Generalized quasi-Baer -rings and Banach -algebras

Abstract

We say that a -ring R is a generalized quasi-Baer -ring if for any ideal I of the right annihilator of is generated, as a right ideal, by a projection, for some positive integer n depending on I. A unital -ring R is left primary if and only if R is a generalized quasi-Baer -ring with no nontrivial central projections. We study basic properties of such rings and we prove their permanence properties such as the Morita invariance. We show that this notion is well-behaved with respect to polynomial extensions and certain triangular matrix extensions and group rings. A sheaf representation for such -rings is also proved. We obtain algebraic examples which are generalized quasi-Baer -rings but are not quasi-Baer -rings. We show that for pre-C*-algebras these latter two notions are equivalent. We obtain classes of both finite and infinite dimensional Banach -algebras which are generalized quasi-Baer *-rings but are not quasi-Baer *-rings. In particular, they do not admit any C*-norms. As applications, we show that for a locally compact abelian group G, the group algebra is a (generalized) quasi-Baer -ring, if and only if so is the group C*-algebra if and only if G is finite, and we prove that the Leavitt path algebra of a finite directed graph E with coefficients in a field K is a (generalized) quasi-Baer *-ring if E is downward directed or a no-exit graph.