We characterize dual spaces and compute hyperdimensions of irreducible representations for two classes of compact hypergroups namely conjugacy classes of compact groups and compact hypergroups constructed by joining compact and finite hypergroups. Also, studying the representation theory of finite hypergroups, we highlight some interesting differences and similarities between the representation theories of finite hypergroups and finite groups. Finally, we compute the Heisenberg inequality for compact hypergroups.

In this work we introduce a class of discrete groups containing subgroups of abstract translations and dilations, respectively. A variety of wavelet systems can appear as $\pi \left( \Gamma \right)\psi $, where $\pi $ is a unitary representation of a wavelet group and $\Gamma $ is the abstract pseudo-lattice $\Gamma $. We prove a sufficent condition in order that a Parseval frame $\pi \left( \Gamma \right)\psi $ can be dilated to an orthonormal basis of the form $\tau \left( \Gamma \right)\Psi $, where $\tau $ is a super-representation of $\pi $. For a subclass of groups that includes the case where the translation subgroup is Heisenberg, we show that this condition always holds, and we cite familiar examples as applications.

Let G and T be topological groups, α : T → Aut(G) a homomorphism defining a continuous action of T on G and G♯ := G ⋊αT the corresponding semidirect product group. In this paper, we address several issues concerning irreducible continuous unitary representations (π♯, ${\mathcal{H}}$) of G♯ whose restriction to G remains irreducible. First, we prove that, for T = ${\mathbb R}$, this is the case for any irreducible positive energy representation of G♯, i.e. for which the one-parameter group Ut := π♯(1,t) has non-negative spectrum. The passage from irreducible unitary representations of G to representations of G♯ requires that certain projective unitary representations are continuous. To facilitate this verification, we derive various effective criteria for the continuity of projective unitary representations. Based on results of Borchers for W*-dynamical systems, we also derive a characterization of the continuous positive definite functions on G that extend to G♯.

Inspired by Aldous' conjecture for the spectral gap of the interchange process and its recent resolution by Caputo, Liggett, and Richthammer, we define an associated order $\prec $ on the irreducible representations of ${{S}_{n}}$. Aldous' conjecture is equivalent to certain representations being comparable in this order, and hence determining the “Aldous order” completely is a generalized question. We show a few additional entries for this order.

A theory of higher rank multiresolution analysis is given in the setting of abelian multiscalings. This theory enables the construction, from a higher rank MRA, of finite wavelet sets whose multidilations have translates forming an orthonormal basis in ${{L}^{2}}({{\mathbb{R}}^{d}})$. While tensor products of uniscaled MRAs provide simple examples we construct many nonseparable higher rank wavelets. In particular we construct Latin square wavelets as rank 2 variants of Haar wavelets. Also we construct nonseparable scaling functions for rank 2 variants of Meyer wavelet scaling functions, and we construct the associated nonseparable wavelets with compactly supported Fourier transforms. On the other hand we show that compactly supported scaling functions for biscaled MRAs are necessarily separable.

We introduce and study several notions of amenability for unitary corepresentations and $*$-representations of algebraic quantum groups, which may be used to characterize amenability and co-amenability for such quantum groups. As a background for this study, we investigate the associated tensor ${{C}^{*}}$-categories.

Pointwise bounds for characters of representations of the compact, connected, simple, exceptional Life groups are obtained. It is a classical result that if μ is a central, continuous measure on such a group, then μdimG is absolutely continuous. Our estimates on the size of characters allow us to prove that the exponent, dimension of G, can be replaced by approximately the rank of G. Similar results were obtained earlier for the classical, compact Lie groups.

Pointwise bounds for characters of representations of the classical, compact, connected, simple Lie groups are obtained with which allow us to study the singularity of central measures. For example, we find the minimal integer k such that any continuous orbital measure convolved with itself k times belongs to L2. We also prove that if k = rank G then μ 2k ∈ L1 for all central, continuous measures μ. This improves upon the known classical result which required the exponent to be dimension of the group G.

Let (ℋ, G, U) be a continuous representation of the Lie group G by bounded operators g ↦ U(g) on the Banach space ℋ and let (ℋ, g, dU) denote the representation of the Lie algebra g obtained by differentiation. If a1,…, ad′ is a Lie algebra basis of g and Ai = dU(ai) then we examine elliptic regularity properties of the subelliptic operators where (cij) is a real-valued strictly positive-definite matrix and c0, c1,…, cd′ ∈ C. We first introduce a family of Lipschitz subspaces ℋγ, γ > 0, of ℋ which interpolate between the Cn-subspaces of the representation and for which the parameter γ is a continuous measure of differentiability. Secondly, we give a variety of characterizations of the spaces in terms of the semigroup generated by the closure of H and the group representation. Thirdly, for sufficiently large values of Re c0 the fractional powers of the closure of H are defined, and we prove that D()γ⊆γ′, for γ′ < 2γ/r where r is the rank of the basis. Further we establish that 2γ/r is the optimal regularity value and it is attained for unitary representations or for the representations obtained by restricting U to ℋγ. Many other regularity properties are obtained.

Let Sbe a topological semigroup, K a compact convex subset of a separated convex space Eand T: S x K → K an affine action (denoted by (s, x) → Ts(x),s ∈ S, x ∈ K) of S as continuous affine maps on K. It is shown in A. Lau and J. Wong [22] that the weakly left uniformly measurable functions WLUM(S) on S has a left invariant mean iff Shas the fixed point property for weakly measurable affine actions, i.e. affine actions such that the scalar function s → x*Ts(x) is measurable for each x ∈ K and x*∈ E* (the dual of E) with respect to the Borel sets in S. It is natural to ask for a “strongly” measurable analogue of this result. There are a number of ways to define such actions and the corresponding functions on S. In this paper, we obtained a neat analogue of this fixed point theorem by a suitable choice of strong measurability which naturally leads to another new fixed point theorem for separable actions. Also, we shall unify these and many known fixed point theorems and extend and generalise them to anti-actions of S as bounded linear operators on Banach spaces