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.