induced representation used below, see [1]. Finally, for any function f on the group G and any xGG, fx and fx are defined by fx(y) =f(x-ly) and fx(y) =f(yx).

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Induced Representations of Locally Compact Groups

2012-11-22

Eberhard Kaniuth , Keith F. Taylor

The dual space of a locally compact group G consists of the equivalence classes of irreducible unitary representations of G. This book provides a comprehensive guide to the theory of … The dual space of a locally compact group G consists of the equivalence classes of irreducible unitary representations of G. This book provides a comprehensive guide to the theory of induced representations and explains its use in describing the dual spaces for important classes of groups. It introduces various induction constructions and proves the core theorems on induced representations, including the fundamental imprimitivity theorem of Mackey and Blattner. An extensive introduction to Mackey analysis is applied to compute dual spaces for a wide variety of examples. Fell's contributions to understanding the natural topology on the dual are also presented. In the final two chapters, the theory is applied in a variety of settings including topological Frobenius properties and continuous wavelet transforms. This book will be useful to graduate students seeking to enter the area as well as experts who need the theory of unitary group representations in their research.

Induced Representations

2014-11-05

H. Fujiwara , Jean Ludwig

Induced representations of discrete groups

1975-02-01

Lawrence Corwin

We determine necessary and sufficient conditions for a unitary representation of a discrete group induced from a finite-dimensional representation to be irreducible, and also briefly examine the question of when … We determine necessary and sufficient conditions for a unitary representation of a discrete group induced from a finite-dimensional representation to be irreducible, and also briefly examine the question of when these induced representations are Type II.

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Induced Representations of Discrete Groups

1975-02-01

Lawrence Corwin

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A Stone‐Weierstrass theorem for group representations

1978-01-01

Joe Repka

It is well known that if G is a compact group and π a faithful (unitary) representation, then each irreducible representation of G occurs in the tensor product of some … It is well known that if G is a compact group and π a faithful (unitary) representation, then each irreducible representation of G occurs in the tensor product of some number of copies of π and its contragredient. We generalize this result to a separable type I locally compact group G as follows: let π be a faithful unitary representation whose matrix coefficient functions vanish at infinity and satisfy an appropriate integrabillty condition. Then, up to isomorphism, the regular representation of G is contained in the direct sum of all tensor products of finitely many copies of π and its contragredient. We apply this result to a symplectic group and the Weil representation associated to a quadratic form. As the tensor products of such a representation are also Weil representations (associated to different forms), we see that any discrete series representation can be realized as a subrepresentation of a Weil representation.

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Representations Induced from Compact Subgroups

1966-07-01

Adam Kleppner

compute the intertwining number of these representations, anld from these results deduce necessary and sufficient conditions for irreducibility and disjointness. For the case of representations induced from compact normlal subgroups … compute the intertwining number of these representations, anld from these results deduce necessary and sufficient conditions for irreducibility and disjointness. For the case of representations induced from compact normlal subgroups we shall obtain certain other kinds of information. Because at one point it becomes necessary to consider projective represenitations and because the treatment of projective representations throughout offers essentially no more difficulty than that of ordinary representations, we shall in fact present these results for projective representations. We shall in general employ the terminology and definitions of [7], with two exceptions. What Mackey calls a r' representation we call a o- representation, and we shall always suppose the multipliers a are normalized, that is, we suppose a (x, x)-' 1, all x, which implies o (x, y)-1 = o(y-1, x-1), all x, y; see [5] for further

Induced representations and a new proof of the imprimitivity theorem

1979-03-01

Bent Ørsted

Unitary Representations of Locally Compact Groups I

1950-01-01

F. I. Mautner

Chapter 5 Induced Representations

1977-01-01

Representations of Finite Groups

2006-01-01

K. N. Srinivasa Rao

Let G be a finite group of order h. We first recall that an operator α is said to be unitary with respect to a scalar product (x, y) if … Let G be a finite group of order h. We first recall that an operator α is said to be unitary with respect to a scalar product (x, y) if it satisfies the condition $$\left( {\alpha x,\alpha y} \right) = \left( {x,y} \right){\text{for all }}x,y$$ of Eq. (3.15.17) and that a unitary operator α is represented by a unitary matrix relative to a basis e which is orthonormal with respect to the given scalar product, as observed at the end of section (3.15).

Induced Representations of Locally Compact Groups and Applications

1970-01-01

George W. Mackey

Representations of finite groups

1988-02-01

Α. I. Kostrikin , I. A. Chubarov

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Induced Representations of Linear Groups

2006-01-01

Induced Representations

1993-01-01

C. Musili

In this chapter, we shall study an important technique due to Frobenius, called “inducing representations” from subgroups, which enables one to construct representations of a group G in terms of … In this chapter, we shall study an important technique due to Frobenius, called “inducing representations” from subgroups, which enables one to construct representations of a group G in terms of those of its subgroups H. Decomposing induced representations is not an easy problem, yet it is one of the most powerful methods for constructing irreducible representations of finite groups.

Weak Containment and Induced Representations of Groups

1962-01-01

J. M. G. Fell

Let G be a locally compact group and G† its dual space, that is, the set of all unitary equivalence classes of irreducible unitary representations of G . An important … Let G be a locally compact group and G† its dual space, that is, the set of all unitary equivalence classes of irreducible unitary representations of G . An important tool for investigating the group algebra of G is the so-called hull-kernel topology of G†, which is discussed in (3) as a special case of the relation of weak containment. The question arises: Given a group G, how do we determine G† and its topology? For many groups G, Mackey's theory of induced representations permits us to catalogue all the elements of G†. One suspects that by suitably supplementing this theory it should be possible to obtain the topology of G† at the same time. It is the purpose of this paper to explore this possibility. Unfortunately, we are not able to complete the programme at present.

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Induced representations of locally compact groups

1970-01-01

Roger Rigelhof

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Induced representations of locally profinite groups

1990-10-01

Colin J. Bushnell

Unitary Representations of Locally Compact Groups II

1950-11-01

F. I. Mautner

We continue the study of unitary representations of locally compact groups begun in (3). Whereas (3) dealt to a large extent with discrete groups, the problems of the present paper … We continue the study of unitary representations of locally compact groups begun in (3). Whereas (3) dealt to a large extent with discrete groups, the problems of the present paper are mainly relevant in the case of nondiscrete groups. Let g -+ U(g) be a unitary representation U of a group G by means of unitary operators U(g) acting in a separable Hilbert space ,. Let us form the smallest weakly closed self-adjoint algebra W of bounded linear operators on ! which

Computing equivariant matrices on homogeneous spaces for geometric deep learning and automorphic Lie algebras

2024-04-01

Vincent Knibbeler

Abstract We develop an elementary method to compute spaces of equivariant maps from a homogeneous space G / H of a Lie group G to a module of this group. … Abstract We develop an elementary method to compute spaces of equivariant maps from a homogeneous space G / H of a Lie group G to a module of this group. The Lie group is not required to be compact. More generally, we study spaces of invariant sections in homogeneous vector bundles, and take a special interest in the case where the fibres are algebras. These latter cases have a natural global algebra structure. We classify these automorphic algebras for the case where the homogeneous space has compact stabilisers. This work has applications in the theoretical development of geometric deep learning and also in the theory of automorphic Lie algebras.

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Notes on unitary theta representations of compact groups

2024-04-17

Chunhui Wang

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Quantum Reference Frames, Measurement Schemes and the Type of Local Algebras in Quantum Field Theory

2024-12-18

Christopher J. Fewster , Daan W. Janssen , Leon Loveridge +2 more

Abstract We develop an operational framework, combining relativistic quantum measurement theory with quantum reference frames (QRFs), in which local measurements of a quantum field on a background with symmetries are … Abstract We develop an operational framework, combining relativistic quantum measurement theory with quantum reference frames (QRFs), in which local measurements of a quantum field on a background with symmetries are performed relative to a QRF. This yields a joint algebra of quantum-field and reference-frame observables that is invariant under the natural action of the group of spacetime isometries. For the appropriate class of quantum reference frames, this algebra is parameterised in terms of crossed products. Provided that the quantum field has good thermal properties (expressed by the existence of a KMS state at some nonzero temperature), one can use modular theory to show that the invariant algebra admits a semifinite trace. If furthermore the quantum reference frame has good thermal behaviour (expressed in terms of the properties of a KMS weight) at the same temperature, this trace is finite. We give precise conditions for the invariant algebra of physical observables to be a type $$\text {II}_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>II</mml:mtext> <mml:mn>1</mml:mn> </mml:msub> </mml:math> factor. Our results build upon recent work of Chandrasekaran et al. (J High Energy Phys 2023(2): 1–56, 2023. arXiv:2206.10780 ), providing both a significant mathematical generalisation of these findings and a refined operational understanding of their model.

Characters of connected lie groups

1974-01-01

L. Pukanszky

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Probing Wigner rotations for any group

2018-03-17

Blagoje Oblak

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Mackey formula for bisets over groupoids

2018-05-31

Laiachi El Kaoutit , Leonardo Spinosa

In this paper, we establish the Mackey formula for groupoids, extending the well-known formula in abstract groups context. This formula involves the notion of groupoid-biset, its orbit set and the … In this paper, we establish the Mackey formula for groupoids, extending the well-known formula in abstract groups context. This formula involves the notion of groupoid-biset, its orbit set and the tensor product over groupoids, as well as cosets by subgroupoids.

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The unitary extension principle for locally compact abelian groups with co-compact subgroups

2020-09-16

Ole Christensen , Say Song Goh

The unitary extension principle by Ron and Shen is one of the cornerstones of wavelet frame theory; it leads to tight frames for<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared left-parenthesis double-struck … The unitary extension principle by Ron and Shen is one of the cornerstones of wavelet frame theory; it leads to tight frames for<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared left-parenthesis double-struck upper R right-parenthesis"><mml:semantics><mml:mrow><mml:msup><mml:mi>L</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">L^{2}(\mathbb {R})</mml:annotation></mml:semantics></mml:math></inline-formula>and associated expansions of functions<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper L squared left-parenthesis double-struck upper R right-parenthesis"><mml:semantics><mml:mrow><mml:mi>f</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mi>L</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">f\in L^{2}(\mathbb {R})</mml:annotation></mml:semantics></mml:math></inline-formula>of similar type as those for orthonormal wavelet bases. In this paper, the unitary extension principle is extended to the setting of a locally compact abelian group, equipped with a collection of nested co-compact subgroups. Unlike all previously known generalizations of the unitary extension principle, the current one is taking place within the setting of continuous frames, which means that the resulting decompositions of functions in the underlying Hilbert space in general are given in terms of integral representations rather than discrete sums. The frame elements themselves appear by letting a collection of modulation operators act on a countable family of basic functions.

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Index

2020-04-14

Didier Arnal , Bradley Currey

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

On the local theta representation

2014-01-01

Chunhui Wang

We study the algebraic framework in which one can define, in the manner of the theta correspondence, a correspondence between representations of two locally profinite groups $H_1$, $H_2$. In particular, … We study the algebraic framework in which one can define, in the manner of the theta correspondence, a correspondence between representations of two locally profinite groups $H_1$, $H_2$. In particular, we examine when and how such a correspondence can be extended to bigger groups $G_1$, $G_2$ containing $H_1$, $H_2$ respectively as normal subgroups. As an application, we discuss the theta correspondence for a reductive dual pair of the similitude groups in the non-archimedean case.

UNITARY REPRESENTATIONS OF NILPOTENT LIE GROUPS

1962-08-31

A. A. Kirillov

CONTENTS Introduction § 1. Induced representations § 2. Representations of Lie algebras and infinitesimal group rings § 3. A special nilpotent group N § 4. Nilpotent Lie groups with one-dimensional … CONTENTS Introduction § 1. Induced representations § 2. Representations of Lie algebras and infinitesimal group rings § 3. A special nilpotent group N § 4. Nilpotent Lie groups with one-dimensional centre § 5. Description of the representations of nilpotent Lie groups § 6. Orbits and representations § 7. Representations of the group ring § 8. Some generalizations and unsolved problems § 9. Examples References

A duality for symmetric spaces with applications to group representations

1970-08-01

Sigurđur Helgason

On Induced Representations of Discrete Groups

1993-05-01

Michael W. Binder

The paper deals with induced representations $\operatorname {ind} _H^G\sigma$ of a locally compact group $G$ where $H$ is an open subgroup. Using "elementary intertwining operators", we first describe the commutant … The paper deals with induced representations $\operatorname {ind} _H^G\sigma$ of a locally compact group $G$ where $H$ is an open subgroup. Using "elementary intertwining operators", we first describe the commutant $\operatorname {ind} _H^G\sigma (G)â$ (also in the case of realizing the induced representations with positive definite measures). Then criteria for irreducibility and pairwise disjointness of induced representations are given. Finally, special attention is devoted to abelian subgroups $H$.

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Inclusions and Noninclusion of Spaces of Convolution Operators

1976-07-01

Michael Cowling , John J. F. Fournier

Let G be an infinite, locally compact group. Denote the space of convolution operators, on G, of strong type $(p,q)$ by $L_p^q(G)$. It is shown that, if $|1/q - 1/2| … Let G be an infinite, locally compact group. Denote the space of convolution operators, on G, of strong type $(p,q)$ by $L_p^q(G)$. It is shown that, if $|1/q - 1/2| < |1/p - 1/2|$, then $L_q^q(G)$ is not included in $L_p^p(G)$. This result follows from estimates on the norms, in these spaces, of Rudin-Shapiro measures. The same method leads to a simple example of a convolution operator that is of strong type (q, q) for all q in the interval $(p,pâ)$ but is not of restricted weak type (p, p) or of restricted weak type $(pâ,pâ)$. Other statements about noninclusion among the spaces $L_p^q(G)$ also follow from various assumptions about G. For instance, if G is unimodular, but not compact, $1 \leqslant p,q,r,s \leqslant \infty$, with $p \leqslant q$, and $\min (s,râ) < \min (q,pâ)$, then $L_p^q(G)$ is not included in $L_r^s(G)$. Using Zafranâs multilinear interpolation theorem for the real method, it is shown that, if $1 < p < 2$, then there exists a convolution operator on G that is of weak type (p, p) but not of strong type (p, p); it is not known whether such operators exist when $p > 2$, but it is shown that if $p \ne 1,2,\infty$, then there exists a convolution operator that is of restricted weak type (p, p) but is not of weak type (p, p). Many of these results also hold for the spaces of operators that commute with left translation rather than right translation. Further refinements are presented in three appendices.

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Square integrable representations of nilpotent groups

1973-01-01

Calvin C. Moore , Joseph A. Wolf

We study square integrable irreducible unitary representations (i.e. matrix coefficients are to be square integrable mod the center) of simply connected nilpotent Lie groups N, and determine which such groups … We study square integrable irreducible unitary representations (i.e. matrix coefficients are to be square integrable mod the center) of simply connected nilpotent Lie groups N, and determine which such groups have such representations. We show that if N has one such square integrable representation, then almost all (with respect to Plancherel measure) irreducible representations are square integrable. We present a simple direct formula for the formal degrees of such representations, and give also an explicit simple version of the Plancherel formula. Finally if $\Gamma$ is a discrete uniform subgroup of N we determine explicitly which square integrable representations of N occur in ${L_2}(N/\Gamma )$, and we calculate the multiplicities which turn out to be formal degrees, suitably normalized.

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SQUARE INTEGRABLE PROJECTIVE REPRESENTATIONS AND SQUARE INTEGRABLE REPRESENTATIONS MODULO A RELATIVELY CENTRAL SUBGROUP

2006-03-01

Paolo Aniello

We introduce the notion of square integrable group representation modulo a relatively central subgroup and, establishing a link with square integrable projective representations, we prove a generalization of a classical … We introduce the notion of square integrable group representation modulo a relatively central subgroup and, establishing a link with square integrable projective representations, we prove a generalization of a classical theorem of Duflo and Moore. As an example, we apply the results obtained to the Weyl–Heisenberg group.

A Notion of Rank for Unitary Representations of General Linear Groups

1990-05-01

Roberto Scaramuzzi

A notion of rank for unitary representations of general linear groups over a locally compact, nondiscrete field is defined. Rank measures how singular a representation is, when restricted to the … A notion of rank for unitary representations of general linear groups over a locally compact, nondiscrete field is defined. Rank measures how singular a representation is, when restricted to the unipotent radical of a maximal parabolic subgroup. Irreducible representations of small rank are classified. It is shown how rank determines to a large extent the asymptotic behavior of matrix coefficients of the representations.

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Weak Containment and Weak Frobenius Reciprocity

1976-01-01

Elliot C. Gootman

We study weak containment relations between unitary representations of a group $G$ and a closed normal subgroup $K$ by exploiting a property of $G$-ergodic quasi-invariant measures on the primitive ideal … We study weak containment relations between unitary representations of a group $G$ and a closed normal subgroup $K$ by exploiting a property of $G$-ergodic quasi-invariant measures on the primitive ideal space of $K$. By this means, we prove that every irreducible representation of $G$ is weakly contained in a representation induced from an irreducible representation of $K$ if the quotient group $G/K$ is amenable; and that the pair $(G,K)$ satisfies a weak Frobenius reciprocity property if and only if $G/K$ is amenable and $G$ acts minimally on the primitive ideal space of $K$. If $G/K$ is compact, $G$ acts minimally if and only if the primitive ideal space of $K$ is ${T_1}$.

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Nonamenability and Borel Paradoxical Decompositions for Locally Compact Groups

1986-01-01

Alan L. T. Paterson

We show that a locally compact group $G$ is not amenable if and only if it admits a Borel paradoxical decomposition. We show that a locally compact group $G$ is not amenable if and only if it admits a Borel paradoxical decomposition.

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The Mackey Obstruction and the Coadjoint Orbits

1994-12-01

Zongyi Li

This paper studies the Mackey obstruction representation theory at the coadjoint orbit level. It shows how to get rid of such obstructions and to get orbits of the "little groups". … This paper studies the Mackey obstruction representation theory at the coadjoint orbit level. It shows how to get rid of such obstructions and to get orbits of the "little groups". Such little group data is essential for inductive construction of coadjoint orbits of general Lie groups.

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On a Result of Baer

1962-02-01

Maxwell Rosenlicht

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Borel structure in groups and their duals

1957-01-01

George W. Mackey

Introduction. In the past decade or so much work has been done toward extending the classical theory of finite dimensional representations of compact groups a theory of (not necessarily finite … Introduction. In the past decade or so much work has been done toward extending the classical theory of finite dimensional representations of compact groups a theory of (not necessarily finite dimensional) unitary representations of locally compact groups. Among the obstacles interfering with various aspects of this program is the lack of a suitable natural topology in the object; that is in the set of equivalence classes of irreducible representations. One can introduce natural topologies but none of them seem have reasonable properties except in extremely special cases. When the group is abelian for example the dual object itself is a locally compact abelian group. This paper is based on the observation that for certain purposes one can dispense with a topology in the dual object in favor of a and that there is a wide class of groups for which this weaker structure has very regular properties. If S is any topological space one defines a Borel (or Baire) subset of S be a member of the smallest family of subsets of S which includes the open sets and is closed with respect the formationi of complements and countable unions. The structure defined in S by its Bore! sets we may call the Borel structure of S. It is weaker than the topological structure in the sense that any one-to-one transformation of S onto S which preserves the topological structure also preserves the Borel structure whereas the converse is usually false. Of course a Borel structure may be defined without any reference a topology by simply singling out an arbitrary family of sets closed with respect the formation of complements and countable unions. Giving a set of mathematical objects a topology amounts giving it a sufficiently space-like structure so that one can speak of certain of the objects being near to or far away from certain others. Giving it a Borel structure amounts distinguishing a family of well behaved or definable subsets. In using the terms

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Spectral multiplicities for ergodic flows

2013-01-01

Alexandre I. Danilenko , Mariusz Lemańczyk

Let $E$ be a subset of positive integers such that $E\cap\{1,2\}\ne\emptyset$. A weakly mixing finite measure preserving flow $T=(T_t)_{t\in\Bbb R}$ is constructed such that the set of spectral multiplicities (of … Let $E$ be a subset of positive integers such that $E\cap\{1,2\}\ne\emptyset$. A weakly mixing finite measure preserving flow $T=(T_t)_{t\in\Bbb R}$ is constructed such that the set of spectral multiplicities (of the corresponding Koopman unitary representation generated by $T$) is $E$. Moreover, for each non-zero $t\in\Bbb R$, the set of spectral multiplicities of the transformation $T_t$ is also $E$.These results are partly extended to actions of some other locally compact second countable Abelian groups.

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Masslessness in n-Dimensions

1998-04-01

E. Angelopoulos , Mourad Laoues

We determine the representations of the ``conformal'' group ${\bar{SO}}_0(2, n)$, the restriction of which on the ``Poincar\'e'' subgroup ${\bar{SO}}_0(1, n-1).T_n$ are unitary irreducible. We study their restrictions to the ``De … We determine the representations of the ``conformal'' group ${\bar{SO}}_0(2, n)$, the restriction of which on the ``Poincar\'e'' subgroup ${\bar{SO}}_0(1, n-1).T_n$ are unitary irreducible. We study their restrictions to the ``De Sitter'' subgroups ${\bar{SO}}_0(1, n)$ and ${\bar{SO}}_0(2, n-1)$ (they remain irreducible or decompose into a sum of two) and the contraction of the latter to ``Poincar\'e''. Then we discuss the notion of masslessness in $n$ dimensions and compare the situation for general $n$ with the well-known case of 4-dimensional space-time, showing the specificity of the latter.

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Co-compact Gabor Systems on Locally Compact Abelian Groups

2015-05-11

Mads S. Jakobsen , Jakob Lemvig

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On the theory of exponential groups

1967-03-01

Lajos Pukánszky

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Square-Integrability of Induced Representations of Semidirect Products

1998-04-01

Paolo Aniello , Gianni Cassinelli , Ernesto De Vito +1 more

We consider a semidirect product G=A×′H, with A abelian, and its unitary representations of the form [Formula: see text] where x 0 is in the dual group of A, G … We consider a semidirect product G=A×′H, with A abelian, and its unitary representations of the form [Formula: see text] where x 0 is in the dual group of A, G 0 is the stability group of x 0 and m is an irreducible unitary representation of G 0 ∩H. We give a new selfcontained proof of the following result: the induced representation [Formula: see text] is square-integrable if and only if the orbit G[x 0 ] has nonzero Haar measure and m is square-integrable. Moreover we give an explicit form for the formal degree of [Formula: see text].

Action of algebraic groups of automorphisms on the dual of a class of type ${\rm I}$ groups

1972-01-01

L. Pukanszky

chier doit contenir la présente mention de copyright. chier doit contenir la présente mention de copyright.

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Representations of algebraic unipotent groups over a self-dual field

1974-01-01

Eugène Gutkin

The Representations of the Poincaré Group in the Framework of Free Quantum Fields

1978-01-01

Jan T. Łopuszański

Abstract This review article arose while the author tried to get acquainted with the recent developments in the domain of the representations of the Poincaré group in the quantum field … Abstract This review article arose while the author tried to get acquainted with the recent developments in the domain of the representations of the Poincaré group in the quantum field theory. Special emphasis was put in this article on the case of massless free fields. Some ideas concerning general restrictions imposed upon the spinorial fields in order to get irreducible representations seem to be new and original.

Plancherel Inversion as Unified Approach to Wavelet Transforms and Wigner functions

2001-01-01

S. Twareque Ali , Hartmut Fuehr , Anna E. Krasowska

We demonstrate that the Plancherel transform for Type-I groups provides one with a natural, unified perspective for the generalized continuous wavelet transform, on the one hand, and for a class … We demonstrate that the Plancherel transform for Type-I groups provides one with a natural, unified perspective for the generalized continuous wavelet transform, on the one hand, and for a class of Wigner functions, on the other. The wavelet transform of a signal is an $L^2$-function on an appropriately chosen group, while the Wigner function is defined on a coadjoint orbit of the group and serves as an alternative characterization of the signal, which is often used in practical applications. The Plancherel transform maps $L^2$-functions on a group unitarily to fields of Hilbert-Schmidt operators, indexed by unitary irreducible representations of the group. The wavelet transform can essentiallly be looked upon as restricted inverse Plancherel transform, while Wigner functions are modified Fourier transforms of inverse Plancherel transforms, usually restricted to a subset of the unitary dual of the group. Some known results both on Wigner functions and wavelet transforms, appearing in the literature from very different perspectives, are naturally unified within our approach. Explicit computations on a number of groups illustrate the theory.

Regularly related lattices in Lie groups

1978-09-01

Marc A. Rieffel

Modular algebras in geometric quantization

1994-12-01

S. Twareque Ali , Amine M. El Gradechi , Gérard G. Emch

Presented here is a discussion on the connection between geometric quantization and algebraic quantization. The former procedure relies on a construction of unitary irreducible representations that starts from co-adjoint orbits … Presented here is a discussion on the connection between geometric quantization and algebraic quantization. The former procedure relies on a construction of unitary irreducible representations that starts from co-adjoint orbits and uses polarizations, while the latter depends on the purely algebraic characterization of unitary irreducible representations, which is based on central decompositions of von Neumann algebras in involutive duality, and their decompositions in terms of maximal Abelian subalgebras. Intermediate stages of these two quantization methods turn out to be complementary, leading thus to a new characterization of the so-called discrete series representations.

A theory of induction and classification of tensor C*-categories

2012-10-12

Claudia Pinzari , John E. Roberts

This paper addresses the problem of describing the structure of tensor C*-categories \mathcal{M} with conjugates and irreducible tensor unit. No assumption on the existence of a braided symmetry or on … This paper addresses the problem of describing the structure of tensor C*-categories \mathcal{M} with conjugates and irreducible tensor unit. No assumption on the existence of a braided symmetry or on amenability is made. Our assumptions are motivated by the remark that these categories often contain non-full tensor C*-subcategories with conjugates and the same objects admitting an embedding into the Hilbert spaces. Such an embedding defines a compact quantum group by Woronowicz duality. An important example is the Temperley–Lieb category canonically contained in a tensor C*-category generated by a single real or pseudoreal object of dimension ≥ 2 . The associated quantum groups are the universal orthogonal quantum groups of Wang and Van Daele. Our main result asserts that there is a full and faithful tensor functor from \mathcal{M} to a category of Hilbert bimodule representations of the compact quantum group. In the classical case, these bimodule representations reduce to the G -equivariant Hermitian bundles over compact homogeneous G -spaces, with G a compact group. Our structural results shed light on the problem of whether there is an embedding functor of \mathcal{M} into the Hilbert spaces. We show that this is related to the problem of whether a classical compact Lie group can act ergodically on a non-type I von Neumann algebra. In particular, combining this with a result of Wassermann shows that an embedding exists if \mathcal{M} is generated by a pseudoreal object of dimension 2.

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Subrepresentations of direct integrals and finite volume homogeneous spaces

1983-07-01

Elliot C. Gootman

We prove a result on representations of separable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{C^*}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebras which … We prove a result on representations of separable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{C^*}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebras which has application to, and was in fact motivated by, a problem concerning relations between unitary representations of a second countable locally compact group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and those of a closed subgroup <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G slash upper K"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">G/K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is of finite volume. The result is that if an irreducible representation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π</mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is contained in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper X Endscripts pi Subscript x Baseline d mu left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo>∫</mml:mo> <mml:mi>X</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>π</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:mi>d</mml:mi> <mml:mi>μ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\int _X {{\pi _x}} d\mu (x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi subset-of-or-equal-to pi Subscript x"> <mml:semantics> <mml:mrow> <mml:mi>π</mml:mi> <mml:mo>⊆</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>π</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi \subseteq {\pi _x}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a set of positive measure. With <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as above, it follows that for each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi element-of ModifyingAbove upper G With caret"> <mml:semantics> <mml:mrow> <mml:mi>π</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>G</mml:mi> <mml:mo stretchy="false">^</mml:mo> </mml:mover> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi \in \hat G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there exists <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma element-of ModifyingAbove upper K With caret"> <mml:semantics> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>K</mml:mi> <mml:mo stretchy="false">^</mml:mo> </mml:mover> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma \in \hat K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi subset-of-or-equal-to upper U Superscript sigma"> <mml:semantics> <mml:mrow> <mml:mi>π</mml:mi> <mml:mo>⊆</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>U</mml:mi> <mml:mi>σ</mml:mi> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi \subseteq {U^\sigma }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the induced representation. Frobenius reciprocity type results are derived as further consequences.

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The Method of Orbits for Real Lie Groups

2002-01-01

Jae-Hyun Yang

In this paper, we outline a development of the theory of orbit method for representations of real Lie groups. In particular, we study the orbit method for representations of the … In this paper, we outline a development of the theory of orbit method for representations of real Lie groups. In particular, we study the orbit method for representations of the Heisenberg group and the Jacobi group.

Bemerkungen zur unitäräquivalenz von lorentzinvarianten feldern

1961-12-01

H. Reeh , S. Schlieder

The Maxwell Group and the Quantum Theory of Particles in Classical Homogeneous Electromagnetic Fields

1972-01-01

Robert Schrader

Abstract Covariant solutions of the Dirac (and Klein‐Gordon) equation in a homogeneous classical electromagnetic field are constructed. This is done using the symmetry group of the equation, the Maxwell group. … Abstract Covariant solutions of the Dirac (and Klein‐Gordon) equation in a homogeneous classical electromagnetic field are constructed. This is done using the symmetry group of the equation, the Maxwell group. These covariant solutions are obtained starting from solutions in the frame where the electromagnetic field is described by a magnetic field pointing in the 3‐direction and then using the theory of induced representations.

The Structure of Translation-Invariant Spaces on Locally Compact Abelian Groups

2015-03-16

Marcin Bownik , Kenneth A. Ross

Projective Representations of Abelian Groups

1972-11-01

Nigel Backhouse , C. J. Bradley

Let $\omega$ be a factor system for the locally compact abelian group G. Then we show that the finite-dimensional unitary irreducible projective representations of G, having factor system $\omega$, possess … Let $\omega$ be a factor system for the locally compact abelian group G. Then we show that the finite-dimensional unitary irreducible projective representations of G, having factor system $\omega$, possess a common dimension $d(\omega )$. Using a characterisation of $d(\omega )$ as the index in G of a maximal subgroup on which $\omega$ is symmetric we derive a formula for $d(\omega )$ in the case that G is discrete and finitely generated.

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On the irreducibility of an induced representation

1979-01-01

John Quigg

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On A Theorem of Dieudonné

1949-01-01

Paul R. Halmos

Civil infrastructure will be essential to face the interlinked existential threats of climate change and rising resource demands while ensuring a livable Anthropocene for all. However, conventional infrastructure planning largely … Civil infrastructure will be essential to face the interlinked existential threats of climate change and rising resource demands while ensuring a livable Anthropocene for all. However, conventional infrastructure planning largely neglects the ...

Some properties of measurable functions

1943-01-01

Herbert Fédérer , Anthony P. Morse

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Unitary representations of the Lorentz group

1945-02-22

P. A. M. Dirac

Certain quantities are introduced which are like tensors in space-time with an infinite enumerable number of components and with an invariant positive definite quadratic form for their squared length. Some … Certain quantities are introduced which are like tensors in space-time with an infinite enumerable number of components and with an invariant positive definite quadratic form for their squared length. Some of the main properties of these quantities are dealt with, and some applications to quantum mechanics are pointed out

Imprimitivity for Representations of Locally Compact Groups I

1949-09-01

George W. Mackey

Humans become increasingly fragile as they age. We show that something similar may happen to states, although for states, the risk of termination levels off as they grow older, allowing … Humans become increasingly fragile as they age. We show that something similar may happen to states, although for states, the risk of termination levels off as they grow older, allowing some to persist for millennia. Proximate causes of their ...How states and great powers rise and fall is an intriguing enigma of human history. Are there any patterns? Do polities become more vulnerable over time as they age? We analyze longevity in hundreds of premodern states using survival analysis to help ...

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Spectral resolution of groups of unitary operators

1944-09-01

W. Ambrose

On Rings of Operators. Reduction Theory

1949-04-01

John von Neumann

On One-Parameter Unitary Groups in Hilbert Space

1932-07-01

M. H. Stone

Unitary Representations of Locally Compact Groups II

1950-11-01

F. I. Mautner

Sur la Theorie des Representations Unitaires

1951-01-01

Roger Godement

Le but de ce travail est de pr6senter quelques r6sultats nouveaux-ainsi que d'autres-relatifs A la th6orie des representations unitaires des algebres involutives. Le Chapitre I concerne la th6orie des decompositions … Le but de ce travail est de pr6senter quelques r6sultats nouveaux-ainsi que d'autres-relatifs A la th6orie des representations unitaires des algebres involutives. Le Chapitre I concerne la th6orie des decompositions spectrales, que nous exposons sur des bases apparemment nouvelles (quoique l'auteur ne se fasse aucune illusion sur l'originalit6 r6elle de ses r6sultats), et dont nous faisons par la suite un usage constant. Que notre m6thode soit plus naturelle que celles qu'on connait d6jh, c'est ce que l'avenir seul d6cidera. Le Chapitre II traite des propri6t6s 6l6mentaires des representations unitaires, d'une fagon du reste sommaire. II contient une g6n6ralisation du Th6oreme de Plancherel, ainsi qu'un r6sultat partiel sur la structure topologique de l'ensemble des formes positives '6l6mentaires. Le Chapitre III est consacr6 A la th6orie des sommes continues d'espaces de Hilbert, notion qui joue dans les travaux r6cents de I. Gelfand et M. Neumark un r6le fondamental et naturel, et qui vient de faire l'objet d'un m6moire important de J. von Neumann. Contrairement A ce qui se passe chez ce dernier, nous nous sommes efforc6 de prendre en consideration l'aspect topologique des problemes, en sorte que notre th6orie repose sur la notion de champ de vecteurs continu d'une part, et que nous ne nous restreignons pas, d'autre part, aux fonctions d6finies sur la droite. Nous pensons aussi avoir montr6 que cette th6orie est le domaine naturel de la theorie de l'intdgration, et que, loin de se construire A l'aide de celle-ci, la th6orie des sommes continues doit redonner comme simples cas particuliers les r6sultats connus relatifs aux fonctions num6riques ou vectorielles. Le Chapitre IV applique les notions pr6c6dentes A la demonstration d'un r6sultat remarquable annonc6 r6cemment par F. I. Mautner; A vrai dire, ce n'est pas exactement ce r6sultat que nous d6montrons, puisque les sommes continues utilis6es ici sont relatives A des espaces g6n6raux, et non pas seulement A la droite; mais il est parfaitement clair qu'une th6orie limit6e A la droite est inutilisable pratiquement: la meilleure preuve en est administr6e par F. I. Mautner lui-meme qui, appliquant son th6oreme aux fonctions de type positif d6finies sur un groupe non ab6lien, obtient un r6sultat qui ne se reduit pas, dans le cas ab6lien, au classique th6oreme de S. Bochner: et ce parce que, comme on 1'a d6jA dit, la th6orie de J. von Neumann n6glige completement les circonstances topologiques du probleme. En ce qui concerne l'originalit6 des r6sultats expos6s dans ces deux derniers Chapitres, elle est 6videmment douteuse-et du reste c'est la lecture de la Note