Annihilators and Type Decomposition in C*-Algebras

Abstract

We initiate the study of annihilators in C*-algebras, showing that they are, in many ways, the best C*-algebra analogs of projections in von Neumann algebras. Using them, we obtain a type decomposition for arbitrary C*-algebras that is symmetric and completely consistent with the classical von Neumann algebra type decomposition. We also show that annihilators admit a very simple notion of equivalence that is again completely consistent with the notion of Murray-von Neumann equivalence in von Neumann algebras, sharing many of its general order theoretic properties.