SOME REDUCED FREE PRODUCTS OF ABELIAN C*-ALGEBRAS AND SOME C*-SUBALGEBRA IN A FREE PRODUCT

Abstract

We prove that the reduced free product of k x k matrix algebras over abelian C*-algebras is not the minimal tensor product of reduced free products of k x k matrix algebras over abelian C*-algebras. It is shown that the reduced group C*-algebra associated with a group having the property T of Kazhdan is not isomorphic to a reduced free product of abelian C*-algebras or the minimal tensor product of such reduced free products. The infinite tensor product of reduced free products of abelian C*-algebras is not isomorphic to the tensor product of a nuclear C*-algebra and a reduced free product of abelian C*-algebra. We discuss the freeness of free product Ill-factors and solidity of free product II(1)-factors weaker than that of Ozawa. We show that the freeness in a free product is related to the existence of Cartan subalgebras in free product II(1)-factors. Finally, we give a free product factor which is not solid in the weak sense.