Smooth Homogeneous Structures in Operator Theory
Features
Summary
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
