Let G be a locally compact (second countable) group and π:G → U(ℋ) a unitary representation of G on the (separable) Hilbert space ℋ.Of course,ℋ may or may not contain non-trivial vectors invariant under π(G). We wish to define a notion of π almost containing invariant vectors.

Kazhdan’s Property (T)

In this article we prove that quasi-multiplicative (with respect to the usual length function) mappings on the permutation group $\SSn$ (or, more generally, on arbitrary amenable Coxeter groups), determined by self-adjoint contractions fulfilling the braid or Yang-Baxter relations, are completely positive. We point out the connection of this result with the construction of a Fock representation of the deformed commutation relations $d_id_j^*-\sum_{r,s} t_{js}^{ir} d_r^*d_s=\delta_{ij}\id$, where the matrix $t_{js}^{ir}$ is given by a self-adjoint contraction fulfilling the braid relation. Such deformed commutation relations give examples for operator spaces as considered by Effros, Ruan and Pisier. The corresponding von Neumann algebras, generated by $G_i=d_i+d_i^*$, are typically not injective.

https://arxiv.org/pdf/funct-an/9408002.pdf

Completely Positive Maps on Coxeter Groups, Deformed Commutation Relations, and Operator Spaces

The free analogues of $U(n)$ in Woronowicz' theory \cite{w2} are the compact matrix quantum groups $\{ A_u(F)|F\in GL(n,\mathbb C)\}$ introduced by Wang and Van Daele. We classify here their irreducible representations. Their fusion rules turn to be related to the combinatorics of Voiculescu's circular variable. If $F\bar{F}\in\mathbb R I_n$ we find an embedding $A_u(F)_{red}\hookrightarrow C(\mathbb T)*_{red}A_o(F)$, where $A_o(F)$ is the deformation of $SU(2)$ studied in \cite{ban}. We use the representation theory and Powers' method for showing that the reduced algebras $A_u(F)_{red}$ are simple, with at most one trace.

Le groupe quantique compact libre U(n)

We will introduce now the notions of correlation functions and their Fourier transforms which play an important role in various applications of the theory of dynamical systems. Suppose {T t } is a flow on a measure space (M,M,μ) and f ∈ L 2 (M,M,μ). By the correlation function corresponding to f we mean the function b f (t) = ∫ M f(T t x)f(x)dμ A number of statistical properties of the dynamical system may be characterized by the limit behavior of the differences [b f (t) — (∫ M f dμ)2]- In the case of mixing, these expressions tend to zero as t → ∞. If the convergence is fast enough one may write the above expressions in the form 

$${b_f}\left( t \right) - {\left( {\int_M {fd\mu } } \right)^2} = \int_{ - \infty }^\infty {\exp \left( {i\lambda t} \right) \cdot {\rho _f}\left( \lambda \right)d\lambda } $$

. The function ρ f (λ) in this representation is called the spectral density of. The set of those λ for which ρ f (λ) is essentially non-zero for typical ∫, characterizes in some sense the frequencies playing the crucial role in the dynamics of the system under consideration. It is sometimes said that the system “produces a noise” on this set. The investigation of the behavior of the functions ρ f (t), as well as of their analogs ρ f (n) n ∈ Z, in the case of discrete time, is very important both for theory and for applications.

Spectral Theory of Dynamical Systems

North-Holland Mathematical Library

We study compact bicrossed product arising from a matched pair of a discrete group Γ and a compact group G. We exhibit an automatic regularity property of the matched pair and describe the representation theory and the fusion rules of such bicrossed product. We characterize the relative Kazhdan property (T) and the relative Haagerup property of the pair given by G and the bicrossed product in terms of the action of Γ on G. We also provide an explicit example of a non-trivial discrete quantum group with the Kazhdan property (T). Finally, we study all the properties mentioned above for the crossed product quantum group given by an action by quantum automorphisms of a discrete group on a compact quantum group, and also estbalish the permanence of rapid decay and weak amenability and some several explicit examples.

/pdf/on-compact-bicrossed-products-9axtn52dbk.pdf

On compact bicrossed products

This paper extends the classical theory of joinings of measurable dynamical systems to the noncommutative setting from several interconnected points of view. Among these is a particularly fruitful identification of joinings with equivariant quantum channels between $W^{\ast}$-dynamical systems that provides noncommutative generalizations of many fundamental results of classical joining theory. We obtain fully general analogues of the main classical disjointness characterizations of ergodicity, primeness and mixing phenomena.

On Noncommutative Joinings

Jolissaint and Stalder introduced definitions of mixing and weak mixing for von Neumann subalgebras of finite von Neumann algebras. In this note, we study various algebraic and analytical properties of subalgebras with these mixing properties. We prove some basic results about mixing inclusions of von Neumann algebras and establish a connection between mixing properties and normalizers of von Neumann subalgebras. The special case of mixing subalgebras arising from inclusions of countable discrete groups finds applications to ergodic theory, in particular, a new generalization of a classical theorem of Halmos on the automorphisms of a compact abelian group. For a finite von Neumann algebra $M$ and von Neumann subalgebras $A$, $B$ of $M$, we introduce a notion of weak mixing of $B\subseteq M$ relative to $A$. We show that weak mixing of $B\subset M$ relative to $A$ is equivalent to the following property: if $x\in M$ and there exist a finite number of elements $x_1,...,x_n\in M$ such that $Ax\subset \sum_{i=1}^nx_iB$, then $x\in B$. We conclude the paper with an assortment of further examples of mixing subalgebras arising from the amalgamated free product and crossed product constructions.

/pdf/mixing-subalgebras-of-finite-von-neumann-algebras-22ucsoea9u.pdf

Kunal Mukherjee

Papers

On compact bicrossed products

On Noncommutative Joinings

Mixing subalgebras of finite von Neumann algebras

Masas and bimodule decompositions of ii1 factors

Generator masas in q-deformed Araki–Woods von Neumann algebras and factoriality