Open projections in operator algebras I: Comparison theory
Volume 208 / 2012
Studia Mathematica 208 (2012), 117-150
MSC: Primary 46L85, 46H10, 46L07, 47L30, 46H30; Secondary 06F25, 17C65, 46L08, 47L07.
DOI: 10.4064/sm208-2-2
Abstract
We begin a program of generalizing basic elements of the theory of comparison, equivalence, and subequivalence, of elements in C$^*$-algebras, to the setting of more general algebras. In particular, we follow the recent lead of Lin, Ortega, Rørdam, and Thiel of studying these equivalences, etc., in terms of open projections or module isomorphisms. We also define and characterize a new class of inner ideals in operator algebras, and develop a matching theory of open partial isometries in operator ideals which simultaneously generalize the open projections in operator algebras (in the sense of the authors and Hay), and the open partial isometries (tripotents) introduced by the authors.
