Equivalence and traces on C∗-algebras

AbstractWe introduce an equivalence relation among the positive elements in a C∗ and show that the algebra is (semi-) finite if and only if there is a separating family of (semi-) finite traces. Concentrating on simple, semi-finite C∗-algebras we relate geometrical properties in the cone of equivalence classes to functional analytic properties of the algebra, such as the number of normalized traces and their possible values on a given element. The paper may be considered as an attempt to extend Murray and von Neumann's type and equivalence theory to C∗-algebras