We use free probability techniques to compute spectra and Brown measures of some non-hermitian operators in finite von Neumann algebras. Examples include $uₙ+u_{\infty }$ where uₙ and $u_{\infty }$ are the generators of ℤₙ and ℤ respectively, in the free product ℤₙ*ℤ, or elliptic elements of the form $S_{\alpha }+i{S}_{\beta }$ where $S_{\alpha }$ and $S_{\beta }$ are free semicircular elements of variance α and β.

To each complex number $\lambda$ is associated a representation ${\pi }_{\lambda }$ of the conformal group $S{O}_0(1,n)$ on ${𝒞}^{\infty }\left(S^{n-1}\right)$ (spherical principal series). For three values ${\lambda }_1,{\lambda }_2,{\lambda }_3$, we construct a trilinear form on ${𝒞}^{\infty }\left(S^{n-1}\right)\times {𝒞}^{\infty }\left(S^{n-1}\right)\times {𝒞}^{\infty }\left(S^{n-1}\right)$, which is invariant by ${\pi }_{{\lambda }_1}\otimes {\pi }_{{\lambda }_2}\otimes {\pi }_{{\lambda }_3}$. The trilinear form, first defined for $({\lambda }_1,{\lambda }_2,{\lambda }_3)$ in an open set of ${ℂ}^3$ is extended meromorphically, with simple poles located in an explicit family of hyperplanes. For generic values of the parameters, we prove uniqueness of trilinear invariant forms.

We introduce and study the notions of w*-approximate Connes amenability and pseudo-Connes amenability for dual Banach algebras. We prove that the dual Banach sequence algebra ℓ¹ is not w*-approximately Connes amenable. We show that in general the concepts of pseudo-Connes amenability and Connes amenability are distinct. Moreover the relations between these new notions are also discussed.

Given a smooth family of vector fields satisfying Chow-Hörmander’s condition of step 2 and a regularity assumption, we prove that the Sobolev spaces of fractional order constructed by the standard functional analysis can actually be “computed” with a simple formula involving the sub-riemannian distance.Our approach relies on a microlocal analysis of translation operators in an anisotropic context. It also involves classical estimates of the heat-kernel associated to the sub-elliptic Laplacian.

Various techniques are presented for constructing $\Lambda$ (p) sets which are not $\Lambda (p+ϵ)$ for all $ϵ\>0$. The main result is that there is a $\Lambda$ (4) set in the dual of any compact abelian group which is not $\Lambda (4+ϵ)$ for all $ϵ\>0$. Along the way to proving this, new constructions are given in dual groups in which constructions were already known of $\Lambda$ (p) not $\Lambda (p+ϵ)$ sets, for certain values of $p$. The main new constructions in specific dual groups are:– there is a $\Lambda$ (2k) set which is not $\Lambda (2k+ϵ)$ in $\mathbf{Z}\left(2\right)\oplus \mathbf{Z}\left(2\right)\oplus \cdots$ for all $2\le k$, $k\in \mathbf{N}$ and $ϵ\>0$, and in $\mathbf{Z}\left(p\right)\oplus \mathbf{Z}\left(p\right)\oplus \cdots$ ($p$ a prime,...