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

I have released a preprint on infinite projections in maximal group C*-algebras of S-arithmetic groups.

Main result

Let K be a number field, let Sigma be a nonempty finite set of nonzero prime ideals of O_K, and let n >= 4.

The paper proves that the maximal group C*-algebra

C*max(SL_n(O{K,Sigma}))

contains an infinite projection and a proper isometry.

Consequently, this C*-algebra is not finite, not stably finite, and not MF.

In particular, the result applies to

SL_n(Z[S^{-1}])

for every nonempty finite set S of rational primes and every n >= 4.

General mechanism

More generally, let G be a countable discrete group containing a property (T) subgroup Gamma.

Suppose there exists an element t in G such that

t Gamma t^{-1} is a proper subgroup of Gamma.

Then the Kazhdan projection p_Gamma is an infinite projection in the maximal group C*-algebra C*_max(G).

More explicitly, the element

u_{t^{-1}} p_Gamma + (1 - p_Gamma)

is a proper isometry.

To the best of our knowledge, these give the first examples of countable discrete groups with a non-finite, and hence non-MF, full group C*-algebra.

Preprint and source

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

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

Archived preprint:
Zenodo DOI: 10.5281/zenodo.21939762

The GitHub repository contains the preprint PDF and LaTeX source.