Jaeseong Heo and Jeong Hee Kim Hanyang University, Hanyang University
Abstract : We prove that the reduced free product of $k \times k$ matrix algebras over abelian $C^*$-algebras is not the minimal tensor product of reduced free products of $k \times k$ 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 II$_1$-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.
Keywords : free product of $C^*$-algebras, Powers' group, minimal tensor product, stable rank 1, prime factor, property $T$, Cartan subalgebra
