| In this dissertation, we define two new classes of operator algebras; matricial operator algebras and scattered operator algebras. The C*-algebras of compact operators play an important role in C*-algebra theory, and they are widely used in mathematical physics and quantum mechanics. We define 1-matricial algebras using a sequence of invertible operators on a Hilbert space, and sigma-matricial algebras are c0-sums of 1-matricial algebras. These operator algebras, in some sense, generalize the class of C*-algebras of compact operators to a non-selfadjoint setting. They possess many properties similar to the properties of the C*-algebras of compact operators. We present a 'Wedderburn type' structure theorem that characterizes the sigma-matricial algebras. We define scattered operator algebras using a composition series where each consecutive quotient is a 1-matricial algebra. Note that a C*-algebra is scattered if and only if it has a composition series where each consecutive quotient is a C*-algebra of compact operators. Hence, our definition of scattered operator algebras is quite natural. We present many results on the structure of the scattered operator algebras and show that they have some properties generalizing the properties of scattered C*-algebras. For example, the dual of a scattered operator algebra has the Radon-Nikodym property and scattered operator algebras are Asplund spaces. Working with a composition series requires us to develop some tools for general operator algebras, and in particular, quotient operator algebras. For example we utilize frequently the isomorphism theorems and a correspondence theorem for operator algebras; as well as the results about the structure of the diagonal of a quotient operator algebra. |
