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The Cauchy kernel for cones

Sławomir Michalik (2004)

Studia Mathematica

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

The extended future tube conjecture for SO(1, 𝑛 )

Peter Heinzner, Patrick Schützdeller (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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

Toeplitz-Berezin quantization and non-commutative differential geometry

Harald Upmeier (1997)

Banach Center Publications

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.

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