Character Kernels of Discrete Groups1

Let G be an arbitrary discrete group and let T = C[G] be its group algebra over the complex numbers C. If 9? is an irreducible representation of the algebra then 9i(r)=P is primitive and hence isomorphic to a dense set of linear transformations over D, the commuting ring of 9J [4, p. 28]. Let L be the center of D. If dimr, P < °o then we say that 9? is finite and since P is central simple over A [4, p. 122] we have dinuP = ?ra2. We set rai = deg 9?, the degree of 9t If G is finite then C is always the commuting ring of 9i so this agrees with the usual definition of degree. Again let P = 9f(r). Then by a theorem of Amitsur [l] deg 9i^ra if and only if for every 2ra elements xi, • • • , x2n in P we have