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 Kahler 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 Beltita is a Principal Researcher at the Institute of Mathematics "Simion Stoilow" of the Romanian Academy, Bucharest, Romania.
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-Kahler 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 Kahler polarizations Admissible pairs of operator ideals Some Kahler homogeneous spaces Notes APPENDICES Differential Calculus and Smooth Manifolds Basic Differential Equations of Lie Theory Topological Groups References Index