In this paper, we give three functors $mathfrak{P}$, $[cdot]_K$ and $mathfrak{F}$ on the category of C$^ast$-algebras. The functor $mathfrak{P}$ assigns to each C$^ast$-algebra $mathcal{A}$ a pre-C$^ast$-algebra $mathfrak{P}(mathcal{A})$ with completion $[mathcal{A}]_K$. The functor $[cdot]_K$ assigns to each C$^ast$-algebra $mathcal{A}$ the Cauchy extension $[mathcal{A}]_K$ of $mathcal{A}$ by ...

Cauchy coeecient integrals and Hankel contours provide a natural generalization of Stirling numbers for unrestricted complex values of their arguments. Many classical identities survive such an extension. Sur les nombres de Stirling d'indice complexe et les contours de Hankel R esum e : Les int egrales de Cauchy et les contours de Hankel fournissent une g en eralisation naturelle des nombres de...

Recently, folk questions on the smoothability of Cauchy hypersurfaces and time functions of a globally hyperbolic spacetime M , have been solved. Here we give further results, applicable to several problems: (1) Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, b...

We develop the basic elements of a Cauchy theory on the complex Levi-Civita field, which constitutes the smallest algebraically closed nonArchimedean extension of the complex numbers. We introduce a concept of analyticity based on differentiation, and show that it leads to local expandability in power series. We show that analytic functions can be integrated over suitable piecewise smooths path...

We deene a residue current of a holomorphic mapping , or more generally a holomorphic section to a holomorphic vector bundle, by means of Cauchy-Fantappie-Leray type formulas , and prove that a holomorphic function that annihilates this current belongs to the corresponding ideal locally. We also prove that the residue current coincides with the Colee-Herrera current in the case of a complete in...

A function f ∈ C(Ω) is holomorphic on Ω, if it satisfies the CauchyRiemann equations: ∂̄f = ∑n k=1 ∂f ∂z̄k dz̄k = 0 in Ω. Denote the set of holomorphic functions on Ω by H(Ω). The Bergman projection, B0, is the orthogonal projection of square-integrable functions onto H(Ω)∩L2(Ω). Since the Cauchy-Riemann operator, ∂̄ above, extends naturally to act on higher order forms, we can as well define Bergm...

Part I: Single Variables: Review and extensions. CR equations and analyticity; Cauchy-Goursat theorem and Cauchy integral formula, Louiville’s theorem; Morera’s theorem; maximum modulus principle; Laurent series and singularities; Riemann extension theorem; residues; Schwartz’s lemma; open mapping theorem; analytic continuation and the dilogarithm; linear fractional transformations; spaces of a...