Mappings on triangular algebras




Cheung, Wai-Shun

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In this dissertation, we study certain types of linear mappings on triangular algebras. Triangular algebras are algebras whose elements can be written in the form of 2 x 2 matrices [special characters omitted]where a ∈ A, b ∈ B, m ∈ M and where A, B are algebras and M is a bimodule. Many widely studied algebras, such as upper triangular matrix algebras and nest algebras, can be viewed as triangular algebras. This dissertation is divided into five chapters. The first chapter is a general account of the basics of triangular algebras, including the unitization of nonunital triangular algebras and the structure of the centre of triangular algebras, as well as a brief introduction to some well-known examples of triangular algebras. In Chapter 2, we study the general structure of derivations on triangular algebras and obtain some results on the first cohomology groups of triangular algebras. The first cohomology group of an algebra is the quotient space of the space of all derivations over the space of all inner derivations, and it is always a main tool in the research of derivations. In addition, we consider the problem of automatic continuity of derivations in the last section of this chapter. In Chapter 3, we consider sufficient conditions on a triangular algebra so that every Lie derivation is a sum of a derivation and a linear map whose image lies in the centre of the triangular algebra. In Chapter 4, we consider sufficient conditions for every commuting map on a triangular algebra to be a sum of a map of the form x ↦ ax and a map whose image lies in the centre of the triangular algebra. In the final chapter, we are concerned with the automorphisms of triangular algebras. The study of automorphism is a most important way to understand the underlying structure of an algebra. We deduce some results on the Skolem-Noether groups, or the outer automorphism groups, of triangular algebras and apply those results to generalize some known results about automorphisms on a triangular matrix algebras.



Algebras, Linear