
Banach Algebras and Harmonic Analysis
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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
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