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 Some reduced free products of abelian $C^*$-algebras and some $C^*$-subalgebra in a free product Bull. Korean Math. Soc. 2010 Vol. 47, No. 5, 997-1010 https://doi.org/10.4134/BKMS.2010.47.5.997Published online September 1, 2010 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 MSC numbers : Primary 46L09, Secondary 46L06 Downloads: Full-text PDF