Infinite Projections in Full Group C*-Algebras of S-Arithmetic Groups

Abstract

Let K be a number field, let Σ be a nonempty finite set of nonzero prime ideals of O_K, and let n ≥ 4. We prove that C*max(SL_n(O_{K,Σ})) contains an infinite projection and a proper isometry. Hence it is not finite and is neither stably finite nor MF, although SL_n(O_{K,Σ}) is a finitely presented, residually finite property-(T) group. In particular, this applies to SL_n(Z[S^{-1}]) for every nonempty finite set S of rational primes. More generally, if a countable discrete group G contains a property-(T) subgroup Γ and some t ∈ G satisfies tΓt^{-1} ⊊ Γ, then the Kazhdan projection p_Γ is infinite in C*max(G); explicitly, u_{t^{-1}}p_Γ + (1 − p_Γ) is a proper isometry. To the best of our knowledge, these are the first examples of countable discrete groups with a non-finite—and hence non-MF—full group C*-algebra. The preparation of this manuscript was AI-assisted. Public source repository: Tseng-math/s-arithmetic-full-group-cstar