Related papers: Kazhdan's Property T for Discrete Quantum Groups
A twist property is developed which imparts certain properties on the twisted group algebra. These include an involution * satisfying (xy)*=y*x* and an inner product satisfying <xy,z> = <x,zy*> and <xy,z>=<y,x*z>. Examples of twisted group…
We show that the semidefinite programs involved in the computer proofs for Kazhdan's property (T) satisfy strong duality and that the dual programs have a geometric interpretation in terms of harmonic cocycles. By dualizing geometric…
This paper discusses `geometric property (T)'. This is a property of metric spaces introduced in earlier work of the authors for its applications to K-theory. Geometric property (T) is a strong form of `expansion property': in particular…
The algebras obtained as fixed points of the action of the cyclic group $Z_N$ on the coordinate algebra of the quantum disc are studied. These can be understood as coordinate algebras of quantum or non-commutative cones. The following…
A semigroup $T$ is called Taimanov if $T$ contains two distinct elements $0,\infty$ such that $xy=\infty$ for any distinct points $x,y\in T\setminus\{0,\infty\}$ and $xy=0$ in all other cases. We prove that any Taimanov semigroup $T$ has…
We investigate when the group $SL_n(\mathcal{O}(X))$ of holomorphic maps from a Stein space $X$ to $SL_n (\C)$ has Kazhdan's property (T) for $n\ge 3$. This provides a new class of examples of non-locally compact groups having Kazhdan's…
The first quantum group cohomology with trivial coefficients of the discrete dual of any unitary easy quantum group is computed. That includes those potential quantum groups whose associated categories of two-colored partitions have not yet…
As a consequence of Kirchberg's work, Connes' Embedding Conjecture is equivalent to the property that every homomorphism of the group $F_\infty\times F_\infty$ into the unitary group $U(\ell^2)$ with the strong topology is pointwise…
In this paper, we provide several instances in which interesting approximation and stability properties are inherited by quotients with respect to finitely generated normal subgroups or, more strongly, normal subgroups with Kazhdan's…
We study the problem of determining when the reduced twisted group C*-algebra associated with a discrete group G is simple and/or has a unique tracial state, and present new sufficient conditions for this to hold. One of our main tools is a…
We address two properties for Abelian topological groups: ``every closed subgroup is dually closed'' and ``every closed subgroup is dually embedded.'' We exhibit a pair of topological groups such that each has both of the properties and the…
We propose a construction of quantum hypergroups using conditional expectations on compact quantum groups. Using this construction, we describe several series of non-trivial finite-dimensional quantum hypergroups via conditional…
In his work on the Novikov conjecture, Yu introduced Property $A$ as a readily verified criterion implying coarse embeddability. Studied subsequently as a property in its own right, Property $A$ for a discrete group is known to be…
We build and investigate a pure gauge theory on arbitrary discrete groups. A systematic approach to the construction of the differential calculus is presented. We discuss the metric properties of the models and introduce the action…
Motivated by the search for new examples of ``noncommutative manifolds'', we study the noncommutative geometry (in the sense of Connes) of the group C*-algebra of the three dimensional discrete Heisenberg group. We present a unified…
We investigate the relation between Cartan decompositions of the unitary group and discrete quantum symmetries. To every Cartan decomposition there corresponds a quantum symmetry which is the identity when applied twice. As an application,…
All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always…
Gromov constructed uncountably many pairwise non-isomorphic discrete groups with Kazhdan's property (T). We will show that no separable II_1-factor can contain all these groups in its unitary group. In particular, no separable II_1-factor…
We use minimal tilting complexes to construct an explicit bijection between the set of thick tensor ideals with the two-out-of-three property in the category of finite-dimensional modules over a quantum group at a root of unity and the set…
We construct the HNN extension of discrete quantum groups, we study their representation theory and we show that an HNN extension of amenable discrete quantum groups is K-amenable.