Convolution Properties of Orlicz Spaces on hypergroups

In this paper, for a locally compact commutative hypergroup $K$ and for a pair $(\Phi_1, \Phi_2)$ of Young functions satisfying sequence condition, we give a necessary condition in terms of aperiodic elements of the center of $K,$ for the convolution $f\ast g$ to exist a.e., where $f$ and $g$ are arbitrary elements of Orlicz spaces $L^{\Phi_1}(K)$ and $L^{\Phi_2}(K)$, respectively. As an application, we present some equivalent conditions for compactness of a compactly generated locally compact abelian group. Moreover, we also characterize compact convolution operators from $L^1_w(K)$ into $L^\Phi_w(K)$ for a weight $w$ on a locally compact hypergroup $K$.Comment: 13 pages. To appear in Proc. Amer. Math. So

Power boundedness in Fourier and Fourier-Stieltjes algebra of an ultraspherical hypergroup

Let $H$ be an ultraspherical hypergroup associated with a locally compact group $G$ and a spherical projector $\pi$ and let $A(H)$ and $B(H)$ denote the Fourier and Fourier-Stieltjes algebras, respectively, associated with $H.$ In this note, we study power boundedness and Ces\`aro boundedness in $B(H)$. We also characterize the power bounded property for both $A(H)$ and $B(H).

Compact Hypergroups from Discrete Subfactors

Conformal inclusions of chiral conformal field theories, or more generally inclusions of quantum field theories, are described in the von Neumann algebraic setting by nets of subfactors, possibly with infinite Jones index if one takes non-rational theories into account. With this situation in mind, we study in a purely subfactor theoretical context a certain class of braided discrete subfactors with an additional commutativity constraint, that we call locality, and which corresponds to the commutation relations between field operators at space-like distance in quantum field theory. Examples of subfactors of this type come from taking a minimal action of a compact group on a factor and considering the fixed point subalgebra. We show that to every irreducible local discrete subfactor $\mathcal{N}\subset\mathcal{M}$ of type ${I\!I\!I}$ there is an associated canonical compact hypergroup (an invariant for the subfactor) which acts on $\mathcal{M}$ by unital completely positive (ucp) maps and which gives $\mathcal{N}$ as fixed points. To show this, we establish a duality pairing between the set of all $\mathcal{N}$-bimodular ucp maps on $\mathcal{M}$ and a certain commutative unital $C^*$-algebra, whose spectrum we identify with the compact hypergroup. If the subfactor has depth 2, the compact hypergroup turns out to be a compact group. This rules out the occurrence of compact \emph{quantum} groups acting as global gauge symmetries in local conformal field theory.Comment: 58 page

Weighted hypergroups and some questions in abstract harmonic analysis

Weighted group algebras have been studied extensively in Abstract Harmonic Analysis.Complete characterizations have been found for some important properties of weighted group algebras, namely, amenability and Arens regularity. Also studies on some other features of these algebras, say weak amenability and isomorphism to operator algebras, have attracted attention. Hypergroups are generalized versions of locally compact groups. When a discrete group has all its conjugacy classes finite, the set of all conjugacy classes forms a discrete commutative hypergroup. Also the set of equivalence classes of irreducible unitary representations of a compact group forms a discrete commutative hypergroup. Other examples of discrete commutative hypergroups come from families of orthogonal polynomials. The center of the group algebra of a discrete finite conjugacy (FC) group can be identified with a hypergroup algebra. For a specific class of discrete FC groups, the restricted direct products of finite groups (RDPF), we study some properties of the center of the group algebra including amenability, maximal ideal space, and existence of a bounded approximate identity of maximal ideals. One of the generalizations of weighted group algebras which may be considered is weighted hypergroup algebras. Defining weighted hypergroups, analogous to weighted groups, we study a variety of examples, features and applications of weighted hypergroup algebras. We investigate some properties of these algebras including: dual Banach algebra structure, Arens regularity, and isomorphism with operator algebras. We define and study Folner type conditions for hypergroups. We study the relation of the Folner type conditions with other amenability properties of hypergroups. We also demonstrate some results obtained from the Leptin condition for Fourier algebras of certain hypergroups. Highlighting these tools, we specially study the Leptin condition on duals of compact groups for some specific compact groups. An application is given to Segal algebras on compact groups

Subfactors and Applications

The theory of subfactors connects diverse topics in mathematics and mathematical physics such as tensor categories, vertex operator algebras, quantum groups, quantum topology, free probability, quantum field theory, conformal field theory, statistical mechanics, condensed matter physics and, of course, operator algebras. We invited an international group of researchers from these areas and many fruitful interactions took place during the workshop