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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.
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.
In several proofs from the theory of finite-dimensional Lie
algebras, an essential contribution comes from the Jordan canonical
structure of linear maps acting on finite-dimensional vector
spaces. On the other hand, there exist classical results concerning
Lie algebras which advise us to use infinite-dimensional vector
spaces as well. For example, the classical Lie Theorem asserts that
all finite-dimensional irreducible representations of solvable Lie
algebras are one-dimensional. Hence, from this point of view, the
solvable Lie algebras cannot be distinguished from one another,
that is, they cannot be classified. Even this example alone urges
the infinite-dimensional vector spaces to appear on the stage. But
the structure of linear maps on such a space is too little
understood; for these linear maps one cannot speak about something
like the Jordan canonical structure of matrices. Fortunately there
exists a large class of linear maps on vector spaces of arbi trary
dimension, having some common features with the matrices. We mean
the bounded linear operators on a complex Banach space. Certain
types of bounded operators (such as the Dunford spectral, Foia
decomposable, scalar generalized or Colojoara spectral generalized
operators) actually even enjoy a kind of Jordan decomposition
theorem. One of the aims of the present book is to expound the most
important results obtained until now by using bounded operators in
the study of Lie algebras."
In several proofs from the theory of finite-dimensional Lie
algebras, an essential contribution comes from the Jordan canonical
structure of linear maps acting on finite-dimensional vector
spaces. On the other hand, there exist classical results concerning
Lie algebras which advise us to use infinite-dimensional vector
spaces as well. For example, the classical Lie Theorem asserts that
all finite-dimensional irreducible representations of solvable Lie
algebras are one-dimensional. Hence, from this point of view, the
solvable Lie algebras cannot be distinguished from one another,
that is, they cannot be classified. Even this example alone urges
the infinite-dimensional vector spaces to appear on the stage. But
the structure of linear maps on such a space is too little
understood; for these linear maps one cannot speak about something
like the Jordan canonical structure of matrices. Fortunately there
exists a large class of linear maps on vector spaces of arbi trary
dimension, having some common features with the matrices. We mean
the bounded linear operators on a complex Banach space. Certain
types of bounded operators (such as the Dunford spectral, Foia
decomposable, scalar generalized or Colojoara spectral generalized
operators) actually even enjoy a kind of Jordan decomposition
theorem. One of the aims of the present book is to expound the most
important results obtained until now by using bounded operators in
the study of Lie algebras."
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