Universal properties of certain crossed products So I was wondering if there are any nice universal properties that the crossed product $C^*$ algebra, $C(\mathbb{T})\times_\alpha \mathbb{Z}_2$ satisfies, where $\alpha$ is the action of conjugation. The reason I am asking is that I am primarily interested in finding an isomorphism between this algebra and $C^\ast(\mathbb{Z}_2\ast\mathbb{Z}_2)$.

Group algebras are not my thing, but here is an attempt. You have, according to Wikipedia, that $\mathbb Z_2*\mathbb Z_2\simeq \mathbb Z\rtimes\mathbb Z_2$, so 
$$
C(\mathbb T)\times_\alpha\mathbb Z_2\simeq C^*(\mathbb Z\rtimes\mathbb Z_2)\simeq C^*(\mathbb Z_2*\mathbb Z_2).
$$
The first isomorphism holds due to the fact that  $\mathbb Z\rtimes\mathbb Z_2$ is amenable, so any faithful representation generates the full C $^*$-algebra.
