SOME REDUCED FREE PRODUCTS OF ABELIAN C*

Title & Authors
SOME REDUCED FREE PRODUCTS OF ABELIAN C*
Heo, Jae-Seong; Kim, Jeong-Hee;

Abstract
We prove that the reduced free product of $\small{k\;{\times}\;k}$ matrix algebras over abelian $\small{C^*}$-algebras is not the minimal tensor product of reduced free products of $\small{k\;{\times}\;k}$ matrix algebras over abelian $\small{C^*}$-algebras. It is shown that the reduced group $\small{C^*}$-algebra associated with a group having the property T of Kazhdan is not isomorphic to a reduced free product of abelian $\small{C^*}$-algebras or the minimal tensor product of such reduced free products. The infinite tensor product of reduced free products of abelian $\small{C^*}$-algebras is not isomorphic to the tensor product of a nuclear $\small{C^*}$-algebra and a reduced free product of abelian $\small{C^*}$-algebra. We discuss the freeness of free product $\small{II_1}$-factors and solidity of free product $\small{II_1}$-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 $\small{II_1}$-factors. Finally, we give a free product factor which is not solid in the weak sense.
Keywords
free product of $\small{C^*}$-algebras;Powers` group;minimal tensor product;stable rank 1;prime factor;property T;Cartan subalgebra;
Language
English
Cited by
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