One-sided M-structure of operator spaces and operator algebras.

摘要

In this thesis, we study the structure and properties of operator spaces and operator algebras which have a one-sided M-structure. We develop a non-commutative theory of operator spaces which are one-sided M-ideals in their bidual. We also investigate the M-ideal structure of the Haagerup tensor product of operator algebras. Further, we consider operator algebras which are, in some sense, a generalization of the algebra of the compact operators. These are called the '1-matricial algebras'. Using the Haagerup tensor product and the 1-matricial algebra, we construct a variety of examples of operator spaces and operator algebras which are one-sided M-ideals in their bidual. In the last part of the thesis, we look at operator spaces over the field of real numbers and generalize a small portion of the existing theory of complex operator spaces. In particular, we show that the injective envelope and C*-envelope for real operator spaces exist. We also briefly consider real operator algebras and their complexification.