# Journal of Operator Theory

Volume 68, Issue 1, Summer 2012 pp. 275-302.

Reduced free products of unital AH algebras and Blackadar and Kirchberg's MF algebras**Authors**: Don Hadwin (1), Jiankui Li (2), Junhao Shen (3), and Liguang Wang (4)

**Author institution:**(1) Department of Mathematics and Statistics, University of New Hampshire, Durham, NH, U.S.A.

(2) Department of Mathematics, East China University of Science and Technology, Shanghai, China

(3) Department of Mathematics and Statistics, University of New Hampshire, Durham, NH, U.S.A.

(4) Department of Mathematics, Qufu Normal University, Qufu, Shangdong, China

**Summary:**In the paper, we prove that reduced free products of unital AH algebras with respect to given faithful tracial states, in the sense of Voiculescu, are Blackadar and Kirhcberg's MF algebras. We also show that reduced free products of unital AH algebras with respect to given faithful tracial states, under mild conditions, are not quasidiagonal. Therefore we conclude, for a large class of AH algebras, that the Brown-Douglas-Fillmore extension semigroups of the reduced free products of these AH algebras with respect to given faithful tracial states are not groups. Our result is based on Haagerup and Thorbjornsen's work on the reduced $C^*$-algebras of free groups.

**Keywords:**MF algebras, reduced free products, BDF semigroups

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