Smooth Homogeneous Structures in Operator Theory: 1st Edition (Hardback) book cover

Smooth Homogeneous Structures in Operator Theory

1st Edition

By Daniel Beltita

Chapman and Hall/CRC

320 pages | 50 B/W Illus.

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pub: 2005-11-01
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Description

Geometric ideas and techniques play an important role in operator theory and the theory of operator algebras. Smooth Homogeneous Structures in Operator Theory builds the background needed to understand this circle of ideas and reports on recent developments in this fruitful field of research.

Requiring only a moderate familiarity with functional analysis and general topology, the author begins with an introduction to infinite dimensional Lie theory with emphasis on the relationship between Lie groups and Lie algebras. A detailed examination of smooth homogeneous spaces follows. This study is illustrated by familiar examples from operator theory and develops methods that allow endowing such spaces with structures of complex manifolds. The final section of the book explores equivariant monotone operators and Kähler structures. It examines certain symmetry properties of abstract reproducing kernels and arrives at a very general version of the construction of restricted Grassmann manifolds from the theory of loop groups.

The author provides complete arguments for nearly every result. An extensive list of references and bibliographic notes provide a clear picture of the applicability of geometric methods in functional analysis, and the open questions presented throughout the text highlight interesting new research opportunities.

Daniel Beltitâ is a Principal Researcher at the Institute of Mathematics "Simion Stoilow" of the Romanian Academy, Bucharest, Romania.

Table of Contents

TOPOLOGICAL LIE ALGEBRAS

Fundamentals

Universal enveloping algebras

The Baker-Campbell-Hausdor series

Convergence of the Baker-Campbell-Hausdor series

Notes

LIE GROUPS AND THEIR LIE ALGEBRAS

Definition of Lie groups

The Lie algebra of a Lie group

Logarithmic derivatives

The exponential map

Special features of Banach-Lie groups

Notes

ENLARGIBILITY

Integrating Lie algebra homomorphisms

Topological properties of certain Lie groups

Enlargible Lie algebras

Notes

Smooth Homogeneous Spaces

Basic facts on smooth homogeneous spaces

Symplectic homogeneous spaces

Some homogeneous spaces related to operator algebras

Notes

QUASIMULTIPLICATIVE MAPS

Supports, convolution, and quasimultiplicativity

Separate parts of supports

Hermitian maps

Notes

COMPLEX STRUCTURES ON HOMOGENEOUS SPACES

General results

Pseudo-Kähler manifolds

Flag manifolds in Banach algebras

Notes

EQUIVARIANT MONOTONE OPERATORS

Definition of equivariant monotone operators

H*-algebras and L*-algebras

Equivariant monotone operators as reproducing kernels

H*-ideals of H*-algebras

Elementary properties of H*-ideals

Notes

L*-IDEALS AND EQUIVARIANT MONOTONE OPERATORS

From ideals to operators

From operators to ideals

Parameterizing L*-ideals

Representations of automorphism groups

Applications to enlargibility

Notes

HOMOGENEOUS SPACES OF PSEUDO-RESTRICTED GROUPS

Pseudo-restricted algebras and groups

Complex polarizations

Kähler polarizations

Admissible pairs of operator ideals

Some Kähler homogeneous spaces

Notes

APPENDICES

Differential Calculus and Smooth Manifolds

Basic Differential Equations of Lie Theory

Topological Groups

References

Index

About the Series

Monographs and Surveys in Pure and Applied Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General
MAT012000
MATHEMATICS / Geometry / General
MAT037000
MATHEMATICS / Functional Analysis