# Journal of Operator Theory

Volume 33, Issue 2, Spring 1995 pp. 201-221.

K-theory for certain reduced free products of C*-algebras**Authors**: Kevin McClanahan

**Author institution:**Department of Mathematics, University of Mississippi, University, MS 38677, U.S.A.

**Summary:**A six-term exact sequence of K-groups for certain reduced free products of C*-algebras is derived. This sequence is used to show that ${\rm{K}}_{\rm{*}} (C^{\rm{*}} (G)*_\mathbb C^{{\rm{red}}} )M_n$ is isomorphic to ${\rm{K}}_{\rm{*}} (C_{{\rm{red}}}^{\rm{*}} (G))$ for any countable discrete group satisfying property $\Lambda$ of Lance. This result is then used to compute the K-groups of the reduced noncommutative unitary C*-algebra $U_{n,{\rm{red}}}^{{\rm{nc}}}$ and the reduced noncommutative Grassmanian C*-algebra $G_{n,{\rm{red}}}^{{\rm{nc}}}$. It is shown that if G is a nontrivial countable discrete group with property $\Lambda$ such that the range of the homomorphism from ${\rm{K}}_0 (C_{{\rm{red}}}^{\rm{*}} (G))$ into $\mathbb R$ induced by the usual trace on $C_{{\rm{red}}}^{\rm{*}} (G)$ is contained in $\frac{1}{n}\mathbb Z$, then the relative commutant of M_n in $C^{\rm{*}} (G)*_\mathbb C^{{\rm{red}}} M_n$ is a simple projectionless C*-algebra.

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