Imprimitivity Theorem Research Articles

Imprimitivity theorems and self-similar actions on Fell bundles

We introduce the notion of self-similar actions of groupoids on other groupoids and Fell bundles. This leads to a new imprimitivity theorem arising from such dynamics, generalizing many earlier imprimitivity theorems involving group and groupoid actions.

KRP and his imprimitivity theorem

KRP and his imprimitivity theorem

An elementary Green imprimitivity theorem for inverse semigroups

A Morita equivalence similar to that found by Green for crossed products by groups will be established for crossed products by inverse semigroups. More precisely, let S be an inverse semigroup, T a finite sub-inverse semigroup of S and A an S-algebra or a T-algebra. Then the crossed product $$A \rtimes T$$ is Morita equivalent to a certain crossed product $$B \rtimes S$$.

Weakly proper group actions, Mansfield’s Imprimitivity and twisted Landstad duality

Using the theory of weakly proper actions of locally compact groups recently developed by the authors, we give a unified proof of both reduced and maximal versions of Mansfield’s Imprimitivity Theorem and obtain a general version of Landstad’s Duality Theorem for twisted group coactions. As one application, we obtain the stabilization trick for arbitrary twisted coactions, showing that every twisted coaction is Morita equivalent to an inflated coaction.

A classification of finite quantum kinematics

Quantum mechanics in Hilbert spaces of finite dimension N is reviewed from the number theoretic point of view. For composite numbers N possible quantum kinematics are classified on the basis of Mackey's Imprimitivity Theorem for finite Abelian groups. This yields also a classification of finite Weyl-Heisenberg groups and the corresponding finite quantum kinematics. Simple number theory gets involved through the fundamental theorem describing all finite discrete Abelian groups of order N as direct products of cyclic groups, whose orders are powers of not necessarily distinct primes contained in the prime decomposition of N. The representation theoretic approach is further compared with the algebraic approach, where the basic object is the corresponding operator algebra. The consideration of fine gradings of this associative algebra then brings a fresh look on the relation between the mathematical formalism and physical realizations of finite quantum systems.

Imprimitivity theorems for weakly proper actions of locally compact groups

In a recent paper the authors introduced universal and exotic generalized fixed-point algebras for weakly proper group actions on$C^{\ast }$-algebras. Here we extend the notion of weakly proper actions to actions on Hilbert modules. As a result we obtain several imprimitivity theorems establishing important Morita equivalences between universal, reduced, or exotic crossed products and appropriate universal, reduced, or exotic fixed-point algebras, respectively. In particular, we obtain an exotic version of Green’s imprimitivity theorem and a very general version of the symmetric imprimitivity theorem by weakly proper actions of product groups$G\times H$. In addition, we study functorial properties of generalized fixed-point algebras for equivariant categories of$C^{\ast }$-algebras based on correspondences.

The Brauer semigroup of a groupoid and a symmetric imprimitivity theorem

In this paper we define a monoid called the Brauer semigroup for a locally compact Hausdorff groupoid E E whose elements consist of Morita equivalence classes of E E -dynamical systems. This construction generalizes both the equivariant Brauer semigroup for transformation groups and the Brauer group for a groupoid. We show that groupoid equivalence induces an isomorphism of Brauer semigroups and that this isomorphism preserves the Morita equivalence classes of the respective crossed products, thus generalizing Raeburn’s symmetric imprimitivity theorem.

FELL BUNDLES AND IMPRIMITIVITY THEOREMS: MANSFIELD’S AND FELL’S THEOREMS

AbstractIn the third and latest paper in this series, we recover the imprimitivity theorems of Mansfield and Fell using our technique of Fell bundles over groupoids. Also, we apply the Rieffel surjection of the first paper in the series to relate our version of Mansfield’s theorem to that of an Huef and Raeburn, and to give an automatic amenability result for certain transformation Fell bundles.

Naturality of symmetric imprimitivity theorems

The first imprimitivity theorems identified the representations of groups or dynamical systems which are induced from representations of a subgroup. Symmetric imprimitivity theorems identify pairs of crossed products by different groups which are Morita equivalent and hence have the same representation theory. Here we consider commuting actions of groupsHHandKKon aC∗C^*-algebra which are saturated and proper as defined by Rieffel in 1990. Our main result says that the resulting Morita equivalence of crossed products is natural in the sense that it is compatible with homomorphisms and induction processes.

Fell bundles and imprimitivity theorems: towards a universal generalized fixed point algebra

We apply the One-Sided Action Theorem from the first paper in this series to prove that Rieffel's Morita equivalence between the reduced crossed product by a proper saturated action and the generalized fixed-point algebra is a quotient of a Morita equivalence between the full crossed product and a universal fixed-point algebra. We give several applications, to Fell bundles over groups, reduced crossed products as fixed-point algebras, and C*-bundles.

Erratum to: Unitary Representations of Super Lie Groups and Applications to the Classification and Multiplet Structure of Super Particles

It is well known that the category of super Lie groups (SLG) is equivalent to the category of super Harish-Chandra pairs (SHCP). Using this equivalence, we define the category of unitary representations (UR's) of a super Lie group. We give an extension of the classical inducing construction and Mackey imprimitivity theorem to this setting. We use our results to classify the irreducible unitary representations of semidirect products of super translation groups by classical Lie groups, in particular of the super Poincare groups in arbitrary dimension and signature. Finally we compare our results with those in the physical literature on the structure and classification of super multiplets.

Unbounded Induced Representations of ∗-Algebras

Induced representations of *-algebras by unbounded operators in Hilbert space are investigated. Conditional expectations of a *-algebra \({{\mathcal{A}}}\) onto a unital *-subalgebra \({{\mathcal{B}}}\) are introduced and used to define inner products on the corresponding induced modules. The main part of the paper is concerned with group graded *-algebras \({{\mathcal{A}}}=\oplus_{g\in G}{{\mathcal{A}}}_g\) for which the *-subalgebra \({{\mathcal{B}}}:={{\mathcal{A}}}_e\) is commutative. Then the canonical projection \(p:{{\mathcal{A}}}\to{{\mathcal{B}}}\) is a conditional expectation and there is a partial action of the group G on the set \({{\mathcal{B}}}p\) of all characters of \({{\mathcal{B}}}\) which are nonnegative on the cone \(\sum{{\mathcal{A}}}^2{{\mathcal{A}}}p{{\mathcal{B}}}.\) The complete Mackey theory is developed for *-representations of \({{\mathcal{A}}}\) which are induced from characters of \({{\widehat{{{\mathcal{B}}}}^+}}.\) Systems of imprimitivity are defined and two versions of the Imprimitivity Theorem are proved in this context. A concept of well-behaved *-representations of such *-algebras \({{\mathcal{A}}}\) is introduced and studied. It is shown that well-behaved representations are direct sums of cyclic well-behaved representations and that induced representations of well-behaved representations are again well-behaved. The theory applies to a large variety of examples. For important examples such as the Weyl algebra, enveloping algebras of the Lie algebras su(2), su(1,1), and of the Virasoro algebra, and *-algebras generated by dynamical systems our theory is carried out in great detail.

Imprimitivity theorem for groupoid representations

Abstract We define and investigate the concept of the groupoid representation induced by a representation of the isotropy subgroupoid. Groupoids in question are locally compact transitive topological groupoids. We formulate and prove the imprimitivity theorem for such representations which is a generalization of the classical Mackey’s theorem known from the theory of group representations.

Full and reduced coactions of locally compact groups on [formula omitted]-algebras

We survey the results required to pass between full and reduced coactions of locally compact groups on C∗-algebras, which say, roughly speaking, that one can always do so without changing the crossed-product C∗-algebra. Wherever possible we use definitions and constructions that are well documented and accessible to non-experts, and otherwise we provide full details. We then give a series of applications to illustrate the use of these techniques. We obtain in particular a new version of Mansfield’s imprimitivity theorem for full coactions, and prove that it gives a natural isomorphism between crossed-product functors defined on appropriate categories.

Hecke C*-Algebras, Schlichting Completions and Morita Equivalence

AbstractThe Hecke algebra of a Hecke pair (G, H) is studied using the Schlichting completion (Ḡ, ), which is a Hecke pair whose Hecke algebra is isomorphic to and which is topologized so that is a compact open subgroup of Ḡ. In particular, the representation theory and C*-completions of are addressed in terms of the projection using both Fell's and Rieffel's imprimitivity theorems and the identity . An extended analysis of the case where H is contained in a normal subgroup of G (and in particular the case where G is a semi-direct product) is carried out, and several specific examples are analysed using this approach.

Covariant representations of Hecke algebras and imprimitivity for crossed products by homogeneous spaces

For discrete Hecke pairs ( G , H ) , we introduce a notion of covariant representation which reduces in the case where H is normal to the usual definition of covariance for the action of G / H on c 0 ( G / H ) by right translation; in many cases where G is a semidirect product, it can also be expressed in terms of covariance for a semigroup action. We use this covariance to characterise the representations of c 0 ( G / H ) which are multiples of the multiplication representation on ℓ 2 ( G / H ) , and more generally, we prove an imprimitivity theorem for regular representations of certain crossed products by coactions of homogeneous spaces. We thus obtain new criteria for extending unitary representations from H to G .

Unitary Representations of Super Lie Groups and Applications to the Classification and Multiplet Structure of Super Particles

It is well known that the category of super Lie groups (SLG) is equivalent to the category of super Harish-Chandra pairs (SHCP). Using this equivalence, we define the category of unitary representations (UR's) of a super Lie group. We give an extension of the classical inducing construction and Mackey imprimitivity theorem to this setting. We use our results to classify the irreducible unitary representations of semidirect products of super translation groups by classical Lie groups, in particular of the super Poincaré groups in arbitrary dimension and signature. Finally we compare our results with those in the physical literature on the structure and classification of super multiplets.

Covariant observables on a nonunimodular group

It is shown that the characterization of covariant positive operator measures on nonunimodular locally compact groups can be obtained by using vector measure theoretic methods, without an application of Mackey's imprimitivity theorem.

A categorical approach to imprimitivity theorems for 𝐶*-dynamical systems

Introduction Right-Hilbert bimodules The categories The functors The natural equivalences Applications Appendix A. Crossed products by actions and coactions Appendix B. The imprimitivity theorems of Green and Mansfield Appendix C. function spaces Appendix. Bibliography.

A Symmetric Imprimitivity Theorem for Commuting Proper Actions

AbstractWe prove a symmetric imprimitivity theoremfor commuting proper actions of locally compact groups H and K on a C*-algebra.