Applications to Rings of Quotients and C^∗-Algebras

Abstract

The focus of the final chapter of the book is on applications of the ideas and results developed in earlier chapters to Functional Analysis and Ring Theory. The chapter begins with the development of necessary and sufficient conditions on a ring so that its maximal right ring of quotients can be decomposed into a direct products of indecomposable rings or into a direct products of prime rings. This relies on results on the idempotent closure class of rings discussed in Chap. 8 and a dimension on bimodules introduced in this chapter. As an application, a quasi-Baer ring hull of a semiprime ring with only finitely many minimal prime ideals is shown to be a finite direct sum of prime rings. Conditions for a ∗-ring to become a Baer ∗-ring or a quasi-Baer ∗-ring are discussed. The quasi-Baer ∗-ring property is shown to transfer from a ring to its various polynomial extensions. Self-adjoint ideals in Baer ∗-rings and quasi-Baer ∗-rings are examined. In applications to the study of C ^∗-algebras, it is shown that a unital C ^∗-algebras A is boundedly centrally closed if and only if A is a quasi-Baer ring. The local multiplier algebra of a C ^∗-algebras is shown to be always quasi-Baer. Characterizations of C ^∗-algebras whose local multiplier algebras are C ^∗-direct products of prime C ^∗-algebras are provided. The quasi-Baer property is discussed for a C ^∗-algebras A with a finite group G of ^∗-automorphisms in terms of the skew group ring A ^∗ G and the fixed ring. Finally, C ^∗-algebras satisfying a polynomial identity with only finitely many minimal prime ideals are characterized.