### 1-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations

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In this paper, we will study the relative complexity of the unitary duals of countable groups. In particular, we will explain that if $G$ and $H$ are countable amenable non-type I groups, then the unitary duals of $G$ and $H$ are Borel isomorphic.

For a locally convex *-algebra A equipped with a fixed continuous *-character ε (which is roughly speaking a generalized F*-algebra), we define a cohomological property, called property (FH), which is similar to character amenability. Let ${C}_{c}\left(G\right)$ be the space of continuous functions with compact support on a second countable locally compact group G equipped with the convolution *-algebra structure and a certain inductive topology. We show that $({C}_{c}\left(G\right),{\epsilon}_{G})$ has property (FH) if and only if G has property (T). On...

Let ${\mathbf{F}}_{k}$ be a free group on $k$ generators. We construct the series of uniformly bounded representations ${\prod}_{z}$ of ${\mathbf{F}}_{k}$ acting on the common Hilbert space, depending analytically on the complex parameter z, $1/(2k-1)\<\left|z\right|\<1$, such that each representation ${\prod}_{z}$ is irreducible. If $z$ is real or $\left|z\right|=1/\left(\sqrt{2k-1}\right)$ then ${\prod}_{z}$ is unitary; in other cases ${\prod}_{z}$ cannot be made unitary. For $z\ne {z}^{\text{'}}$ representations ${\prod}_{z}$ and ${\prod}_{{z}^{\text{'}}}$ are congruent modulo compact operators.

There is constructed a compactly generated, separable, locally compact group G and a continuous irreducible unitary representation π of G such that the image π(C*(G)) of the group C*-algebra contains the algebra of compact operators, while the image $\pi \left({L}^{1}\left(G\right)\right)$ of the ${L}^{1}$-group algebra does not containany nonzero compact operator. The group G is a semidirect product of a metabelian discrete group and a “generalized Heisenberg group”.

Let S be a locally compact (σ-compact) group or semigroup, and let T(t) be a continuous representation of S by contractions in a Banach space X. For a regular probability μ on S, we study the convergence of the powers of the μ-average Ux = ʃ T(t)xdμ(t). Our main results for random walks on a group G are: (i) The following are equivalent for an adapted regular probability on G: μ is strictly aperiodic; ${U}^{n}$ converges weakly for every continuous unitary representation of G; U is weakly mixing for any...

It is proved for Abelian groups that the Reidemeister coincidence number of two endomorphisms ϕ and ψ is equal to the number of coincidence points of ϕ̂ and ψ̂ on the unitary dual, if the Reidemeister number is finite. An affirmative answer to the bitwisted Dehn conjugacy problem for almost polycyclic groups is obtained. Finally, we explain why the Reidemeister numbers are always infinite for injective endomorphisms of Baumslag-Solitar groups.