Researches on the classification of von Neumann algebras by selfdual cones and operator inequalities

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We considered the question how the algebraic structure of a operator algebra determines or is determined by the structure of its underlying Hilbert space. In order to give the answer to this problem, we have introduced the notion of the complete order isomorphisms and complete orthogonal decomposition isomorphisms on the matrix ordered Hilbert space with a family of selfdual cones. Matrix ordered Hilbert spaces are the appropriate objects to which completely positive maps apply and enabled us to handle non-commutative order. In this research, we have proven that a completely positive projection on the Hilbert space induces a conditional expectation on a von Neumann algebra, and a complete order isomorphism induces an automorphism of the von Neumann algebra. We have then investigated the relationship between the type of von Neumann algebras and completely positive maps, and characterized the matrix ordered standard forms of finite von Neumann algebras.We have also introduced the notion … More of the order of operators on a Hilbert space based on the positivity with respect to the selfdual cone. The different point of this order from the usual order of operators is the compatibility with the products. We have investigated the fundamental properties on the operator inequalities with respect to this order, and sharpened the arithmetic and geometric mean inequality in the category of the ordered Hilbert space associated with a finite dimensional commutative von Neumann algebra.Furthermore, we have obtained some results in the geometry and the number theory. One of them is the investigation of the relationship between the sphere theorem and the Dehn lemma. The rank of the group of relative units of a finite Galois extension of the algebraic number field has then been calculated. It is necessary to determine the finite groups all whose abelian subgroups are cyclic. We have described by cyclotomic polynomials quite simply a mechanical system of one-dimensional hard core chain which consists of infinite many particles with two different alternating mass Finally, we have generalized and given very generally applicable proof of the formula of Grant, which is a generalization of product formulae for division values of periods of the exponential function or each elliptic function with complex multiplication in a cyclotomic field, to every hyperelliptic function of cyclotomic type. Less