Smooth Homogeneous Structures in Operator Theory

by Beltita; Daniel

• ISBN13: 9781584886174
• ISBN10: 158488617X
• Format: Hardcover
• Copyright: 2005-11-01
• Publisher: Chapman & Hall/

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

Author Biography

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

Table of Contents

Preface	vii
Introduction	xi
Topological Lie Algebras	3	(22)
Fundamentals	3	(3)
Universal enveloping algebras	6	(3)
The Baker-Campbell-Hausdorff series	9	(10)
Convergence of the Baker-Campbell-Hausdorff series	19	(6)
Notes	24	(1)
Lie Groups and Their Lie Algebras	25	(30)
Definition of Lie groups	25	(5)
The Lie algebra of a Lie group	30	(7)
Logarithmic derivatives	37	(2)
The exponential map	39	(4)
Special features of Banach-Lie groups	43	(12)
Notes	54	(1)
Enlargibility	55	(24)
Integrating Lie algebra homomorphisms	55	(9)
Topological properties of certain Lie groups	64	(10)
Enlargible Lie algebras	74	(5)
Notes	76	(3)
Smooth Homogeneous Spaces	79	(26)
Basic facts on smooth homogeneous spaces	79	(13)
Symplectic homogeneous spaces	92	(6)
Some homogeneous spaces related to operator algebras	98	(7)
Notes	102	(3)
Quasimultiplicative Maps	105	(22)
Supports, convolution, and quasimultiplicativity	105	(5)
Separate parts of supports	110	(11)
Hermitian maps	121	(6)
Notes	125	(2)
Complex Structures on Homogeneous Spaces	127	(20)
General results	127	(5)
Pseudo-Kahler manifolds	132	(5)
Flag manifolds in Banach algebras	137	(10)
Notes	143	(4)
Equivariant Monotone Operators	147	(24)
Definition of equivariant monotone operators	147	(4)
H*-algebras and L*-algebras	151	(6)
Equivariant monotone operators as reproducing kernels	157	(6)
H*-ideals of H*-algebras	163	(4)
Elementary properties of H*-ideals	167	(4)
Notes	169	(2)
L*-ideals and Equivariant Monotone Operators	171	(24)
From ideals to operators	171	(3)
From operators to ideals	174	(9)
Parameterizing L*-ideals	183	(6)
Representations of automorphism groups	189	(2)
Applications to enlargibility	191	(4)
Notes	193	(2)
Homogeneous Spaces of Pseudo-restricted Groups	195	(32)
Pseudo-restricted algebras and groups	195	(9)
Complex polarizations	204	(3)
Kahler polarizations	207	(6)
Admissible pairs of operator ideals	213	(5)
Some Kahler homogeneous spaces	218	(9)
Notes	224	(3)
A Differential Calculus and Smooth Manifolds	227	(28)
Locally convex spaces and algebras	227	(7)
Differential calculus and infinite-dimensional holomorphy	234	(8)
Smooth manifolds	242	(13)
B Basic Differential Equations of Lie Theory	255	(16)
C Topological Groups	271	(10)
References	281	(19)
Index	300