Elements in crossed product $C^*$-algebras Let $A$ be a $C^*$-algebra and $G$ be a discrete group. I am quite confused about the definition of the reduced crossed product $A\rtimes_r G$ and the  full crossed product $A\rtimes G$. 
What are the form of elements in reduced crossed product and full crossed product. Do they have explicit expressions?
 A: As with nearly all constructions for building new $C^*$-algebras out of old ones, there is not an explicit representation of every element of $A\rtimes_rG$ and $A\rtimes G$.  But (again, as with nearly all constructions) there is a dense $*$-subalgebra where elements have an explicit form, and most calculations can be done in this subalgebra.
Let $\alpha:G\to\operatorname{Aut}(A)$ denote the action of $G$ on $A$ (we typically write $\alpha_g=\alpha(g)$ to simplify notation).  Let $C_c(G,A)$ denote the space of all functions $G\to A$ with finite support.  Elements of $C_c(G,A)$ can be written as formal sums $\sum_{g\in G}a_gg$, where $a_g\in A$.  This is made into a $*$-algebra in the following way:  for $\lambda\in\mathbb C$, $a=\sum_{g\in G}a_gg,b=\sum_{h\in G}b_hh\in C_c(G,A)$, we have 
\begin{align*}
\lambda a&=\sum_{g\in G}(\lambda a_g)g,\\
a+b&=\sum_{g\in G}(a_g+b_g)g,\\
a\cdot b&=\sum_{g\in G}\left(\sum_{h\in G}a_h\alpha_h(b_{h^{-1}g})\right)g,\\
a^*&=\sum_{g\in G}\left((\alpha_g(a_{g^{-1}}^*)\right)g.
\end{align*}
(Exercise:  Check that this does make $C_c(G,A)$ into a $*$-algebra.)  Then the full and reduced crossed products $A\rtimes G$ and $A\rtimes_rG$ are defined by choosing suitable $C^*$-norms on this algebra, and taking completions.
