This self-contained encyclopedic monograph gives a detailed introduction to Bézout equations and stable ranks, encompassing and explaining needed topological, analytical, and algebraic tools and methods. Some of the highlights included are Carleson's corona theorem and the Bass, topological, and matricial stable ranks. The first volume focusses on topological structures, Banach algebras, and advanced function theory, thus preparing the stage for the algebraic structures in the second volume towards examining stable ranks with analytic methods. The main emphasis is laid on algebras of holomorphic functions. Often a new approach is presented or at least a different angle of sight, which makes the book attractive both for researchers and students interested in these active fields of research.
Part I. Topological structures and Banach algebras.- A short
introduction into point-set topology.- The Tietze extension theorem.- Smooth
functions and Lebesgue measurable sets.- Approximation theorems.- Maximum
principles, integral formulas, and Blaschke products.- Banach algebra
techniques.- The Stone-Cech compactification and extension of maps.-
Nonunital Banach algebras.- Brouwer's fixed point theorem.- Extension
problems in R^n and mappings into the sphere.- Topology in the plane: a
function theoretic approach.- Local and simple connectedness, curves, arcs,
and continua.- Continuous extensions of conformal maps and homeomorphisms.-
Advanced function-theoretic tools.- Borsuk's extension theorem.- Dimension
theory.- Michael's selection theorem.- Part II. Algebraic structures.-
Algebraic preliminaries.- The Bézout equation.- The algebras C_b(X, K)
and C(X, K) on non-compact spaces.- Polynomial, Noetherian, and von Neumann
regular rings.- The Bass and topological stable ranks.- The stable ranks for
algebras of polynomials and entire functions.- The Bass and topological
stable ranks of C(X, K).- The planar algebras P(K), R(K), A(K), C(K), and
their siblings.- The algebra of bounded analytic functions.- Peak sets in
function algebras.- Lipschitz algebras.- Regular and normal Banach algebras.-
The stable ranks for a zoo of algebras.- Various notions of reducibility and
stable ranks.- Stable ranks of holomorphic function algebras in C^n.-
Real-symmetric function algebras.- The algebra of almost periodic
functions.- Matricial stable ranks.- Part III. The Appendix.- Some results in
number theory and function theory.- The L^p(T)-spaces.- Matrix analysis.-
E-valued functions: integration and holomorphy.- Some tables.- Notes and
Sources.- Bibliography.- Index.
RAYMOND MORTINI, born 1958, is professor emeritus at the Université de Lorraine in Metz (France) and, since 2020, guest researcher at the University of Luxembourg, his home country. He obtained his Ph.D. (1984) and habilitation (1988) at the University of Karlsruhe (Germany) and was hired by the Université de Metz in 1995. From 2017-2020 he hold his courses at the University of Luxembourg, still being affiliated with the Université de Lorraine. His main interests lie in function theory, functional analysis, operator theory and point set topology. Currently he is in the editorial board of three Springer journals. RUDOLF RUPP, born 1960, is professor at the Nuremberg Institute of Technology, Georg-Simon-Ohm in Nuremberg, Germany. He obtained his Ph.D.(1988) and habilitation (1992) at the University of Karlsruhe (Germany), now called KIT. From 1999 until 2003 he persued a job outside the university. His research interests lie in function theory, Banach algebras and point set topology.
