Open projections in operator algebras I: Comparison theory

David P. Blecher¹ ; Matthew Neal²

¹ Department of Mathematics University of Houston Houston, TX 77204-3008, U.S.A.
² Department of Mathematics Denison University Granville, OH 43023, U.S.A.

Studia Mathematica, Tome 208 (2012) no. 2, pp. 117-150

Résumé

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.

Détail

DOI : 10.4064/sm208-2-2

Keywords: begin program generalizing basic elements theory comparison equivalence subequivalence elements * algebras setting general algebras particular follow recent lead lin ortega rdam thiel studying these equivalences etc terms projections module isomorphisms define characterize class inner ideals operator algebras develop matching theory partial isometries operator ideals which simultaneously generalize projections operator algebras sense authors hay partial isometries tripotents introduced authors