Jordan, Real And Lie Structures In Operator Algebras (mathematics And Its Applications)
by Sh. Ayupov /
1997 / English / PDF
8.2 MB Download
The theory of operator algebras acting on a Hilbert space was
initiated in thirties by papers of Murray and von Neumann. In these
papers they have studied the structure of algebras which later were
called von Neu mann algebras or W* -algebras. They are weakly
closed complex *-algebras of operators on a Hilbert space. At
present the theory of von Neumann algebras is a deeply developed
theory with various applications. In the framework of von Neumann
algebras theory the study of fac tors (i.e. W* -algebras with
trivial centres) is very important, since they are comparatively
simple and investigation of general W* -algebras can be reduced to
the case of factors. Therefore the theory of factors is one of the
main tools in the structure theory of von Neumann algebras. In the
middle of sixtieth Topping [To 1] and Stormer [S 2] have ini
tiated the study of Jordan (non associative and real) analogues of
von Neumann algebras - so called JW-algebras, i.e. real linear
spaces of self adjoint opera.tors on a complex Hilbert space,
which contain the identity operator 1. closed with respect to the
Jordan (i.e. symmetrised) product INTRODUCTION 2 x 0 y = ~(Xy + yx)
and closed in the weak operator topology. The structure of these
algebras has happened to be close to the struc ture of von Neumann
algebras and it was possible to apply ideas and meth ods similar
to von Neumann algebras theory in the study of JW-algebras.
The theory of operator algebras acting on a Hilbert space was
initiated in thirties by papers of Murray and von Neumann. In these
papers they have studied the structure of algebras which later were
called von Neu mann algebras or W* -algebras. They are weakly
closed complex *-algebras of operators on a Hilbert space. At
present the theory of von Neumann algebras is a deeply developed
theory with various applications. In the framework of von Neumann
algebras theory the study of fac tors (i.e. W* -algebras with
trivial centres) is very important, since they are comparatively
simple and investigation of general W* -algebras can be reduced to
the case of factors. Therefore the theory of factors is one of the
main tools in the structure theory of von Neumann algebras. In the
middle of sixtieth Topping [To 1] and Stormer [S 2] have ini
tiated the study of Jordan (non associative and real) analogues of
von Neumann algebras - so called JW-algebras, i.e. real linear
spaces of self adjoint opera.tors on a complex Hilbert space,
which contain the identity operator 1. closed with respect to the
Jordan (i.e. symmetrised) product INTRODUCTION 2 x 0 y = ~(Xy + yx)
and closed in the weak operator topology. The structure of these
algebras has happened to be close to the struc ture of von Neumann
algebras and it was possible to apply ideas and meth ods similar
to von Neumann algebras theory in the study of JW-algebras.