Morita Theory for Dynamical von Neumann Algebras

We consider compact matrix quantum groups whose $N$-dimensional fundamental representation decomposes into an $(N-1)$-dimensional and a one-dimensional subrepresentation. Even if we know that the compact matrix quantum group associated to this $(N-1)$-dimensional subrepresentation is isomorphic to the given $N$-dimensional one, it is a priori not clear how the intertwiner spaces transform under this isomorphism. In the context of so-called easy and non-easy quantum groups, we are able to define a transformation of linear combinations of partitions and we explicitly describe the transformation of intertwiner spaces. As a side effect, this enables us to produce many new examples of non-easy quantum groups being isomorphic to easy quantum groups as compact quantum groups but not as compact matrix quantum groups.

On square roots of the Haar state on compact quantum groups

This is a transcript of a series of eight lectures, 90 min each, held at IMSc Chennai, India from 5–24 January 2015. We give basic definitions, properties and examples of compact quantum groups and compact matrix quantum groups such as the existence of a Haar state, the representation theory and Woronowicz’s quantum version of the Tannaka–Krein theorem. Building on this, we define Banica–Speicher quantum groups (also called easy quantum groups), a class of compact matrix quantum groups determined by the combinatorics of set partitions. We sketch the classification of Banica–Speicher quantum groups and we list some applications. We review the state-of-the-art regarding Banica–Speicher quantum groups and we list some open problems.

We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf ∗-algebra with positive invariant functionals, then also the dual is a multiplier Hopf ∗-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C∗-algebras.

In this paper, we will prove that if A is a C*-algebra with an effective coaction ε by a compact quantum group, then the fixed point algebra and the reduced crossed product are Morita equivalent. As an application, we prove an imprimitivity type theorem for crossed products of coactions by discrete Kac C*-algebras.

We prove that the L 1-algebra of any non-Kac type compact quantum group is not operator biflat. Since operator amenability implies operator biflatness, this result shows that any co-amenable, non-Kac type compact quantum group gives a counterexample to the conjecture that L 1(𝔾) is operator amenable if and only if 𝔾 is amenable and co-amenable for any locally compact quantum group 𝔾. The result also implies that the L 1-algebra of a locally compact quantum group is operator biprojective if and only if 𝔾 is compact and of Kac type.

Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory

In this paper, we investigate the co-amenability of compact quantum groups. Combining with some properties of regular C*-norms on algebraic compact quantum groups, we show that the quantum double of co-amenable compact quantum groups is unique. Based on this, this paper proves that co-amenability is preserved under formulation of the quantum double construction of compact quantum groups, which exhibits a type of nice symmetry between the co-amenability of quantum groups and the amenability of groups.

We present versions of several classical results on harmonic functions and Poisson boundaries in the setting of locally compact quantum groups. In particular, the Choquet–Deny theorem holds for compact quantum groups; also, the result of Kaimanovich–Vershik and Rosenblatt, which characterizes group amenability in terms of harmonic functions, admits a noncommutative analogue in the separable case. We also explore the relation between classical and quantum Poisson boundaries by investigating the spectrum of the quantum group. We apply this machinery to find a concrete realization of the Poisson boundaries of the compact quantum group SU q(2) arising from measures on its spectrum.

We compare algebraic objects related to a compact quantum group action on a unital C*-algebra in the sense of Podleś and Baum et al. and show that they differ by the kernel of the morphism describing the action. Then we address ways to remove the kernel without changing the Podleś algebraic core. A minimal such procedure is described. We end the paper with a natural example of an action of a reduced compact quantum group with non-trivial kernel.

For a non-degenerate pair of compact quantum groups, we first construct the quantum double as an algebraic compact quantum group in an algebraic framework. Then by adopting some completion procedure, we give the universal and reduced quantum double constructions in the correspondence C*-algebraic settings, which generalize Drinfeld’s quantum double construction and yield new C*-algebraic compact quantum groups.

For a locally compact quantum group $\mathbb{G}$ we define its center, $\mathscr{Z}(\mathbb{G})$, and its quantum group of inner automorphisms, $\mathrm{Inn}(\mathbb{G})$. We show that one obtains a natural isomorphism between $\mathrm{Inn}(\mathbb{G})$ and $\mathbb{G}/\!\mathscr{Z}(\mathbb{G})$, we characterize normal quantum subgroups of a compact quantum group as those left invariant by the action of the quantum group of inner automorphisms and discuss several examples.

These lecture notes are intended as an introduction to the theory of locally compact quantum groups that are studied in the framework of operator algebras, i.e. C*-algebras and von Neumann algebras. The presentation revolves around the definition of a locally compact quantum group as given in [KuV00a] and [KuV03].KeywordsHilbert SpaceQuantum GroupHaar MeasureNorm TopologyCompact Quantum GroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

To be able to comprehend and study noncommutative Lp and Orlicz spaces for general possibly non-semifinite von Neumann algebras, some background in modular theory and crossed products is essential. Starting with a review of the GNS-construction we give a brief overview of the essentials of modular theory. We then present various extensions of the Radon-Nikodym theorem proved in Chapter 3 including the enigmatic Connes cocycle derivatives, after which we review the existence criteria for faithful normal conditional expectation in the general setting. Next we briefly describe how modular theory may also be approached from a Hilbert space rather than operator algebraic perspective. Here the natural positive cone plays a crucial role. In conclusion we take the trouble to also give a thumbnail introduction to group von Neumann algebras for locally compact groups boasting amongst other features a new proof of the existence of the Plancherel weight and culminating in the definition of locally compact Quantum Groups.

Non-existence of genuine (compact) quantum symmetries of compact, connected smooth manifolds