Projections in free product C*-algebras

TL;DR

This paper investigates the structure of projections in free product C*-algebras, establishing conditions under which the K_0-group is weakly unperforated or the algebra is properly infinite, depending on the nature of the states involved.

Contribution

It provides new criteria for the positivity and infiniteness of projections in free product C*-algebras based on state properties and Avitzour conditions.

Findings

K_0-group is weakly unperforated under certain conditions.

A is properly infinite if states are not traces and weaker conditions hold.

Characterizes projections in free product C*-algebras based on state faithfulness and nuclearity.

Abstract

Consider the reduced free product of C*-algebras, (A,\phi)=(A_1,\phi_1)*(A_2,\phi_2), with respect to states \phi_1 and \phi_2 that are faithful. If \phi_1 and \phi_2 are traces, if the so-called Avitzour conditions are satisfied, (i.e. A_1 and A_2 are not ``too small'' in a specific sense) and if A_1 and A_2 are nuclear, then it is shown that the positive cone of the K_0-group of A consists of those elements g in K_0(A) for which g=0 or K_0(\phi)(g)>0. Thus, the ordered group K_0(A) is weakly unperforated. If, on the other hand, \phi_1 or \phi_2 is not a trace and if a certain condition weaker than the Avitzour conditions hold, then A is properly infinite.