Extension Problems and Stable Ranks A Space Odyssey 1st Edition by Raymond Mortini, Rudolf Rupp ISBN 9783030738723 3030738728

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ISBN 10: 3030738728
ISBN 13: 9783030738723
Author: Raymond Mortini, Rudolf Rupp

Extension Problems and Stable Ranks A Space Odyssey 1st Edition Table of contents:

Part I Topological structures and Banach algebras

Chapter 1 A short introduction into point-set topology

1.1 Compact and connected spaces

1.1.A Basic topological notions

1.1.B Discrete subsets in topological spaces

1.1.C Compact sets

1.1.D Exhaustion sequences in Rn

1.1.E Separable metric spaces

1.1.F Baire spaces

1.1.G Continuous functions between general topological spaces

1.1.H Connected sets

1.1.I Connected components

1.1.J Compact connected sets

1.1.K Simply connected planar domains

L Perfect sets

1.2 The axiom of choice and Zorn’s Lemma

1.2.A The size or cardinal of some distinguished sets

1.3 Nets and compactness

1.3.A Nets in topological spaces

1.3.B Subnets

1.4 Filters and compact sets

1.5 Lindelöf spaces

1.6 Infinite products of topological spaces

1.6.A Basic properties of infinite products

1.6.B Tychonov’s theorem

1.6.C Countable products of metric spaces

1.6.D Completely metrizable spaces

1.6.E The Alexandroff-Hausdorff theorem

Chapter 2 The Tietze extension theorem

Chapter 3 Smooth functions and Lebesgue measurable sets

3.1 Smooth partitions of unity

3.2 Interpolation and approximation by smooth functions

3.2.A Interpolation and zero sets

3.2.B Friedrich mollifiers and approximation

3.3 Smooth extensions

3.4 Rudiments of measure theory

3.4.A Lebesgue-measurable sets

3.4.B Measurable functions

Chapter 4 The ∂bar-calculus

4.1 The Wirtinger derivatives

4.2 Some basics in function theory/complex analysis

4.3 Planar Cauchy integrals and the dbar-equation

4.4 The Cauchy, Gauss, Green, Stokes, and Pompeiu formulas

4.5 Weak solutions of the ∂-equation

4.6 Explicit values of some integrals

4.6.A Planar Cauchy integrals

4.6.B Other types of planar integrals

Chapter 5 Approximation theorems

5.1 The Runge approximation theorems

5.1.A Approximation by rational functions

5.1.B The polynomial convex hull

5.2 The ∂bar-equation revisited

5.3 Montel’s approximation theorem

5.4 The Stone-Weierstrass approximation theorem

5.5 Compactness criteria in function spaces

5.5.A The Arzelà-Ascoli theorems

5.5.B Montel’s criteria for normality

Chapter 6 Maximum principles, integral formulas, and Blaschke products

6.1 Maximum principle for holomorphic functions in several complex variables

6.2 Cauchy and Poisson representations

6.3 Holomorphic functions with smooth boundary values

6.4 Poisson representation on half-planes

6.5 The argument principle

6.5.A Winding numbers, roots and logarithms

6.5.B Index of maps on the unit circle and the winding number

6.6 Blaschke products

6.6.A Blaschke products and Riesz factorization

6.6.A.1 The pseudohyperbolic distance

6.6.A.2 Infinite products and Blaschke products

6.6.B Cluster sets

Chapter 7 Banach algebra techniques

7.1 Basics in linear algebra and functional analysis

7.1.A Linear maps and norms

7.1.B Finite-dimensional normed spaces

7.1.C Extension theorems for linear functionals

7.1.D Quotient spaces

7.1.E The open mapping theorem

7.1.F Projections and complemented subspaces

7.2 Basic theory of Banach algebras

7.2.A Renormings and examples of Banach algebras

7.2.B Completions of normed algebras

7.2.C The spectrum of a Banach algebra element

7.2.D Commutative Banach algebras

7.2.E The maximal ideal space or spectrum of a Banach algebra

7.2.F Separability and metrizability of spectra

7.2.G The spectrum of several classical Banach algebras

7.2.H The Gelfand transform

7.2.I Semi-simple algebras

7.2.J Uniform algebras and function algebras

7.2.K Natural Banach function algebras

7.3 Finitely generated Banach algebras and joint spectra

7.3.A Joint spectra

7.3.B The basic theory on finitely generated Banach algebras

7.3.C Polynomial convexity in Cn

7.4 The Shilov boundary

7.4.A Minimal closed boundaries

7.4.B The Shilov extension theorem

7.5 The A-convex hull, L-sets and support sets

7.5.A The A-convex hull

7.5.B L-sets and the A-rational convex hull

7.5.C Support sets for characters

7.6 The holomorphic functional calculus

7.6.A The one-dimensional functional calculus

7.6.B The holomorphic functional calculus II

7.7 Exponentials and logarithms in Banach algebras

7.7.A Exponentials in general Banach algebras

7.7.B Exponentials and logarithms in the matrix algebra

7.7.B.1 The algebra Mn(K)

7.7.B.2 The matrix algebra Mn(A)

7.7.B.3 The exponential of a matrix in M(A)

7.7.B.4 Logarithms of the identity matrix in M2(A)

7.7.B.5 Logarithms of triangular matrices in M(A)

7.7.B.6 Representations with few exponential matrices

7.8 Invertible tuples in a unital Banach algebra

7.8.A The connected components of Un(A)

7.8.B The Arens-Royden-Novodvorski-Taylor theorems

7.9 The group of units in the algebra C(T,C)

7.10 Real Banach algebras

7.10.A Complexifications of real algebras

7.10.A.1 Ideals in real algebras and their complexifications

7.10.B The real-symmetric spectrum of elements in real Banach algebras

7.10.C Real division algebras

7.10.D Functional calculus for real Banach algebras

7.10.E Maximal ideals and characters of real Banach algebras

7.10.E.1 The complexification of the algebra CR

7.10.E.2 The Gelfand transform in real Banach algebras

7.10.E.3 Some relations between the invertible tuples of a real Banach algebra and its complexificat

7.10.F Semi-simple algebras and the real Arens-Royden theorem

7.10.G The Shilov idempotent theorem for real Banach algebras

7.10.H Different levels of reality

7.10.I Natural real Banach function algebras

Chapter 8 The Stone-Čech compactification and extension of maps

8.1 Completely regular spaces

8.2 Compactifications

8.2.A The Stone-Čech compactification

8.2.B The Alexandroff compactification

8.3 Extensions of maps from dense subsets

Chapter 9 Nonunital Banach algebras

9.1 Adjoining an identity element

9.2 Bounded approximate identities

9.2.A Bounded approximate identities

9.2.B Cohen’s factorization theorem

9.3 Spectra and maximal ideals

9.3.A Spectra of nonunital Banach algebras and one-point compactications

9.3.B Maximal ideals in nonunital Banach algebras

9.3.C Direct products of Banach algebras

9.3.D Banach algebras with empty spectrum

9.3.E Nonunital Banach algebras with compact spectrum

9.3.F Maximal ideals in Banach algebras II

9.3.G Radical and hyperradical algebras

9.3.H The Volterra and Dales algebras

9.3.I Ideals not contained in any maximal ideal

Chapter 10 Brouwer’s fixed point theorem

10.1 The fixed point theorem

10.2 Non-extendability

10.3 Extensions, retracts and homotopy

10.4 The hairy ball theorem

10.5 The Borsuk-Ulam theorems

Chapter 11 Extension problems in Rn and mappings into the sphere

11.1 Self-mappings of the sphere

11.2 More general extension problems

Chapter 12 Topology in the plane: a function theoretic approach

12.1 Zero-free extensions, logarithms, and Eilenberg’s Theorem

12.2 Simple connectivity and logarithms on open sets

12.3 Harmonic functions and logarithms

12.4 The open mapping theorem for injective maps

12.5 Unbounded continuous logarithms

12.6 The Jordan curve theorem

12.6.A The complementary components of a Jordan curve

12.6.B Three Applications of the Jordan curve theorem

12.7 Level sets

12.8 Rouché type theorems

12.8.A Counting zeros and poles

12.8.B A homotopic variant of Rouché’s theorem

12.8.C Rouché for continuous-holomorphic pairs

12.8.D Rouché in abstract Banach algebras

Chapter 13 Local and simple connectedness, curves, arcs, and continua

13.1 Peano curves

13.2 Locally connected spaces

13.3 The Hahn-Mazurkiewicz theorem

13.4 Metric continua

13.4.A Local connectedness versus local path-connectedness

13.4.B A characterization of Jordan arcs and Jordan curves

13.4.C Local connectedness versus local arc-connectedness

13.4.D Jordan arcs passing through prescribed points

13.5 Connectivity: examples and counterexamples

13.6 Homotopies of curves

13.7 The continuous lifting theorem and logarithms on psc-spaces

13.8 Simply connected planar domains: a résumé

13.9 The Pochhammer curve

Chapter 14 Continuous extensions of conformal maps and homeomorphisms

14.1 Extensions of conformal maps

14.2 Extension theorems of Schoenflies type

14.3 The Schwarz reflection principle

14.4 A conformal map with strange properties

14.5 Interpolation by homeomorphisms

Chapter 15 Advanced function-theoretic tools

15.1 Normal families of holomorphic functions revisited

15.2 Montel’s normality criterion via conformal metrics

15.3 The Ahlfors-Shimizu characteristic and normal functions

Chapter 16 Borsuk’s extension theorem

16.1 Retracts, fixed points and contractibility

16.2 The extension theorem

Chapter 17 Dimension theory

17.1 Dimension of normal spaces

17.2 The topological dimension and extension properties

17.3 Some distinguished properties for the covering dimension

17.4 The covering dimension of Rn

17.5 The dimension of the Stone-Čech compactification

17.6 Spaces of dimension 0 and total disconnectedness

17.6.A The relations

17.6.B Examples of connected and totally disconnected sets in R2

17.6.C The dimension of topologically thin subsets in Rn

17.6.D Cantor sets and zero-dimensionality

17.7 The Stone-Cech compactification of the integers

Chapter 18 Michael’s selection theorem

18.1 Michael’s theorem for compact spaces

18.2 Right inverses in functional analysis

Part II Algebraic structures

Chapter 19 Algebraic preliminaries

19.1 Maximal ideals and prime ideals

19.2 Nakayama’s lemma

19.3 PIDs and UFDs

Chapter 20 The Bézout equation

20.1 The solution set

20.2 Some examples

20.2.A The polynomial ring C[z]

20.2.B The ring H(U) of holomorphic functions

20.2.C The Bézout equation in C[z] with degree estimate

20.2.D The Bézout equation for rational functions

20.3 The Bézout equation in general function algebras

Chapter 21 The algebras Cb(X,K) and C(X,K) on non-compact spaces

21.1 Maximal ideals in C(X,K)

21.2 Free ideals

21.3 The inner and outer corona

21.4 Maximal ideals in C(X,K) versus Cb(X,K)

21.5 The Q (or realcompact) spaces

Chapter 22 Polynomial, Noetherian, and von Neumann regular rings

22.1 Euclidean division

22.2 Noetherian rings and minimal prime ideals

22.3 Hilbert’s Nullstellensatz

22.4 The polynomial ring R[x1, … , xn]

22.4.A Generators for maximal ideals in real polynomial rings

22.5 Algebraic dependence of polynomials

22.6 The Krull dimension of polynomial rings over fields

22.7 Krull’s intersection theorem

22.8 Von Neumann regular rings and zero-dimensional rings

Chapter 23 The Bass and topological stable ranks

23.1 Some fundamental properties of the stable ranks

23.2 Stable ranks in the nonunital case

23.3 Reducibility in C(M(A),C) versus reducibility in A

23.4 Reducibility to the principal component

23.5 Stable rank and Krull dimension

23.5.A Stable rank of Noetherian rings

23.5.B Bézout domains

Chapter 24 The stable ranks for algebras of polynomials and entire functions

24.1 The algebras C[z], H(U), and R0(K)

24.2 Real- and complex-valued polynomials

Chapter 25 The Bass and topological stable ranks of C(X,K)

25.1 Vasershtein’s Theory

25.2 Examples of non-reducible tuples

25.3 Reducibility in C(X,K)

25.3.A Reducibility versus extensions

25.3.B Reducibility of pairs and the boundary principle

25.3.C Reducibility of tuples in the several variables setting

25.3.D Complex Banach algebras with spectrum in Cn

25.4 Stable ranks of subrings of C(X,K)

25.4.A Conditions for equality

25.4.B Subalgebras of C(X, K) with prescribed stable ranks

25.5 Mappings between spheres

25.5.A Applications to totally disconnected sets in Rn

Chapter 26 The planar algebras P(K), R(K), A(K), C(K), and their siblings

26.1 The disk algebra

26.1.A The Herglotz kernel and the conjugate Poisson kernel

26.1.B Zero sets and interpolation in A(D)

26.2 Some approximation theorems relating P(K), R(K), A(K), and C(K)

26.3 Generators for P(K), R(K), A(K), and C(K)

26.3.A R(K) is two-generated

26.4 Injective holomorphic extensions of P(K)-functions

26.5 Algebraic properties: spectra and Bézout equation

26.5.A Stability of algebras on planar sets

26.5.B The algebra P(K)

26.5.C A constructive proof of the Nullstellensatz for A(D)

26.5.D The cylinder algebra and parametrized families of disk algebra functions

26.5.E The algebra R(K)

26.5.F The algebras A(K) and A(G)

26.5.G The Nullstellensatz for A(S,K)

26.5.H The Nullstellensatz for subalgebras of A(K)

26.5.I The algebra A1(K)

26.6 The Shilov boundaries for P(K), R(K), A(K) and beyond

26.7 The stable ranks of C(K)

26.7.A Sard’s lemma

26.7.B The stable ranks of C(K) revisited

26.8 The stable ranks for P(K), R(K), A1(K)

26.8.A The stable ranks for R(K) and P(K)

26.8.B The stable ranks for the cylinder algebra

26.8.C The stable ranks for A1(K)

26.9 The stable ranks for A(K), A(G) and A(S,K)

26.10 Stable ranks for subalgebras of A(K)

26.11 The spectrum and stable ranks for M(K)

Chapter 27 The algebra of bounded analytic functions

27.1 Fatou-Lindelöf theorems

27.1 Some uniqueness theorems and trigonometric series

27.3 Interpolation problems

27.3.A Interpolation of boundary data by finite Blaschke products

27.3.B Interpolating sequences

27.3.C Perturbation of interpolating sequences and approximative interpolation

27.3.D Interpolating Blaschke products

27.3.E Finite unions of interpolating sequences

27.3.F Interpolation on Carleson-Newman sequences

27.3.G Special interpolation sets in M(H)∞

27.3.H The interpolation sets for the disk algebra

27.4 The algebra of essentially bounded functions

27.4.A Bounded measurable functions

27.4.B Essentially bounded functions

27.5 Hardy spaces on the circle and disk

27.5.A Boundary values of bounded analytic functions

27.5.B Basic results in Hp-theory

27.5.B.1 The Hardy-Hilbert space H2( D)

27.5.B.2 The Hardy space H1( T) and some dual spaces

27.5.B.3 The spaces Hp( D) and Hp( T)

27.5.B.4 The Fejér-Riesz, Hilbert, and Hardy inequalities

27.5.B.5 Harmonic conjugates, Fourier series and the Gleason-Whitney theorem

27.5.C Inner-outer factorization in Hardy spaces

27.6 The uncomplementedness of the disk algebra

27.7 The functions exp[(1+z)p/(1-z)p] for p=2,4

27.8 The spectrum of H∞ and the Corona Theorem

27.8.A The proof of the corona theorem

27.8.A.1 Littlewood-Paley-type inequalities

27.8.A.2 Estimating solutions to the -equation

27.8.A.3 The proof of the corona theorem via a -problem with bounds

27.8.B Additional properties of H( D) and its spectrum.

27.9 The Sarason algebra

27.9.A The spectrum of the Sarason algebra

27.9.B The stable ranks for the Sarason algebra

27.10 Non-tangential limits

27.10.A Hardy-Littlewood maximality and Fatou’s theorem

27.10.B The Axler-Shields-Ivanov extension theorem

Chapter 28 Peak sets in function algebras

28.1 Peak sets and peak points

28.2 The Jarosz algebra and its peak points

Chapter 29 Lipschitz algebras

29.1 Basic properties of Lipschitz algebras

29.1.A Holomorphic Hölder-Lipschitz algebras

29.1.B The Hölder-Lipschitz algebra on the disk

29.1.C Extension theorems for Hölder-Lipschitz functions

29.1.D Pointwise Lipschitz functions

29.2 Algebraic properties of Lipschitz algebras

29.2.A Spectra of planar Lipschitz algebras

29.2.B Stable ranks of holomorphic Lipschitz algebras

29.3 Lipschitz classes on metric spaces

Chapter 30 Regular and normal Banach algebras

30.1 Normal subalgebras of C(X,K)

30.2 Stable rank of regular Banach algebras

Chapter 31 The stable ranks for a zoo of algebras

31.1 Lower and upper estimates for the stable ranks

31.1.A Lower estimates for the Bass stable rank

31.1.B Lower estimates for the topological stable rank

31.1.C Upper estimates for stable ranks

31.2 Stable ranks for algebras of smooth functions

31.2.A The algebra C∞(Ω,K)

31.2.B The algebra C∞bb(Ω,K)

31.3 Stable ranks of finitely generated Banach algebras

31.4 Two-generated algebras

31.4.A The cone algebra

31.5 Sequence algebras

31.6 The algebra of A-valued continuous functions

Chapter 32 Various notions of reducibility and stable ranks

32.1 The Bass stable rank via homomorphisms

32.1.A The surjective and dense stable rank

32.1.B Criteria for the connectedness

32.1.C The connected stable rank

32.1.D The stable ranks for homomorphic images of algebras

32.2 The dense stable rank of several classical algebras

32.2.A Vasershtein’s theorem revisited

32.2.B Regular and approximatively regular algebras

32.2.C The dense stable rank for subalgebras of A(K)

32.2.D The dense stable ranks of finitely generated Banach algebras

32.3 Three classes of stable ranks

32.3.A Unitary M-stable rank versus all-units rank

32.3.B Norm-one rank

32.4 Total reducibility

32.4.A Some sufficient conditions for total reducibility

32.4.B Total reducibility in H(U)

32.4.C Total reducibility in A(D)

32.4.D Exponential reducibility

32.5 The absolute stable rank

32.6 Generalized E-stable ranks

32.6.A E-stable rank and L-sets

32.6.B The E-stable rank of the disk algebra

32.7 Stable ranks of quotient rings and the almost stable rank

32.7.A Quotients of rings and the Bass stable rank

32.7.B The almost stable rank

32.7.C Calculating alsr for some rings

Chapter 33 Stable ranks of holomorphic function algebras in Cn

33.1 The dbar-calculus in several complex variables

33.1.A The chain rule for the dbar-calculus

33.2 The dbar-equation in polydisks

33.2.A Conditions for the existence of solutions to dbar f=u

33.2.B Hartogs phenomenon

33.3 The polydisk and ball algebras and beyond

33.4 The infinite polydisk algebra

33.5 The Bézout equation for entire functions

33.5.A Bézout equation for two generators

33.5.B Cartan’s Nullstellensatz for entire functions

33.5.C Stable ranks for H(Cn)

33.6 Subalgebras of A(D) via several complex variables

Chapter 34 Real-symmetric function algebras

34.1 Symmetric continuous functions

34.1.A Maximal ideals and the Bézout equation in C(X,tau)

34.1.B Asymmetric and -invariant sets

34.1.C The stable ranks for C(X,)

34.2 The algebra A( D)sym revisited

34.2.A Reducibility in A( D)sym

34.2.B The E-stable rank for A( D)sym

34.3 The real-symmetric algebra A(K)sym and its siblings

34.3.A Characters and ideals in real-symmetric function algebras

34.3.B Reducibility in real-symmetric function algebras

34.3.C Stable ranks of P(K)sym and R(K)sym

34.3.D The Bass stable rank for A(K)sym

34.4 Real-symmetric bounded analytic functions

Chapter 35 The algebra of almost periodic functions

35.1 Kronecker’s theorems in ergodic theory

35.2 Equidistributed sequences

35.3 Basic function theoretic properties of AP

35.4 Fourier-Bohr series

35.5 The Banach algebra AP

35.5.A The spectrum of AP

35.5.B The Bohr compactification as a group

35.5.C Interpolating sequences in AP

35.5.D The Bochner-Fejér operators

35.5.E Further algebraic properties of AP

35.6 AP+

35.6.A Function-theoretic properties of AP+

35.7 The stable ranks of AP and AP+

35.8 Subalgebras of AP

35.8.A Algebras of type AP-Lambda

35.8.B General subalgebras of AP

35.8.C Further examples of subalgebras of AP

Chapter 36 Matricial stable ranks

36.1 Some elementary matrix operations

36.2 Extension to invertible matrices

36.2.A The Hermite property

36.2.B The K-Hermite property

36.3 The stable ranks of non-commutative rings

36.3.A The Bass stable rank of matricial rings

36.3.B The topological stable rank of matricial rings

36.4 The Bézout equation in matricial rings

Part III The Appendix

Appendix A Some results in number theory and function theory

A.1 Frobenius numbers

A.2 An application of the Vandermonde matrix

A.3 Some trigonometric series

A.4 Absolute convergence of infinite products

A.5 An entire function with vanishing radial limits

A.6 A special Hölder-Lipschitz function

A.7 Conformal maps and the Riemann mapping theorem

A.8 The monodromy theorem

A.9 Convex sets in the plane

A.10 Branches of the inverse of cosh z

A.11 A brief glimpse on hyperbolic geometry

A.12 The chordal and spherical metrics

A.13 A criterion in real analysis for the existence of potentials

A.14 Poisson integral estimates

A.15 Some fascinating identities

A.16 A useful improper integral

A.17 Two elementary limits involving factorials

A.18 The Faà di Bruno formula

The Cantor sets

Appendix B The Lp( T)-spaces

Appendix C Matrix analysis

C.1 Differentiability and M-differentiability

C.2 Diagonalization and Elementary matrices

C.3 Exponentials and logarithms of matrices

C.4 Schur’s reduction theorem

C.5 Diagonal dominant matrices

Appendix D E-valued functions: integration and holomorphy

D.1 Riemann integrable functions with values in normed spaces

D.2 A-valued holomorphic functions

Appendix E Some tables

E.1 The Bass and topological stable ranks of various algebras/rings: a résumé

Appendix F Notes and Sources

Index

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