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  "body": "\n## Definition\n\nLet $A$ be a unital C*-algebra, and let $G$ be a discrete group.\nLet $\\alpha \\colon G \\to \\Aut(A)$ be a [[group action|0WCY42EV9V]] of $G$ on $A$ by $*$-automorphisms.\nLet $C_c(G, A)$ be the space of finitely supported functions $f \\colon G \\to A$, with the product and involution\n$$(f * g)(s) = \\sum_{t \\in G} f(t) \\, \\alpha_t\\big(g(t^{-1} s)\\big), \\qquad f^*(s) = \\alpha_s\\big(f(s^{-1})^*\\big).$$\nLet $H$ be a [[Hilbert space|HM2HHE3J51]].\nA *covariant representation* of $(A, G, \\alpha)$ on $H$ is a pair $(\\pi, u)$ with two parts.\nThe first, $\\pi$, is a unital $*$-representation of $A$ on $H$; the second, $u$, is a unitary representation of $G$ on $H$.\nThe pair must satisfy $u_s \\pi(a) u_s^* = \\pi(\\alpha_s(a))$ for all $s \\in G$ and $a \\in A$.\nIt gives the $*$-representation $(\\pi \\rtimes u)(f) = \\sum_{s \\in G} \\pi(f(s)) u_s$ of $C_c(G, A)$.\nThe *crossed product* $A \\rtimes_\\alpha G$ is the completion of $C_c(G, A)$ in the norm\n$$\\norm{f} = \\sup \\set{\\norm{(\\pi \\rtimes u)(f)} : (\\pi, u) \\text{ a covariant representation}}.$$\n\n## Remarks\n\nThe 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 $A \\rtimes_\\alpha G$.\nThe *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.\nFor locally compact groups, $C_c(G, A)$ is replaced by continuous compactly supported functions and sums by integrals against Haar measure.\n\nExample: take $A = \\C$ and the trivial action.\nThen $C_c(G, \\C)$ is the group algebra of $G$ with convolution, and the covariant representations are the unitary representations of $G$.\nSo $\\C \\rtimes G$ is the full group C*-algebra $C^*(G)$.\n\nNon-example: let $\\Z$ act on $A = C(\\mathbb{T})$ by an irrational rotation.\nIf $\\alpha$ is replaced by the identity in the formulas for the product and the involution, the result is the crossed product for the trivial action.\nThat algebra is $C(\\mathbb{T}) \\otimes C^*(\\Z) \\cong C(\\mathbb{T}^2)$, which is commutative.\nThe twisted product gives instead the noncommutative irrational rotation algebra.\n",
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