C*-algebras Arising from Group Actions on the Boundary of a Triangle Building
نویسندگان
چکیده
A subgroup of an amenable group is amenable. The C*-algebra version of this fact is false. This was first proved by M.-D. Choi [9] who proved that the non-nuclear C*-algebra C*(Z2 * Z3) is a subalgebra of the nuclear Cuntz algebra €2. A. Connes provided another example, based on a crossed product construction. More recently J. Spielberg [23] showed that these examples were essentially the same. In fact he proved that certain of the C*-algebras studied by J. Cuntz and W. Krieger [10] can be constructed naturally as crossed product algebras. For example, if the group F acts simply transitively on a homogeneous tree of finite degree with boundary Q then C(Q) X F is a Cuntz-Krieger algebra. Such trees may be regarded as affine buildings of type Ax. The present paper is devoted to the study of the analogous situation where a group F acts simply transitively on the vertices of an affine building of type A2 with boundary Q [8]. The corresponding crossed product algebra C(Q) X F is then generated by two Cuntz-Krieger algebras. (See § 3.) Moreover, we show that C(Q) X F is simple and nuclear. This is a consequence of the facts that the action of F on Q is minimal, topologically free, and amenable. (See § 4.)
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