Banach Algebras and Harmonic Analysis
Description
The book includes recent articles on various topics studied recently in Banach algebras and abstract harmonic analysis. On the Banach algebra side, the reader will find idempotents and socle of a Banach algebra; closed subspaces of a Banach space, including the classical sequence spaces, which are realized as the kernel of a bounded operator; the connection between the stable rank one and Dedekind-finite property of the algebra of operators on a Banach space; spectral synthesis properties in convolution Sobolev algebras on the real line.
The harmonic analysis side includes a generalization of the famous Beurling theorem; groups with few finite-dimensional unitary representations; relations between ideals of the Figa-Talamanca Herz algebra of a locally compact group and ideals of Figa-Talamanca Herz algebra of its closed subgroup; the tame functionals on Banach algebras and in harmonic analysis; and cancellation, factorization, and isometries in algebras on a locally compact group.
The book also includes four surveys written by leaders in the area of full sheaf cohomology theory for noncommutative C - algebras; one-parameter semigroups of bounded operators on a Banach space which are weakly continuous in the sense of Arveson.
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Mahmoud Filali obtained his Phd in 1990 at the university of Sheffield, UK. He has since then been working at the university of Oulu holding different positions; currently he is a lecturer. His research focuses on Banach algebras, semigroup compactifications and harmonic analysis. He has supervised four Phds at his home university and co-supervised one Phd outside his university. He is the editor of a number of journals including his own university journal Acta Universitatis Ouluensis. He has published over sixty articles, co-authored one book and edited two conference proceedings.
Content
- Intro
- Preface
- Contents
- Kernels of bounded operators on the classical transfinite Banach sequence spaces
- 1 Introduction and main results
- 2 Preliminaries and the proof of Theorem 1.2
- 3 Proof of Theorem 1.3
- 4 Wark's Banach space admits a closed subspace that is not the kernel of any bounded operator on it
- Bibliography
- Beurling's theorem on locally compact Abelian groups
- 1 Introduction
- 2 The theorem
- Bibliography
- On idempotents and the socle of a Banach algebra
- 1 Introduction
- 2 Idempotents and the socle
- Bibliography
- Fréchet algebras with a dominating Hilbert algebra norm
- 1 Introduction
- 2 Preliminaries
- 3 Fréchet subalgebras of L*(s)
- 4 Examples
- Bibliography
- Relations between ideals of the Figà-Talamanca Herz algebra Ap(G) of a locally compact group G and ideals of Ap(H) of a closed subgroup
- 1 Introduction
- 2 On the multipliers of Ap(G)
- 3 Ideals with bounded approximate unit
- 4 The image of a closed ideal under the restriction map
- Bibliography
- The composition of conditional expectation and multiplication operators
- 1 Introduction and preliminaries
- 2 Centered and normal weighted conditional type operators
- 3 Applications
- Bibliography
- On the generation of Arveson weakly continuous semigroups
- 1 Introduction
- 2 Arveson weakly continuous one-parameter semigroups
- 3 The Arveson ideal
- 4 The generator as a weakly densely defined operator
- 5 Quasimultipliers on the Arveson ideal and Arveson spectrum
- 6 Quasimultipliers on the Arveson ideal as weakly densely defined operators
- 7 The normalized Arveson ideal
- 8 The generator as a quasimultiplier on the Arveson ideal
- 9 The Arveson spectrum
- 10 The resolvent
- 11 Holomorphic functional calculus
- Bibliography
- The semigroup algebra of a foundation semigroup with locally convex topologies and its Arens regularity
- 1 Introduction
- 2 Preliminaries
- 3 Dual of Ma(S) with a locally convex topology
- 4 The second dual of Ma(S) with a locally convex and the topological center of (Ma(S),tLUC)**
- Bibliography
- Bohr compactification and Chu duality of non-Abelian locally compact groups
- 1 Introduction
- 2 Preliminaries
- 3 Self-Bohrifying groups
- 4 Bohr compactification versus Chu duality
- Bibliography
- l1-bases in Banach algebras and Arens irregularities in harmonic analysis
- 1 Introduction
- 2 Some basic definitions and notation
- 2.1 The space LUC(A)
- 3 Non-Arens regularity
- 3.1 What is known on non-Arens regularity in harmonic analysis
- 3.2 The definitions
- 4 Extreme non-Arens regularity
- 4.1 What is known on extreme non-Arens regularity
- 4.2 The new approach: triangles and l1-bases
- 4.3 l1(?)-bases in A
- 4.4 EnArity of general Banach algebras
- 4.5 Applications: algebras with bai's
- 4.6 Algebras with weak TI's
- 4.7 Algebras in harmonic analysis I
- 4.8 EnArity of preduals of a von Neuman algebras
- 4.9 Algebras in harmonic analysis II
- 5 On the strong Arens irregularity of Banach algebras
- 5.1 What is known on strong Arens irregularity and topological centers
- 5.2 The new approach: Fl1(?)-bases and factorizations
- 5.3 Fl1(?)-bases
- 6 Examples
- 6.1 Group algebra and measure algebra
- 6.2 Fourier algebra and Fourier-Stieltjes algebra
- 7 The algebraic structure of A**, LUC(A)*, and M(A)**
- Bibliography
- Right cancellation, factorization, and right isometries
- 1 Introduction
- 2 Results
- Bibliography
- On spectral synthesis in convolution Sobolev algebras on the real line
- 1 Introduction
- 2 Derivatives, convolution, and Fourier transform
- 3 Weak spectral synthesis in Sobolev algebras
- 4 Approximation by ideals of spectral synthesis
- Bibliography
- Projective and free matricially normed spaces
- 1 Introduction
- 2 Initial definitions
- 3 Construction of the matricially normed spaces M"310Mn
- 4 Projectivity and freeness in rigged categories
- 5 The rig n and its free objects with the one-point base
- 6 Characterization of free and projective spaces
- 7 Some properties of the matricially normed space M"310Mn
- Bibliography
- Banach spaces whose algebras of operators are Dedekind-finite but they do not have stable rank one
- 1 Introduction and basic terminology
- 2 Algebras of operators with stable rank one and their connection to Dedekind-finiteness
- Bibliography
- Invariant complementation property and fixed-point sets on power bounded elements in the group von Neumann algebra
- 1 Introduction
- 2 Some preliminaries
- 3 Invariant complementation property of the group von Neumann algebra
- 4 Fixed-point sets of power bounded elements in VN(G)
- 5 Some remarks and open problems
- 5.1 Invariant complementation property
- 5.2 Natural projections
- 5.3 Geometric form of the Hahn-Banach separation theorem
- 5.4 Sets of spectral synthesis
- Bibliography
- Subspaces that can and cannot be the kernel of a bounded operator on a Banach space
- 1 Introduction
- 2 Results
- Bibliography
- Towards a sheaf cohomology theory for C*-algebras
- 1 Introduction
- 2 Topology
- 3 Exact categories
- 4 Sheaves
- 5 Our kind of categories
- 6 OMod8A(X) is exact
- 7 Homological algebra in OMod8A(X)
- Bibliography
- Tame functionals on Banach algebras
- 1 Introduction
- 2 Some definitions: independence and tameness
- 2.1 Independence and tame systems
- 2.2 Tame functionals
- 3 Tame functionals on the group algebra l1(G)
- 3.1 Tame functions on groups
- 4 Appendix
- 4.1 Background on fragmentability and tame families
- 4.2 Independent sequences of functions
- Bibliography
- Left ideals of Banach algebras and dual Banach algebras
- 1 Introduction
- 2 Preliminaries
- 3 Examples from abstract harmonic analysis
- 4 Left and right ideals of approximable operators on a Banach space
- 5 Multiplier algebras and dual Banach algebras
- 6 Some classification results
- Bibliography
- Index
