Marcel de Jeu

PALEY-WIENER THEOREMS FOR THE DUNKL TRANSFORM

We conjecture a geometrical form of the Paley-Wiener theorem for the Dunkl transform and prove three instances thereof, by using a reduction to the one-dimensional even case, shift operators, and a limit transition from Opdams results for the graded Hecke algebra, respectively. These Paley- Wiener theorems are used to extend Dunkls intertwining operator to arbitrary smooth functions. Furthermore, the connection between Dunkl operators and the Cartan mo- tion group is established. It is shown how the algebra of radial parts of in- variant differential operators can be described explicitly in terms of Dunkl operators. This description implies that the generalized Bessel functions coin- cide with the spherical functions. In this context of the Cartan motion group, the restriction of Dunkls intertwining operator to the invariants can be inter- preted in terms of the Abel transform. We also show that, for certain values of the multiplicities of the restricted roots, the Abel transform is essentially inverted by a differential operator. 1. Introduction and overview In recent times the study of special functions associated with root systems has developed to a considerable degree. Starting with a number of conjectures by Mac- donald, and the work of Heckman and Opdam on multivariable hypergeometric functions in the late 1980s, the development of the theory was greatly enhanced by the introduction of rational Dunkl operators by Dunkl (6). Through various intermediate steps of generalization, these operators can even be said to have ulti- mately provided crucial building blocks for Cheredniks work on double affine Hecke algebras and the q-Macdonald conjectures. Originally, before the introduction of Dunkl operators, the idea when studying special functions related to root systems was to consider root multiplicities in the theory of spherical functions on Lie groups as parameters, and then to develop a theory for Weyl group invariant objects in this more general situation, without the aid of the presence of the group. It was this point of view which underlay the Macdonald conjectures and which led Heckman and Opdam to the development of their theory of hypergeometric functions in higher dimension. One of the main technical problems in this context is the description of the generalized radial parts of invariant differential operators. Apart from an explicit formula for the generalized radial part of the Laplacian—an expression which was in fact the starting point for Heckman and Opdam—the other operators remain somewhat intangible.

Real Paley-Wiener theorems and local spectral radius formulas

We systematically develop real Paley-Wiener theory for the Fourier transform on ℝ d for Schwartz functions, L p -functions and distributions, in an elementary treatment based on the inversion theorem. As an application, we show how versions of classical Paley-Wiener theorems can be derived from the real ones via an approach which does not involve domain shifting and which may be put to good use for other transforms of Fourier type as well. An explanation is also given as to why the easily applied classical Paley-Wiener theorems are unlikely to be able to yield information about the support of a function or distribution which is more precise than giving its convex hull, whereas real Paley—Wiener theorems can be used to reconstruct the support precisely, albeit at the cost of combinatorial complexity. We indicate a possible application of real Paley-Wiener theory to partial differential equations in this vein, and furthermore we give evidence that a number of real Paley-Wiener results can be expected to have an interpretation as local spectral radius formulas. A comprehensive overview of the literature on real Paley― Wiener theory is included.

Elementary proofs of Paley-Wiener theorems for the Dunkl transform on the real line

We give an elementary proof of the Paley-Wiener theorem for smooth functions for the Dunkl transforms on the real line, establish a similar theorem for L^2-functions and prove identities in the spirit of Bang for L^p-functions. The proofs seem to be new also in the special case of the Fourier transform.

DYNAMICAL SYSTEMS AND COMMUTANTS IN CROSSED PRODUCTS

In this paper, we describe the commutant of an arbitrary subalgebra A of the algebra of functions on a set X in a crossed product of A with the integers, where the latter act on A by a composition automorphism defined via a bijection of X. The resulting conditions which are necessary and sufficient for A to be maximal abelian in the crossed product are subsequently applied to situations where these conditions can be shown to be equivalent to a condition in topological dynamics. As a further step, using the Gelfand transform, we obtain for a commutative completely regular semi-simple Banach algebra a topological dynamical condition on its character space which is equivalent to the algebra being maximal abelian in a crossed product with the integers.

Connections Between Dynamical Systems and Crossed Products of Banach Algebras by ℤ

Starting with a complex commutative semi-simple regular Banach algebra A and an automorphism σ of A, we form the crossed product of A by the integers, where the latter act on A via iterations of σ. The automorphism induces a topological dynamical system on the character space Δ(A) of A in a natural way. We prove an equivalence between the property that every nonzero ideal in the crossed product has non-zero intersection with the subalgebra A, maximal commutativity of A in the crossed product, and density of the non-periodic points of the induced system on the character space. We also prove that every non-trivial ideal in the crossed product always intersects the commutant of A non-trivially. Furthermore, under the assumption that A is unital and such that Δ(A) consists of infinitely many points, we show equivalence between simplicity of the crossed product and minimality of the induced system, and between primeness of the crossed product and topological transitivity of the system.

Algebraic curves for commuting elements in the

In this paper we extend the eliminant construction of Burchnall and Chaundy for commuting differential operators in the Heisenberg algebra to the q-deformed Heisenberg algebra and show that it again provides annihilating curves for commuting elements, provided q satisfies a natural condition. As a side result we obtain estimates on the dimensions of the eigenspaces of elements of this algebra in its faithful module of Laurent series.

q

We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michaels Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; a strong version of the usual Open Mapping Theorem is then a special case. As another consequence, an improved version of the analogue of Andos Theorem for an ordered Banach space is obtained for a Banach space that is, more generally than in Andos Theorem, a sum of possibly uncountably many closed not necessarily proper cones. Applications are given for a (pre)-ordered Banach space and for various spaces of continuous functions taking values in such a Banach space or, more generally, taking values in an arbitrary Banach space that is a finite sum of closed not necessarily proper cones.

-deformed Heisenberg algebra

We argue that a proof of the geometrical form of the Paley–Wiener theorems for the Dunkl transform in the literature is not correct.

A strong open mapping theorem for surjections from cones onto Banach spaces

Abstract If X is a compact Hausdorff space and σ is a homeomorphism of X, then a Banach algebra l 1 ( Σ ) of crossed product type is naturally associated with this topological dynamical system Σ = ( X , σ ) . If X consists of one point, then l 1 ( Σ ) is the group algebra of the integers. We study the algebraically irreducible representations of l 1 ( Σ ) on complex vector spaces, its primitive ideals, and its structure space. The finite dimensional algebraically irreducible representations are determined up to algebraic equivalence, and a sufficiently rich family of infinite dimensional algebraically irreducible representations is constructed to be able to conclude that l 1 ( Σ ) is semisimple. All primitive ideals of l 1 ( Σ ) are selfadjoint, and l 1 ( Σ ) is Hermitian if there are only periodic points in X. If X is metrizable or all points are periodic, then all primitive ideals arise as in our construction. A part of the structure space of l 1 ( Σ ) is conditionally shown to be homeomorphic to the product of a space of finite orbits and T . If X is a finite set, then the structure space is the topological disjoint union of a number of tori, one for each orbit in X. If all points of X have the same finite period, then it is the product of the orbit space X / Z and T . For rational rotations of T , this implies that the structure space is homeomorphic to T 2 .

Some remarks on a proof of geometrical Paley–Wiener theorems for the Dunkl transform

In this paper, which is part of a study of positive representations of locally compact groups in Banach lattices, we initiate the theory of positive representations of finite groups in Riesz spaces. If such a representation has only the zero subspace and possibly the space itself as invariant principal bands, then the space is Archimedean and finite-dimensional. Various notions of irreducibility of a positive representation are introduced and, for a finite group acting positively in a space with sufficiently many projections, these are shown to be equal. We describe the finite-dimensional positive Archimedean representations of a finite group and establish that, up to order equivalence, these are order direct sums, with unique multiplicities, of the order indecomposable positive representations naturally associated with transitive G-spaces. Character theory is shown to break down for positive representations. Induction and systems of imprimitivity are introduced in an ordered context, where the multiplicity formulation of Frobenius reciprocity turns out not to hold.