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http://dx.doi.org/10.4134/BKMS.2010.47.5.997

SOME REDUCED FREE PRODUCTS OF ABELIAN C*  

Heo, Jae-Seong (DEPARTMENT OF MATHEMATICS RESEARCH INSTITUTE FOR NATURAL SCIENCES HANYANG UNIVERSITY)
Kim, Jeong-Hee (DEPARTMENT OF MATHEMATICS RESEARCH INSTITUTE FOR NATURAL SCIENCES HANYANG UNIVERSITY)
Publication Information
Bulletin of the Korean Mathematical Society / v.47, no.5, 2010 , pp. 997-1000 More about this Journal
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;
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