We investigate the fundamental concept of a closed quantum subgroup of a locally compact quantum group. Two definitions -one due to S. Vaes and one due to S.L. Woronowicz -are analyzed and relations between them discussed. Among many reformulations we prove that the former definition can be phrased in terms of quasi-equivalence of representations of quantum groups while the latter can be related to an old definition of Podleś from the theory of compact quantum groups. The cases of classical groups, duals of classical groups, compact and discrete quantum groups are singled out and equivalence of the two definitions is proved in the relevant context. A deep relationship with the quantum group generalization of Herz restriction theorem from classical harmonic analysis is also established, in particular, in the course of our analysis we give a new proof of Herz restriction theorem.

Abstract. Quantum families of maps between quantum spaces are defined and studied. We prove that quantum semigroup (and sometimes quantum group) structures arise naturally on such objects out of more fundamental properties. As particular cases we study quantum semigroups of maps preserving a fixed state and quantum commutants of given quantum families of maps.

We introduce a non commutative analog of the Bohr compactification. Starting from a general quantum group G we define a compact quantum group bG which has a universal property such as the universal property of the classical Bohr compactification for topological groups. We study the object bG in special cases when G is a classical locally compact group, dual of a classical group, discrete or compact quantum group as well a quantum group arising from a manageable multiplicative unitary. We also use our construction to give new examples of compact quantum groups.

It is shown that all important features of a C * -algebraic quantum group (A, Δ) defined by a modular multiplicative W depend only on the pair (A, Δ) rather than the multiplicative unitary operator W . The proof is based on thorough study of representations of quantum groups. As an application we present a construction and study properties of the universal dual of a quantum group defined by a modular multiplicative unitary-without assuming existence of Haar weights.

Abstract. Given a quantum subgroup G ⊂ U n and a number k ≤ n we can form the homogeneous space X = G/(G ∩ U k ), and it follows from the Stone-Weierstrass theorem that C(X) is the algebra generated by the last n − k rows of coordinates on G. In the quantum group case the analogue of this basic result doesn't necessarily hold, and we discuss here its validity, notably with a complete answer in the group dual case. We focus then on the "easy quantum group" case, with the construction and study of several algebras associated to the noncommutative spaces of type X = G/(G ∩ U + k ).