A new representation of the Cauchy kernel ${}_{\Gamma }$ for an arbitrary acute convex cone Γ in ℝⁿ is found. The domain of holomorphy of ${}_{\Gamma }$ is described. An estimation of the growth of ${}_{\Gamma }$ near the singularities is given.

Let $C$ be the open upper light cone in ${ℝ}^{1+n}$ with respect to the Lorentz product. The connected linear Lorentz group ${\mathrm{SO}}_{ℝ}{(1,n)}^0$ acts on $C$ and therefore diagonally on the $N$-fold product $T^N$ where $T={ℝ}^{1+n}+iC\subset {ℂ}^{1+n}.$ We prove that the extended future tube ${\mathrm{SO}}_{ℂ}(1,n)·T^N$ is a domain of holomorphy.

In this survey article we describe how the recent work in quantization in multi-variable complex geometry (domains of holomorphy, symmetric domains, tube domains, etc.) leads to interesting results and problems in C*-algebras which can be viewed as examples of the "non-commutative geometry" in the sense of A. Connes. At the same time, one obtains new functional calculi (of pseudodifferential type) with possible applications to partial differential equations and group representations.

On introduit une classe de domaines dans ${\mathbf{C}}_{\left(z\right)}^n={\mathbf{R}}_{\left(x\right)}^n\times {\mathbf{R}}_{\left(y\right)}^n$ appelés tuboïdes. Un tuboïde $D={\cup }_{x\in \Omega }(x,D_x)$ de profil $\Lambda ={\cup }_{x\in \Omega }(x,{\Lambda }_x)$ est un domaine de ${\mathbf{C}}_{\left(z\right)}^n$ dont chaque fibre $D_x$ (dans ${\mathbf{R}}_{\left(y\right)}^n)$ admet ${\Lambda }_x$ comme cône tangent à l’origine.On montre dans la première partie que l’enveloppe d’holomorphie d’un tuboïde $\widehat{D}$ de profil $\widehat{\Lambda }={\cup }_{x\in \Omega }(x,{\widehat{\Lambda }}_x)$ où ${\widehat{\Lambda }}_x$ est pour tout $x$ l’enveloppe convexe de ${\Lambda }_x$. dans la deuxième partie, l’on montre alors que tout tuboïde $D$ dont le profil $\Lambda$ a toutes ses fibres ${\Lambda }_x$ convexes contient un tuboïde $D^{\text{'}}$ de même profil qui est de plus un domaine d’holomorphie....