We establish two conditions equivalent to coamenability for type I locally compact quantum groups. The first condition is concerned with the spectra of certain convolution operators on the space L 2 (Irr(G)) of functions square integrable with respect to the Plancherel measure. The second condition involves spectra of character-like operators associated with direct integrals of irreducible representations. As examples we study special classes of quantum groups: classical, dual to classical, compact or given by a certain bicrossed product construction.

We provide mathematicaly rigorous justification of using term probability in connection to the so called non-signalling theories, known also as Popescu's and Rohrlich's box worlds. No only do we prove correctness of these models (in the sense that they describe composite system of two independent subsystems) but we obtain new properties of non-signalling boxes and expose new tools for further investigation. Moreover, it allows strightforward generalization to more complicated systems.

We show that the quantum disk, i.e. the quantum space corresponding to the Toeplitz C * -algebra does not admit any compact quantum group structure. We prove that if such a structure existed the resulting compact quantum group would simultaneously be of Kac type and not of Kac type. The main tools used in the solution come from the theory of type I locally compact quantum groups, but also from the theory of operators on Hilbert spaces.

We prove a number of results having to do with equipping type-I C * -algebras with compact quantum group structures, the two main ones being that such a compact quantum group is necessarily co-amenable, and that if the C * -algebra in question is an extension of a non-zero finite direct sum of elementary C * -algebras by a commutative unital C * -algebra then it must be finite-dimensional.

The center of the algebra of continuous functions on the quantum group SUq(2) is determined as well as centers of other related algebras. Several other results concerning this quantum group are given with direct proofs based on concrete realization of these algebras as algebras of operators on a Hilbert space. Dedicated toMarek Bożejko on the occasion of his 70th birthday. IntroductionThe aim of this paper is to provide very direct and relatively elementary proofs of certain facts concerning the quantum SU(2) group introduced by S.L. Woronowicz in the seminal paper [10]. The issues addressed in this paper are the following ◮ faithfulness of the representation π introduced in [10, Proof of Theorem 1.2], ◮ determining the center of the algebras Pol(SU q (2)), C(SU q (2)) and L ∞ (SU q (2)) as well as the commutant of π(C(SU q (2))), ◮ giving direct proofs of faithfulness of Haar measure and continuity of the counit. The above tasks are interrelated and the relations between them will be explained in detail.Most results of this work are taken from the first author's BSc thesis submitted at the Faculty of Physics, University of Warsaw. These results are known and in most cases proofs are published, but our approach is rather elementary and direct.The quantum groups SU q (2) were introduced in [10] and later studied in numerous papers in mathematics and theoretical physics. Apart from [10] our approach will be based on fundamental texts [9, 12] and more specialized [1,4,5,11]. Methods of functional analysis and operator algebras are covered in textbooks such as [2,6,8].The paper is organized as follows: in the next subsections we briefly introduce terminology and notation needed in the remainder of the paper. Section 2 is devoted to a detailed proof of faithfulness of a particular representation of the algebra of functions on the quantum group SU q (2) defined in [10]. In Section 3 we introduce the additional structure on the C * -algebra studied in Section 2 which defines the quantum group SU q (2). We also list some objects needed for later sections and recall the formula for the Haar measure. Section 4 provides the proof of the main result of the paper, namely that the center of the algebra of continuous functions on SU q (2) is trivial. This is achieved by examining the commutant of this algebra in the faithful representation studied earlier. These results are used in Section 5 to prove that the Haar measure of SU q (2) is faithful and its co-unit is continuous (the latter fact is justified in two different ways). Section 6 is devoted to determining the center of the von Neumann algebra generated by the image of the algebra of continuous functions on SU q (2) in the GNS representation for the Haar measure, i.e. the center of the algebra L ∞ (SU q (2)). Finally in Section 7 we sketch a way to use some of our results and some major results from the literature to obtain an alternative proof of faithfulness of π.