The concept of a quantum group of unitary operators is relevant for the theory of non-compact locally compact quantum groups. It plays similar role as the concept of a quantum matrix group in the compact case. To show the usefulness of this notion we present an approach to a construction of quantum ‘az + b’ group based on this idea. A brief survey of the present status of quantum group theory i...

A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory.

Let G be a locally compact group. Then G is known to be amenable if and only if it has Reiter’s property (P1), i.e., there is a net (mα)α of non-negative norm one functions in L(G) such that limα supx∈K ‖Lx−1mα−mα‖ = 0 for each compact subset K ⊂ G (Lx−1mα stands for the left translate of mα by x). We give a formulation of property (P1) that extends naturally to locally compact quantum groups i...

Let G be a locally compact group. Then G is known to be amenable if and only if it has Reiter’s property (P1), i.e., there is a net (mα)α of non-negative norm one functions in L(G) such that limα supx∈K ‖Lx−1mα−mα‖ = 0 for each compact subset K ⊂ G (Lx−1mα stands for the left translate of mα by x). We give a formulation of property (P1) that extends naturally to locally compact quantum groups i...

In this paper we associate to every reduced C∗-algebraic quantum group (A,∆) (as defined in [18]) a universal C∗-algebraic quantum group (Au,∆u). We fine tune a proof of Kirchberg to show that every ∗-representation of a modified L-space is generated by a unitary corepresentation. By taking the universal enveloping C∗-algebra of a dense sub ∗-algebra of A we arrive at the C∗-algebra Au. We show...

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The Haagerup property for locally compact groups is generalised to the context of locally compact quantum groups, with several equivalent characterisations in terms of the unitary representations and positive-definite functions established. In particular it is shown that a locally compact quantum group G has the Haagerup property if and only if its mixing representations are dense in the space ...

in this paper, we investigate the relation between l-projections and conditional expectations on subalgebras of the fourier-stieltjes algebra b(g), and we will show that compactness of g plays an important role in this relation.