Extension Problems and Stable Ranks A Space Odyssey

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.