Let ${\mathbb{Z}}^{1}[t,t^{-1}]$ be the subring of ${\mathbb{Z}}[t,t^{-1}]$ of polynomials such that if $p(t)\in{\mathbb{Z}}^{1}[t,t^{-1}]$ then

1. (1)

   $p(t)=p(t^{-1}),$

2. (2)

   $p(1)=\pm 1.$

Let $G$ be a locally compact group. A $G$ - $C^{*}$ -algebra is a $C^{*}$ -algebra endowed with a continuous action $\alpha$ of $G$ . Specifically, $\alpha$ is a homomorphism from $G$ to the group of automorphisms of $A$ , such that for $a\in A$ the map $s\mapsto\alpha_{s}(a)$ is norm continuous. We shall often write $s.a$ for $\alpha_{s}(a)$ .

A covariant representation of the $G$ - $C^{*}$ -algebra $A$ is a pair $(\pi,\sigma)$ where $\pi$ and $\sigma$ are representations of $A$ and $G$ respectively in a Hilbert space $H$ , such that

$$\sigma(s)\pi(a)\sigma(s)^{-1}=\pi(s.a)$$

for every $a\in A,s\in G$ .