From the author’s introduction: Let and be two -algebras, a completely positive map from to . The map gives a Hilbert -module and a *-homomorphism from into the -algebra of all bounded -module maps with adjoints on . In the case that , it is well known that is pure if and only if is irreducible. It is natural to ask whether it is also true for general -algebras .
In this note we give a negative answer to the problem in general. We also show that for many -algebras , the maps are never pure and are never irreducible. However, for some -algebras , the purity of does imply the irreducibility of and for some -algebras , the irreducibility of implies is pure.