On Representability of *-Regular and Regular Involutive Rings in Endomorphism Rings of Vector Spaces

Niklas Niemann

ISBN 978-3-8325-1563-8
219 pages, year of publication: 2007
price: 40.50 €

The origins of von Neumann-regular rings and *-regular rings are the works of John von Neumann and F. J. Murray during the 30ies of the last century. They constitute a connection of the areas of operator theory, ring theory and lattice theory. Starting from this historical origin, one can speculate that von Neumann was inspired by both operator theory and lattice theory to introduce the notion of a *-regular ring: On the one hand, the requirement of positivity of the involution can be seen as the appropriate generalisation of the involution of operator algebras. On the other hand, *-regular rings give rise to a strong class of lattices which are closely connected to operator algebras.

This thesis shows that this speculation might have some substance, that is, the concept of a *-regular ring indeed gives an adequate axiomatic framework for regular rings of operators, if one is prepared to deal with vector spaces over general involutive skew fields, equipped with a scalar product. The main results of this thesis are that every *-regular ring is representable in this sense, and that every variety of *-regular rings is generated by its simple Artinian members.

Furthermore, the thesis deals with the larger class of regular involutive rings and questions of their representability. In the context of rings without involution, Jacobson proved that representability as subrings of endomorphism rings of vector spaces is captured by primitivity. Dealing with involutive rings, one can introduce the notion of *-primitivity and representations in terms of bi-vector spaces, as done by Rowen and Wiegandt. Alternatively, one can examine primitive rings endowed with an involution, with the aim to construct an appropriate non-degenerated form on the vector space to capture the involution. In this context, the thesis presents a continuation of previous research of Herrmann, Micol and Niemann. A complete characterisation of representability of regular involutive rings is given.