Crossed product

Remarks

The element $a \delta_s$ that takes the value $a$ at $s$ and $0$ elsewhere is written $a u_s$; then $u_s a u_s^* = \alpha_s(a)$ in $A \rtimes_\alpha G$. The reduced crossed product $A \rtimes_{\alpha, r} G$ uses only the regular covariant representations on $\ell^2(G, H)$; it equals the full one when $G$ is amenable. For locally compact groups, $C_c(G, A)$ is replaced by continuous compactly supported functions and sums by integrals against Haar measure.

Example: take $A = \C$ and the trivial action. Then $C_c(G, \C)$ is the group algebra of $G$ with convolution, and the covariant representations are the unitary representations of $G$. So $\C \rtimes G$ is the full group C*-algebra $C^*(G)$.

Non-example: let $\Z$ act on $A = C(\mathbb{T})$ by an irrational rotation. If $\alpha$ is replaced by the identity in the formulas for the product and the involution, the result is the crossed product for the trivial action. That algebra is $C(\mathbb{T}) \otimes C^*(\Z) \cong C(\mathbb{T}^2)$, which is commutative. The twisted product gives instead the noncommutative irrational rotation algebra.