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We introduce Hecke algebras associated to discrete quantum groups with commensurated quantum subgroups. We study their modular properties and the associated Hecke operators. In order to investigate their analytic properties we adapt the construction of the Schlichting completion to the quantum setting, thus obtaining locally compact quantum groups with compact open quantum subgroups. We study in detail a class of examples arising from quantum HNN extensions. [ABSTRACT FROM AUTHOR]

We discuss just infiniteness of C∗‐algebras associated to discrete quantum groups and relate it to the C∗‐uniqueness of the quantum groups in question, that is, to the uniqueness of a C∗‐completion of the underlying Hopf ∗‐algebra. It is shown that duals of q‐deformations of simply connected semisimple compact Lie groups are never C∗‐unique. On the other hand, we present an example of a discrete quantum group which is not locally finite and yet is C∗‐unique. [ABSTRACT FROM AUTHOR]

For a locally compact quantum group [ABSTRACT FROM AUTHOR]

We develop fully noncommutative Feynman–Kac formulae by employing quantum stochastic processes. To this end, we establish some theory for perturbing quantum stochastic flows on von Neumann algebras by multiplier cocycles. Multiplier cocycles are constructed via quantum stochastic differential equations whose coefficients are driven by the flow. The resulting class of cocycles is characterized under alternative assumptions of separability or Markov regularity. Our results generalize those obtained using classical Brownian motion on the one hand, and results for unitarily implemented flows on the other. [ABSTRACT FROM AUTHOR]

Motivated by the work of Goswami on quantum isometry groups of non-commutative manifolds, we define the quantum symmetry group of a unital C*-algebra A equipped with an orthogonal filtration as the universal object in the category of compact quantum groups acting on A in a filtration-preserving fashion. The existence of such a universal object is proved and several examples discussed. In particular, we study the universal quantum group acting on the dual of the free group and preserving both the word length and the block length. [ABSTRACT FROM AUTHOR]

We construct explicit approximating nets for crossed products of C*-algebras by actions of discrete quantum groups. This implies that C*-algebraic approximation properties such as nuclearity, exactness or completely bounded approximation are preserved by taking crossed products by the actions of amenable discrete quantum groups. We also show that the noncommutative topological entropy of a transformation commuting with the quantum group action does not change when we pass to the canonical extension to the crossed product. Both these results are extended to twisted crossed products via a stabilization trick. [ABSTRACT FROM PUBLISHER]