Order properties of generalized projections
Anil Khairnar, B. N. Waphare · Linear and Multilinear Algebra · 2016
In this paper, we introduce a concept of a generalized projection in a -ring (ring with an involution). A generalized projection is a self-adjoint element a such that for some integer . We also introduce a partial order on the set of generalized projections which is a generalization of the partial order on the set of projections. We prove that, the set of generalized projections GP(R) of a Rickart -ring R forms a lattice. We characterize Baer -rings in terms of completeness of the lattice of generalized projections. A sufficient condition for a Rickart -ring R is given so that the lattice of GP(R) is complemented. The concept of generalized comparability is extended to involve all elements of GP(R) of a -ring R. Further, we characterize generalized comparability of generalized projections in terms of their orthogonal decomposition.