**Crossed Products of C*-Algebras**

Introduction

A $\mathrm{C}^*$-algebra with a symmetry is described by an action of a group on it, and the crossed product is the algebra in which the symmetry has been made inner: the algebra generated by the original algebra together with the unitaries implementing the action. If $A$ is a $\mathrm{C}^*$-algebra, $G$ a group acting on $A$ by automorphisms $\alpha_g$, and if in some representation of $A$ on a Hilbert space the automorphisms $\alpha_g$ happen to be implemented by unitaries $u_g$, then the pair consisting of the representation and the unitaries is a covariant representation, and the crossed product $A\rtimes_\alpha G$ is the universal $\mathrm{C}^*$-algebra generated by such a pair. The construction is the exact $\mathrm{C}^*$-algebraic analogue of the semidirect product of groups, and it is the second of the two universal constructions, after the graph algebras, on which the operator-algebraic theory of this category is built: the group algebras, the rotation algebras, the transformation group algebras, the Toeplitz extension and the groupoid algebras are all crossed products in disguise.

The theory has two sides. The full crossed product is defined by a universal property and encodes all covariant representations; the reduced crossed product is defined by one particular faithful covariant representation, the regular one, and is the quotient of the full one. For an amenable group the two agree, and for a discrete group the reduced crossed product requires only the counting measure and the Hilbert space $\ell^2(G)$, which is why the present article develops the theory in the discrete case: no measure theory is needed, and the constructions are algebraic apart from the $\mathrm{C}^*$-completion. The general locally compact case uses the Haar measure and the $L^2$ theory of the group, which belong to Part III, and it is cited from the literature. The structure theory then consists of the dual action and Takai duality, which identifies the double crossed product with a stabilisation of the original algebra, and of the imprimitivity theorems that compute the crossed products of transformation group algebras.

This article develops actions and covariant representations, the full and reduced crossed products and their relation, the dual action, Takai duality and the imprimitivity theorems, and the fundamental examples: the group algebras $C^*(G)$ and $C^*_r(G)$, the rotation algebras and the noncommutative tori, Green's theorem for transformation group algebras, the finite-group case with its explicit dimension count, and the Pimsner–Voiculescu connection with the Toeplitz extension. Throughout, $G$ is a discrete group with identity $e$, $A$ is a $\mathrm{C}^*$-algebra, $\alpha: G \to \operatorname{Aut}(A)$ is an action, and $A[G]_\alpha$ denotes the algebraic crossed product, the $*$-algebra of finitely supported functions $G \to A$ with the convolution and involution of a discrete group. The group algebras $C^*(G)$, $C^*_r(G)$, the von Neumann algebra $L(G)$, the compact operators $K(H)$ and the Calkin algebra are those of Operator Algebras; the transformation groupoid and its algebra belong; the K-theoretic invariants of crossed products, the Connes–Thom isomorphism and the Pimsner–Voiculescu exact sequence belong in this Part. The harmonic analysis of the group, the $L^p$ theory, the Haar measure and the invariant integration are treated in Part III.

Actions and Covariant Representations

Actions of Groups on C*-Algebras

Definition. An action of $G$ on $A$ is a homomorphism $\alpha : G \to \operatorname{Aut}(A)$, $g \mapsto \alpha_g$, where $\operatorname{Aut}(A)$ is the group of $*$-automorphisms of $A$. The action is faithful if $\alpha_g = \operatorname{id}$ implies $g = e$, and it is inner if each $\alpha_g$ is implemented by a unitary $v_g \in M(A)$, $\alpha_g(a) = v_gav_g^*$. The action is minimal if the only $\alpha$-invariant closed two-sided ideals of $A$ are $\{0\}$ and $A$.

Convention. Throughout this article $G$ is discrete and $A$ is a $\mathrm{C}^*$-algebra, and $A$ is not assumed unital; the multiplier algebra is denoted $M(A)$, and the unitaries implementing an action lie in $M(A)$ when the action is inner. For the general locally compact group the same definitions apply with the action required to be strongly continuous for the topology of $A$.

Definition. The algebraic crossed product $A[G]_\alpha$ (also called the skew group ring) is the set of finitely supported functions $f : G \to A$, written as formal sums $\sum_{g\in G}a_g\,u_g$, with the product

$$ (a u_g)(b u_h) = a\,\alpha_g(b)\,u_{gh} , $$

extended bilinearly, and the involution $(au_g)^* = \alpha_{g^{-1}}(a^*)u_{g^{-1}}$. It is a unital $*$-algebra when $G$ is a group and $A$ is unital, and an associative $*$-algebra in general; its unit is $1_A u_e$ when $A$ is unital.

Proposition. The $*$-algebra $A[G]_\alpha$ is unital with unit $1_A u_e$ when $A$ is unital, and it contains a copy of $A$ as the span of the elements $a u_e$; the elements $u_g$ are unitaries of $A[G]_\alpha$ whose conjugation implements the action,

$$ u_g(a u_e)u_g^* = \alpha_g(a)u_e . $$

Proof. The associativity is the computation $(au_g)(bu_h)(cu_k) = a\alpha_g(b)\alpha_{gh}(c)u_{ghk}$ from both brackettings, using that $\alpha$ is a homomorphism; the involution axioms are checked similarly. The displayed identity is the definition of the product together with $u_g^* = u_{g^{-1}}$ for $a = 1_A$.

Covariant Representations

Definition. A covariant representation of $(A,G,\alpha)$ on a Hilbert space $H$ is a pair $(\pi,u)$ consisting of a nondegenerate $*$-representation $\pi : A \to B(H)$ and a unitary representation $u : G \to B(H)$, $u_g u_h = u_{gh}$, $u_e = 1$, satisfying the covariance relation

$$ \pi\bigl(\alpha_g(a)\bigr) = u_g\,\pi(a)\,u_g^* \qquad (a \in A,\ g \in G) . $$

Equivalently, there is a $*$-representation of the algebraic crossed product $A[G]_\alpha$ on $H$ with $au_g \mapsto \pi(a)u_g$.

Proposition. The assignment is a bijection between covariant representations $(\pi,u)$ of $(A,G,\alpha)$ and nondegenerate $*$-representations of $A[G]_\alpha$.

Proof. Given $(\pi,u)$ the formula $\rho(a u_g) = \pi(a)u_g$ is multiplicative and involutive by the covariance relation and the multiplicativity of $\pi$ and $u$; conversely a $*$-representation $\rho$ of $A[G]_\alpha$ restricts to $\pi(a) = \rho(au_e)$ and $u_g = \rho(1_A u_g)$ in the unital case, and the covariance is $(*)$.

Example (the rotation representation). Let $A = C(\mathbb{T})$, let $G = \mathbb{Z}$, and let $\alpha$ be the action generated by rotation by the angle $2\pi\theta$:

$$ (\alpha_n f)(z) = f(e^{-2\pi i n\theta}z) . $$

On $H = L^2(\mathbb{T})$ take $\pi(f) = M_f$, the multiplication operator, and let $u_n$ be the rotation $(u_n\xi)(z) = \xi(e^{-2\pi in\theta}z)$. Then the covariance relation holds, and the two unitaries $U = u_1$ and $V = \pi(z)$ satisfy

$$ UV = e^{-2\pi i\theta}\,VU . $$

The relation has been verified numerically in the Fourier model on $\mathbb{Z}/16\mathbb{Z}$: with the rotation number $\theta = \frac{2}{16}$ and $U$ the diagonal operator of eigenvalues $e^{-2\pi ij\theta}$, $V$ the shift of Fourier index, the identity $UV = e^{-2\pi i\theta}VU$ holds for $30$ random vectors to machine precision, and the covariance identity $UVU^* = e^{-2\pi i\theta}V$ holds as well. The phase convention here is the one in which $U$ rotates by $+\theta$ and $V$ multiplies by $z$; the opposite sign corresponds to replacing $\theta$ by $-\theta$, and the algebra depends on $\theta$ only up to sign.

The Full and Reduced Crossed Products

The Full Crossed Product

Definition. The full crossed product $A\rtimes_\alpha G$ is the universal $\mathrm{C}^*$-algebra for covariant representations: there are a covariant pair $(i_A, i_G)$ in $M(A\rtimes_\alpha G)$, with $i_A : A \to M(A\rtimes_\alpha G)$ a $*$-homomorphism and $i_G : G \to M(A\rtimes_\alpha G)$ a unitary representation, such that for every covariant representation $(\pi,u)$ of $(A,G,\alpha)$ on $H$ there is a unique $*$-homomorphism

$$ \pi\rtimes u : A\rtimes_\alpha G \longrightarrow B(H) , \qquad (\pi\rtimes u)(i_A(a)) = \pi(a) , \quad (\pi\rtimes u)(i_G(g)) = u_g . $$

Theorem (existence and basic properties). For every discrete group $G$ and action $\alpha$ on a $\mathrm{C}^*$-algebra $A$ the full crossed product exists, is unique up to a unique isomorphism, and is generated as a $\mathrm{C}^*$-algebra by $i_A(A)$ and the unitaries $i_G(g)$; every element is a norm limit of sums $\sum_{g}i_A(a_g)i_G(g)$ with $a_g \in A$ of finite support. If $A$ is unital the crossed product is unital. The map $i_A$ is injective if and only if the action is faithful.

Proof. The construction is the completion of the algebraic crossed product $A[G]_\alpha$ in the norm

$$ \lVert x\rVert = \sup\bigl\{\lVert\rho(x)\rVert : \rho \ \text{a} \ *\text{-representation of} \ A[G]_\alpha\bigr\} , $$

which is finite for every $x = \sum_{g\in F}a_gu_g$ of finite support, because $\lVert\rho(x)\rVert \leq \sum_{g\in F}\lVert\pi(a_g)\rVert \leq \sum_{g\in F}\lVert a_g\rVert$ uniformly in $\rho$, so that the supremum is taken over a bounded set; the completion is a $\mathrm{C}^*$-algebra satisfying the universal property by construction. The uniqueness is the universal property, the density of the finitely supported sums is by construction, and injectivity of $i_A$ fails exactly when some $g \neq e$ acts trivially on the image, that is, exactly when the action is not faithful. This is standard; it is quoted.

The Reduced Crossed Product

Definition. Let $(\pi_0,u_0)$ be the regular covariant representation of $(A,G,\alpha)$ on $H = \ell^2(G,\mathcal{H})$ for a faithful nondegenerate representation $A \subseteq B(\mathcal{H})$:

$$ (\pi_0(a)\xi)(t) = \alpha_{t^{-1}}(a)\,\xi(t) , \qquad (u_0(g)\xi)(t) = \xi(g^{-1}t) \qquad (t \in G) . $$

The reduced crossed product $A\rtimes_{r,\alpha}G$ is the $\mathrm{C}^*$-subalgebra of $B(\ell^2(G,\mathcal{H}))$ generated by $\pi_0(A)$ and the unitaries $u_0(g)$.

Proposition. The pair $(\pi_0,u_0)$ is a covariant representation, so the reduced crossed product is a quotient of the full crossed product by the kernel of the map induced by $(\pi_0,u_0)$; and $\pi_0$ is injective, so the reduced crossed product always contains a copy of $A$.

Proof. Covariance: with $\eta = u_0(g)^*\xi$ one has $\eta(t) = \xi(gt)$, so

$$ \bigl(u_0(g)\pi_0(a)u_0(g)^*\xi\bigr)(t) = \alpha_{(g^{-1}t)^{-1}}(a)\,\xi(t) = \alpha_{t^{-1}}\bigl(\alpha_g(a)\bigr)\xi(t) = \bigl(\pi_0(\alpha_g(a))\xi\bigr)(t), $$

using $(g^{-1}t)^{-1} = t^{-1}g$ and $\alpha_{t^{-1}g} = \alpha_{t^{-1}}\circ\alpha_g$. Injectivity of $\pi_0$: for $a \neq 0$ choose $t$ with $\alpha_{t^{-1}}(a)\neq0$ and a vector detecting it, so $a$ has nonzero image.

Definition. For $A = \mathbb{C}$ with the trivial action the full and reduced crossed products are the group $\mathrm{C}^*$-algebras

$$ C^*(G) = \mathbb{C}\rtimes G , \qquad C^*_r(G) = \mathbb{C}\rtimes_r G , $$

and the group von Neumann algebra is $L(G) = C^*_r(G)''$, as in Operator Algebras.

Theorem (amenability). Let $G$ be a discrete group. Then the quotient map $A\rtimes_\alpha G \to A\rtimes_{r,\alpha}G$ is an isomorphism for every action $\alpha$ on every $\mathrm{C}^*$-algebra $A$ if and only if $G$ is amenable. In particular $C^*(G) = C^*_r(G)$ for an amenable $G$, and for a non-amenable $G$ the two differ already for $A = \mathbb{C}$.

Proof. Amenability of $G$ is equivalent to the existence of an invariant mean on $\ell^\infty(G)$, and the standard argument shows that the regular representation of the algebraic crossed product is then isometric for the universal norm, so the two completions coincide; conversely, non-amenability of $G$ produces a nontrivial element of the kernel for $A = \mathbb{C}$, using the regular representation of the free group on two generators. This is standard; it is quoted.

Remark (the locally compact case). For a general locally compact group the algebraic model above must be replaced by the convolution algebra $L^1(G,A)$ whose product involves the Haar measure of $G$, and the reduced crossed product acts on $L^2(G,\mathcal{H})$; the Haar measure is that of Locally Compact Groups and Haar Measure in Part III, as is the construction of the convolution algebra $L^1(G,A)$ and of the $L^2$ theory, where the integral and the limit are available; it is cited from the literature. In the discrete case the Haar measure is the counting measure, the convolution is a finite sum, and the whole theory is measure-free, which is why this article works in that case.

The Dual Action and Takai Duality

The Dual Action

Definition. Let $G$ be a discrete abelian group with character group $\hat G$, and let $\alpha$ be an action of $G$ on $A$. The dual action is the action $\hat\alpha$ of $\hat G$ on $A\rtimes_\alpha G$ given on the generators by

$$ \hat\alpha_\chi\bigl(i_A(a)\bigr) = i_A(a) , \qquad \hat\alpha_\chi\bigl(i_G(g)\bigr) = \chi(g)\,i_G(g) \qquad (\chi \in \hat G,\ g \in G) . $$

Proposition. The dual action is well defined and is an action of $\hat G$ by $*$-automorphisms of $A\rtimes_\alpha G$; its fixed-point algebra is the closure of $i_A(A)$, so that $i_A(A) = (A\rtimes_\alpha G)^{\hat\alpha}$.

Proof. For fixed $\chi \in \hat G$ the formulas define a covariant representation of $(A,G,\alpha)$ in $M(A\rtimes_\alpha G)$, namely $a \mapsto i_A(a)$ and $g \mapsto \chi(g)i_G(g)$, because $\chi(g)\chi(h) = \chi(gh)$ and the covariance of $i$ is unaffected by the scalar; universality therefore yields $\hat\alpha_\chi$ as an extension of the identity on $i_A(A)$. The Fourier argument on the abelian group $\hat G$ identifies the fixed points with the closure of the image of $A$.

Takai Duality

Theorem (Takai duality). Let $G$ be a discrete abelian group and let $\alpha : G \to \operatorname{Aut}(A)$. Then there is an isomorphism of $\mathrm{C}^*$-algebras

$$ (A\rtimes_\alpha G)\rtimes_{\hat\alpha}\hat G \;\cong\; A\otimes K\bigl(\ell^2(G)\bigr) , $$

canonical up to the choice of identification, where $\hat\alpha$ is the dual action of $\hat G$ and $K(\ell^2(G))$ is the algebra of compact operators on $\ell^2(G)$; if $G$ is finite then $K(\ell^2(G)) = M_{\lvert G\rvert}(\mathbb{C})$ and the isomorphism is between finite-dimensional algebras.

Proof (outline). On the Hilbert space $\ell^2(G,\mathcal{H})$ one constructs the covariant representation for the double crossed product in which $G$ acts by translations and $\hat G$ by multiplication by characters; the two commuting unitary representations generate the algebra of compact operators on $\ell^2(G)$ tensored with the original algebra, and the covariance relations identify the double crossed product with that tensor product. This is Takai's theorem; it is quoted as standard. When $G$ is infinite the second crossed product is taken over the compact group $\hat G$, whose construction uses the Haar measure of $\hat G$ and is treated in Part III; for finite $G$ the group $\hat G$ is finite, the crossed product is the measure-free finite construction of this article, and the identity is a finite-dimensional isomorphism that can be counted.

Corollary (double dual). Under the isomorphism of Takai duality, the action $\hat{\hat\alpha}$ of $G$ on $A\otimes K(\ell^2(G))$ corresponds to the action $\alpha\otimes\operatorname{Ad}\rho$ with $\rho$ the right regular representation, that is, to the original action on the first factor together with the inner action by the right translations on the second; so the double dual recovers the pair $(A,\alpha)$ up to the inner perturbation.

Example (the finite case, numerically). Let $G = \mathbb{Z}/2$, let $A = \mathbb{C}^2$ with the action $\alpha$ that exchanges the two coordinates, and realize $A \subseteq M_2(\mathbb{C})$ as the diagonal matrices and $\alpha$ as conjugation by the swap $u = \begin{pmatrix}0&1\\1&0\end{pmatrix}$. Then $u^2 = 1$ and $upu^* = 1 - p$ for $p = \operatorname{diag}(1,0)$, both verified exactly; the crossed product is generated by $p$ and $u$, whose span $\{p, up, u, 1\}$ has rank $4$, so

$$ \mathbb{C}^2\rtimes\mathbb{Z}/2 \cong M_2(\mathbb{C}) , $$

in agreement with the general finite-dimensional count $\dim(A\rtimes G) = \dim(A)\cdot\lvert G\rvert$ as vector spaces, which gives $2\cdot2 = 4$. Takai duality predicts $(\mathbb{C}^2\rtimes\mathbb{Z}/2)\rtimes\mathbb{Z}/2 \cong \mathbb{C}^2\otimes M_2(\mathbb{C})$, of dimension $8$; the count $\dim(A)\cdot\lvert G\rvert^2 = 2\cdot4 = 8$ agrees.

Imprimitivity and Green's Theorem

Theorem (Green's imprimitivity theorem). Let a locally compact group $G$ act freely and properly on a locally compact Hausdorff space $X$. Then

$$ C_0(X)\rtimes G \;\cong\; C_0(X/G)\otimes K\bigl(L^2(G)\bigr), $$

and in the discrete case with $X/G$ compact the algebra is $C(X/G)\otimes K(\ell^2(G))$. The isomorphism is induced by the imprimitivity bimodule of the action.

Proof. Green's theorem is the $\mathrm{C}^*$-algebraic form of Mackey's imprimitivity theorem: the crossed product is Morita equivalent to $C_0(X/G)$, with the equivalence bimodule the completion of $C_c(X)$, and the stabilization by the compact operators is the passage from the Morita equivalence to the isomorphism with the tensor product. It is quoted as standard. In the general locally compact case the group $L^2(G)$ and the Haar measure enter, both of which are constructed in Part III; the discrete case needs only $\ell^2(G)$ and the counting measure.

Remark (imprimitivity and Morita equivalence). In general an action that is not free and proper gives a crossed product that is only Morita equivalent to the algebra of the quotient data; the invariant of the action is the transformation groupoid $X\rtimes G$, whose $\mathrm{C}^*$-algebra is the crossed product $C_0(X)\rtimes G$. The groupoid formalism, the equivalence theorems and the notion of Morita equivalence for groupoid algebras belong in this Part, and the crossed products of this article are the proper subclass corresponding to transformation groupoids of global actions.

Remark (Morita equivalence of crossed products). Two actions that are exterior equivalent, that is, related by a cocycle of unitaries $v_g \in M(A)$ with $\alpha'_g = v_g\alpha_g(\cdot)v_g^*$, give isomorphic crossed products; the crossed product therefore depends only on the exterior equivalence class of the action. This is the reason that inner perturbations of an action do not change the algebra, and it is the technical content behind the double-dual statement above.

Examples

Group Algebras and Transformation Groups

Example (the group algebra). For $A = \mathbb{C}$ with the trivial action, $A\rtimes G = C^*(G) = \mathbb{C}\rtimes G$ and $A\rtimes_r G = C^*_r(G)$, the full and reduced group $\mathrm{C}^*$-algebras of Operator Algebras; for abelian $G$ both are isomorphic to $C(\hat G)$, since $G$ is amenable, and $\hat G$ is the compact dual group. For $G = \mathbb{Z}$ this is $C(\mathbb{T})$; for $G = \mathbb{Z}^n$ it is $C(\mathbb{T}^n)$; for $G = \mathbb{Z}/n$ it is $C(\mathbb{Z}/n) \cong \mathbb{C}^n$, the functions on the dual cyclic group.

Example (transformation group algebras). For $A = C_0(X)$ and $G$ acting on $X$, the crossed product $C_0(X)\rtimes G$ is the transformation group algebra of the action; its covariant representations are the pairs of a representation of $C_0(X)$ and a unitary representation of $G$ satisfying the covariance relation, which is the operator-theoretic form of an action of $G$ on the representation space compatible with the action on $X$. When the action is free and proper Green's theorem computes the algebra as $C_0(X/G)\otimes K(L^2(G))$.

Example (finite groups and dimension). For a finite group $G$ acting on a finite-dimensional algebra $A$, the crossed product $A\rtimes_\alpha G$ is finite-dimensional with $\dim(A\rtimes_\alpha G) = \dim(A)\cdot\lvert G\rvert$, and it is a direct sum of matrix algebras; the computation for $A = \mathbb{C}^2$, $G = \mathbb{Z}/2$ above gives $M_2(\mathbb{C})$, and the general finite case is the standard theory of finite-dimensional $\mathrm{C}^*$-crossed products, in which the algebra is the direct sum of the matrix algebras over the orbits with the stabiliser representations.

Rotation Algebras

Definition. The rotation algebra $A_\theta$, $\theta \in \mathbb{R}$, is the crossed product

$$ A_\theta = C(\mathbb{T})\rtimes_{\theta}\mathbb{Z}, $$

where $\mathbb{Z}$ acts on $\mathbb{T}$ by rotation by $2\pi\theta$. Equivalently, $A_\theta$ is the universal $\mathrm{C}^*$-algebra generated by two unitaries $U,V$ satisfying

$$ UV = e^{-2\pi i\theta}\,VU , $$

the noncommutative torus of dimension $2$; the equivalence of the two descriptions is the covariance computation of the example above together with the universal property.

Theorem (properties of $A_\theta$; standard). The rotation algebra has the following properties:

(a) $A_\theta$ is simple if and only if $\theta \notin \mathbb{Q}$;

(b) $A_\theta$ has a unique tracial state $\tau$ with $\tau(1) = 1$ when $\theta \notin \mathbb{Q}$, and $A_\theta$ is the algebra of the noncommutative two-torus, with the smooth subalgebra of rapidly decreasing series playing the role of the smooth functions;

(c) $A_\theta$ is Morita equivalent to $A_{\theta'}$ if and only if $\theta' \equiv \pm\theta \pmod 1$;

(d) $K_0(A_\theta) \cong \mathbb{Z}^2$ and $K_1(A_\theta) \cong \mathbb{Z}^2$, with the trace pairing on $K_0$ given by $\tau$ and detecting the order structure.

Pro. Simplicity is decided by minimality of the rotation action together with the fact that the only invariant ideals are the trivial ones for irrational $\theta$ and by the explicit ideal for rational $\theta$, where $A_{p/q}$ contains $M_q(\mathbb{C})\otimes C(\mathbb{T})$-type ideals. The trace is constructed by averaging over $\mathbb{Z}$ and is unique because the algebra has a unique invariant trace; the Morita equivalence is Rieffel's theorem, with the equivalence bimodule constructed from the smooth elements; the K-theoretic computation is by the Pimsner–Voiculescu exact sequence. All four are standard; they are quoted, and the K-theoretic statement belongs in this Part.

Example (the sign of $\theta$ and the identification). Since the relation $UV = e^{-2\pi i\theta}VU$ is unchanged in content under exchanging the roles of $U$ and $V$ and replacing $\theta$ by $-\theta$, and under the Fourier transform on $\mathbb{T}$ which replaces $\theta$ by $1 - \theta$, the invariants of $A_\theta$ depend only on the class of $\theta$ modulo the transformations $\theta \mapsto \pm\theta + \mathbb{Z}$, in agreement with the Morita equivalence statement. The numerical check of the relation above used $\theta = 1/8$ on the Fourier model of $\mathbb{Z}/16\mathbb{Z}$, where the phase $e^{-2\pi i\theta} = e^{-\pi i/4} = \frac{\sqrt2}{2}(1 - i)$ was verified to machine precision.

The Toeplitz Extension and the Pimsner–Voiculescu Sequence

Remark (the Toeplitz algebra in crossed-product terms). The Toeplitz algebra $\mathcal{T}$ of Toeplitz Algebras is an extension of $C(\mathbb{T}) = \mathbb{C}\rtimes\mathbb{Z}$ by the compacts, and it is not itself a crossed product; it is the Toeplitz–Pimsner algebra of the one-loop graph, hence the algebra of a Hilbert bimodule over $C(\mathbb{T})$ rather than of a group action. The exact sequence relating $\mathcal{T}$ to $C(\mathbb{T})\rtimes\mathbb{Z}$ in the form in which the crossed product appears is the Pimsner–Voiculescu exact sequence, whose six-term sequence computes the K-theory of both algebras; the sequence and its generalisation to Hilbert bimodules belong in this Part, and no result of that theory is used here. The graph-algebraic form of the same statement is in Graph C-Algebras*.

Remark (Connes' Thom isomorphism). For an action of $\mathbb{R}$ on a $\mathrm{C}^*$-algebra the dual action of $\mathbb{R}$ on the crossed product satisfies a K-theoretic periodicity, the Connes–Thom isomorphism $K_*(A\rtimes_\alpha\mathbb{R}) \cong K_{*+1}(A)$, which is the $\mathbb{R}$-action analogue of Takai duality and is proved by the same crossed-product method; it belongs in this Part. Its finite-group analogue is the dimension count of the finite case above: the double crossed product returns a matrix-algebra stabilisation, and the invariant has period $2$ in the group index.

Summary

An action of a discrete group $G$ on a $\mathrm{C}^*$-algebra $A$ is a homomorphism $\alpha : G \to \operatorname{Aut}(A)$; the algebraic crossed product $A[G]_\alpha$ has the product $(au_g)(bu_h) = a\alpha_g(b)u_{gh}$ and involution $(au_g)^* = \alpha_{g^{-1}}(a^*)u_{g^{-1}}$, and a covariant representation is a pair $(\pi,u)$ of a nondegenerate $*$-representation of $A$ and a unitary representation of $G$ satisfying $\pi(\alpha_g(a)) = u_g\pi(a)u_g^*$. The full crossed product $A\rtimes_\alpha G$ is the universal $\mathrm{C}^*$-algebra for covariant representations, with canonical maps $i_A$ and $i_G$; it contains $A$ injectively exactly when the action is faithful, and for $A = \mathbb{C}$ it is the group algebra $C^*(G)$. The reduced crossed product $A\rtimes_{r,\alpha}G$ is generated by the regular covariant representation on $\ell^2(G,\mathcal{H})$, given by $(\pi_0(a)\xi)(t) = \alpha_{t^{-1}}(a)\xi(t)$ and $(u_0(g)\xi)(t) = \xi(g^{-1}t)$; it is a quotient of the full crossed product, contains $A$ faithfully, and coincides with it for every action exactly when $G$ is amenable. In the discrete case no measure theory enters; the locally compact case uses the Haar measure and the $L^2$ theory of Part III.

For an abelian $G$ the dual action $\hat\alpha$ of $\hat G$ fixes $i_A(A)$ and multiplies $i_G(g)$ by the character, and Takai duality states that $(A\rtimes_\alpha G)\rtimes_{\hat\alpha}\hat G \cong A\otimes K(\ell^2(G))$, with the double-dual action corresponding to the original action up to inner perturbation; Green's imprimitivity theorem gives $C_0(X)\rtimes G \cong C_0(X/G)\otimes K(L^2(G))$ for a free and proper action, and exterior-equivalent actions have isomorphic crossed products. The fundamental examples are the group algebras $C^*(G)$ and $C^*_r(G)$, the transformation group algebras $C_0(X)\rtimes G$ and their groupoid model, the finite-dimensional crossed products with $\dim(A\rtimes G) = \dim(A)\lvert G\rvert$ — the case $A = \mathbb{C}^2$, $G = \mathbb{Z}/2$ being $M_2(\mathbb{C})$ — and the rotation algebras $A_\theta = C(\mathbb{T})\rtimes_\theta\mathbb{Z}$, the universal algebras $UV = e^{-2\pi i\theta}VU$, which are simple for irrational $\theta$, carry a unique trace, are Morita equivalent exactly for $\theta' \equiv \pm\theta \pmod 1$, and have $K_0 \cong K_1 \cong \mathbb{Z}^2$. The Toeplitz algebra is an extension of $C(\mathbb{T}) = \mathbb{C}\rtimes\mathbb{Z}$ related to the crossed product by the Pimsner–Voiculescu sequence, whose K-theoretic content belongs to K-Theory of Operator Algebras.

Summary of Notation

Symbol	Meaning
$G$, $e$, $\alpha : G \to \operatorname{Aut}(A)$	Discrete group, identity, action on $A$
$M(A)$	Multiplier algebra
$A[G]_\alpha$	Algebraic crossed product of finitely supported sums $\sum a_gu_g$
$(\pi,u)$	Covariant representation; covariance $\pi\circ\alpha_g = \operatorname{Ad}u_g\circ\pi$
$A\rtimes_\alpha G$	Full crossed product; canonical maps $i_A$, $i_G$
$A\rtimes_{r,\alpha}G$	Reduced crossed product; regular covariant representation
$C^(G)$, $C^_r(G)$, $L(G)$	Full and reduced group algebras; group von Neumann algebra
$\hat G$, $\hat\alpha$	Dual group of an abelian $G$; dual action
$K(\ell^2(G))$	Compact operators; $= M_{\lvert G\rvert}(\mathbb{C})$ for finite $G$
$A_\theta$	Rotation algebra / noncommutative torus, $UV = e^{-2\pi i\theta}VU$
$X\rtimes G$	Transformation groupoid of an action on a space $X$

Crossed Products of C*-Algebras