Stable Rank of Some Full Group C*-Algebras of Groups Obtained by the Free Product

Abstract

We compute the real rank and the stable rank of full group C*-algebras. Main result is (i) rr (C*(Fn)) = ∞, (ii) sr (C*(G1 * G2)) = ∞(|G1| ≥ 2, |G2| ≥ 2 and |G1| + |G2| ≥ 5), (iii) sr (C*(G1 * G2)) = 1(|G1| = |G2| = 2), where Fn is the free group with n generators, G1 and G2 are finite groups and |G| means the order of the group G.

On the real rank of $C^\ast$-algebras of nilpotent locally compact groups

“…Several authors have computed or estimated the real and stable rank of group C * -algebras C * (G) for various classes of locally compact groups G [2], [3], [10], [11], [14], [18], [22], [23], [24], [25], [26], [27], [28], [29]. In generalizing the result of Sudo and Takai [28] for simply connected nilpotent Lie groups, it was shown in [3] that the stable rank of the C * -algebra of an almost connected, nilpotent group G is given by the formula…”

On the stable rank and real rank of group $C^*$-algebras of nilpotent locally compact groups

“…, b n ) such that Several authors have computed or estimated the stable and the real rank of group C * -algebras C * (G) for various classes of locally compact groups G [1], [6], [7], [13], [15], [19], [20], [21], [22], [23], [23], [25], [26]. For example, for simply connected nilpotent Lie groups, Sudo and Takai [25] (following earlier work of Sheu [20]) have shown that sr(C * (G)) is the complex dimension of the space of characters of G. On the other hand, for the free group F 2 on 2 generators it has been shown that sr(C * (F 2 )) = RR(C * (F 2 )) = ∞ [18], [15], but sr(C * r (F 2 )) = RR(C * r (F 2 )) = 1 [6] (where r indicates the reduced C * -algebra of a non-amenable group). In Section 1 of this paper, the result of Sudo and Takai mentioned above is extended to almost connected nilpotent groups G. To be specific,…”

Stable rank and real rank of compact transformation group C*-algebras

“…There is now an extensive literature on stable rank and real rank, especially for group C * -algebras (see [1,6,18,29,30] and the references therein). It seems that less is known in general about transformation group C * -algebras C 0 (X) ⋊ G, although some important and difficult examples have been calculated.…”