Simplicity of the C∗-algebra associated with the free group on two generators

C*-algebras associated with amalgamated products of groups

“…In this paper, we show that C*(G) is simple and has a unique tracial state if G e V, thus generalizing the results of [7] and [6] except when G = G X *G 2 where G x or G 2 only has elements of order 1 or 2. Related work for other classes of groups is treated in [1], [2].…”

“…Hence C*(G) is nonnuclear by Theorem 4.2 of [4]. As in [7] and [6], parts (ii) and (iii) of Theorem 3.1 are direct consequences of the following two lemmas, the first of which is a variant of Lemma 2.1 in [3] with G e V. …”

“…the symmetric group on three objects, then the free product of A and B with the cyclic group generated by c and d amalgamated is a group in V. Let C*(G) denote the C*-algebra generated by the left regular representation of a discrete group G. If G = Z * Z, Z 2 * Z 3 or G x * G 2> where Z is the infinite cyclic group, Z 2 the cyclic group of order 2, Z 3 the cyclic group of order 3, and G lt G 2 are not both of order 2, then it is known that C*(G) is simple and has a unique tracial state ( [7], [6], [3]). In this paper, we show that C*(G) is simple and has a unique tracial state if G e V, thus generalizing the results of [7] and [6] except when G = G X *G 2 where G x or G 2 only has elements of order 1 or 2.…”

“…The proof of the theorem above is based on the techniques of [7], [6], [3] and generalizes [7] and [6] except for groups of the form A * B, where A or B only has elements of order 1 or 2.…”

Pure States on Free Group C*-Algebras

“…[11,Corollary 6.7], where it is shown that C * r (F R ) is inseparable, but that every abelian subalgebra is separable. Powers [12] showed that for Card(R) = 2, C * r (F R ) is simple and has unique trace. Powers' method extends to general R. For general free products of groups, simplicity and uniqueness of trace follow by results of Avitzour [7].…”