The full group C*-algebra of the modular group is primitive

TL;DR

This paper investigates the primitivity of the full group C*-algebra of the modular group, showing it is primitive for PSL(2,Z) but not for higher n, and constructs many inequivalent irreducible representations.

Contribution

It establishes the primitivity status of the full group C*-algebra for PSL(n,Z) and constructs an uncountable family of irreducible representations for PSL(2,Z).

Findings

C*-algebra of PSL(2,Z) is primitive.

C*-algebra of PSL(n,Z) is not primitive for n ≥ 3.

Existence of uncountably many inequivalent irreducible representations for PSL(2,Z).

Abstract

We show that the full group C$^*$-algebra of $PSL(n, \Z)$ is primitive when $n=2$, and not primitive when $n\geq 3$. Moreover, we show that there exists an uncountable family of pairwise inequivalent, faithful irreducible representations of $C^*(PSL(2,\Z))$.