Beltita Smooth Homogeneous Structures in Operator Theory

Erscheinungsjahr 2010
ISBN: 978-1-4200-3480-6
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 320 Seiten

Reihe: Monographs and Surveys in Pure and Applied Mathematics

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.

Zielgruppe

Graduate students and researchers in functional analysis, Lie groups, representation theory, and complex geometry

Autoren/Hrsg.

Beltita, Daniel

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein

Inhaltsverzeichnis

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