Complex analysis for Fréchet manifolds in algebras of Fourier operators
by
Bernhard Gramsch
Gutenberg-Univ. Mainz, Germany

Related to common work with W. Kaballo we develop some aspects of the theory of holomorphic Fredholm and semi-Fredholm functions having values in Fréchet algebras with local spectral invariance in the L^p-theory and for special algebras of Fourier operators.
A denotes a submultiplicative subalgebra of the Banach algebra L(E) for a Banach space E over C, (e=Id_E Î A) such that the inverse of a Î A with ||a-e||_L(E) < e < 1 is an element of A (A is a Y₀-algebra).
Let W be a holomorphy region in a DFN-space D and T: W® F^l(A) a holomorphic map with values in the set of semi-Fredholm operators of A.

Theorem 1: If T(z^*) is left invertible in A for some z^* Î W, then there exists a meromorphic left inverse M of T on W with a (global) decomposition on W

M(z)=H(z)+S(z),     z Î W.

where S is meromorphic with values in the ideal S(A) of operators with fast decreasing approximation numbers.

Theorem 2: In addition to the assumptions of Theorem 1 let us assume that the nuclear space D has a basis, W is contractible and T is a Fredholm function of index 0, then there exist a (global) multiplicative decomposition

T(z)=J(z)(I+h(z)),     z Î W

where J: W® A^-1 is invertible and holomorphic ("z Î W) and h(z) is holomorphic with values in S(A).

Remark: 1) Theorem 1 seems to be new for closed subalgebras of L(E) (and for C^*-algebras) for functions on finite dimensional holomorphy regions.
2) Theorem 2 depends on new results concerning the Oka principle in the non abelian cohomology for infinite dimensional holomorphy regions.
3.) The results can be applied to deformation quantization and to the Chern-Connes character.
4) An Oka-homotopy principle can be derived for the space of periodic geodesics in the Fréchet manifold of idempotent elements of A. 5) A general construction of Y₀-algebras involving also Fourier integral operators is derived.
6) The Oka principle leads to the solution of a Riemann-Hilbert problem.
7) Some problems and counterexamples are presented.

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Date received: June 27, 2005

Copyright © 2005 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # carf-68.