Crossed Products of C*-Algebras

The Full Crossed Product Algebra

The Algebra of Test Functions

Let \newcommand{\A}{\mathcal{A}}\newcommand{\C}{\mathbb{C}}\newcommand{\Z}{\mathbb{Z}}\newcommand{\T}{S^{1}}(\A, G, \alpha) be a C*-dynamical system. We define the vector space of $\text{A}$-valued functions on $G$ with finite support as: $C_c(G,\text{A}):=\{h:G\to \text{A}\mid h\text{ has finite support}\}$
We equip $C_c(G,\text{A})$ with a convolution product and an involution, making it a $\ast$-algebra.

• Convolution: For $h,k\in C_c(G,\text{A})$, the product $h\ast k$ is defined as: $(h\ast k)(t)={\sum }_{s\in G}h(s){\alpha }_s(k(s^{-1}t))\text{for }t\in G$
• Involution: The involution $h^{\ast }$ is defined as: $h^{\ast }(t)={\alpha }_t(h(t^{-1}{)}^{\ast })$

An element $h\in C_c(G,\text{A})$ can be formally written as a sum $h={\sum }_{t\in G}h(t){\delta }_t$, where ${\delta }_t$ is the function that is $1_{\text{A}}$ at $t$ and $0$ elsewhere.

Integrated Form of a Covariant Representation

Given a covariant representation $(\pi ,U)$ of $(\text{A},G,\alpha )$ on a Hilbert space $\mathcal{H}$, we can define a map $\pi \times U:C_c(G,\text{A})\to B(\mathcal{H})$ by: $(\pi \times U)(h)={\sum }_{t\in G}\pi (h(t))U_t\in B(\mathcal{H})$

Proposition

The map $\pi \times U$ is a $\ast$-representation of the algebra $C_c(G,\text{A})$.

The Full Crossed Product

We define the universal norm on $C_c(G,\text{A})$ as: $||h|{|}_u={\sup }_{(\pi ,U)}||(\pi \times U)(h)||$where the supremum is taken over all covariant representations $(\pi ,U)$ of $(\text{A},G,\alpha )$. This norm is bounded by the $L^1$-norm:$||h|{|}_u\le {\sum }_{t\in G}||\pi (h(t))U_t||\le {\sum }_{t\in G}||h(t)|{|}_{\text{A}}=||h|{|}_{L^1(G,\text{A})}$

The (full) crossed product C*-algebra, denoted $\text{A}{⋊}_{\alpha }G$, is the completion of $C_c(G,\text{A})$ with respect to the universal norm $||\cdot |{|}_u$.

Universal Property

Canonical Maps

There exist canonical maps from $\text{A}$ and $G$ into the (multiplier algebra of) $\text{A}{⋊}_{\alpha }G$.

1. The map $i_{\text{A}}:\text{A}\to \text{A}{⋊}_{\alpha }G$ defined by $i_{\text{A}}(a)=a{\delta }_e$ is an isometric *-homomorphism.
2. The map $i_G:G\to \text{A}{⋊}_{\alpha }G$ defined by $i_G(s)=1_{\text{A}}{\delta }_s$ is an injective group homomorphism into the group of unitaries.

Proof Sketch For the isometry of $i_{\text{A}}$, we first note that for any covariant representation $(\pi ,U)$: $(\pi \times U)(i_{\text{A}}(a))=(\pi \times U)(a{\delta }_e)=\pi (a)U_e=\pi (a)$This implies $||i_{\text{A}}(a)|{|}_u=\sup ||(\pi \times U)(i_{\text{A}}(a))||=\sup ||\pi (a)||\le ||a||$.
To show the reverse inequality, one can construct a specific covariant representation using the universal representation ${\pi }_0$ of $\text{A}$ and the left regular representation of $G$ on ${\ell }^2(G,\mathcal{H})$. For this representation, one finds that $||({\pi }_0\times U)(i_{\text{A}}(a))||=||a||$. Therefore, $||i_{\text{A}}(a)|{|}_u=||a||$.
For the map $i_G$:

• It is a homomorphism since $i_G(st)=1_{\text{A}}{\delta }_{st}$ and $(i_G(s))(i_G(t))=(1_{\text{A}}{\delta }_s)\ast (1_{\text{A}}{\delta }_t)=1_{\text{A}}{\delta }_{st}$.
• To show injectivity, we can use the same covariant representation $({\pi }_0,U)$ on ${\ell }^2(G,\mathcal{H})$. We have $({\pi }_0\times U)(i_G(s))={\pi }_0(1_{\text{A}})U_s=U_s$. Since the left regular representation $s\mapsto U_s$ is injective, $i_G$ must also be injective.
  $\square$

Universal Property of the Crossed Product

For every covariant representation $(\pi ,U)$ of $(\text{A},G,\alpha )$ on $B(\mathcal{H})$, there exists a unique $\ast$-representation of the crossed product, which we also denote by $\pi \times U$, from $\text{A}{⋊}_{\alpha }G$ to $B(\mathcal{H})$ such that:

• $(\pi \times U)(i_{\text{A}}(a))=\pi (a)$ for all $a\in \text{A}$.
• $(\pi \times U)(i_G(s))=U_s$ for all $s\in G$.

Conversely, every non-degenerate representation of $\text{A}{⋊}_{\alpha }G$ arises in this way from a covariant representation of $(\text{A},G,\alpha )$.

The Reduced Crossed Product

Reduced vs. Full

We can also define a reduced norm by taking the supremum only over regular covariant representations: $||h|{|}_r={\sup }_{(\pi ,U)\text{ regular}}||(\pi \times U)(h)||$
The completion of $C_c(G,\text{A})$ with respect to this norm is the reduced crossed product, denoted $\text{A}{⋊}_{\alpha ,r}G$.

In general, there is a surjective *-homomorphism $\text{A}{⋊}_{\alpha }G\to \text{A}{⋊}_{\alpha ,r}G$. However, this map may not be injective.

Examples

Trivial Group Action

If the group is trivial, $G=\{e\}$, and the action is the identity, then $\text{A}{⋊}_{id}\{e\}\cong \text{A}$.

Group C*-Algebras

Let the C*-algebra be the complex numbers, $\text{A}=\text{C}$, and the action be the trivial action, $\alpha =id$. The crossed product $\text{C}{⋊}_{id}G$ is called the group C-algebra* of G, denoted $C^{\ast }(G)$.

• In this case, $C_c(G,\text{C})$ is just the space of complex-valued functions with finite support, $C_c(G)$.
• The convolution product becomes $(h\ast k)(t)={\sum }_{s\in G}h(s)k(s^{-1}t)$.
• The involution is $h^{\ast }(t)=\overline{h(t^{-1})}$.
• $C^{\ast }(G)$ is the universal C*-algebra generated by a set of unitaries $\{u_s{\}}_{s\in G}$ satisfying the group relations $u_s{u}_t=u_{st}$.

Abelian Groups

Consider the group C*-algebra $C^{\ast }(G)$ where $G$ is an abelian group.

• The algebra $C_c(G)$ is abelian.
• Since $C^{\ast }(G)$ is an abelian, unital C*-algebra, the Gelfand-Naimark theorem states that there exists a compact Hausdorff space $X$ such that $C^{\ast }(G)\cong C(X)$.

This space $X$ can be identified with the Pontryagin dual of G, denoted $\hat{G}$.

Character Group (Pontryagin Dual)

A character of an abelian group $G$ is a group homomorphism $\psi :G\to \text{T}$. The set of all characters is denoted by $\hat{G}$. When endowed with the topology of pointwise convergence, $\hat{G}$ becomes a compact Hausdorff space.

Proposition

For an abelian group $G$, we have the isomorphism $C^{\ast }(G)\cong C(\hat{G})$.

Proof Sketch
The proof involves showing a correspondence between the character space (maximal ideal space) of the C*-algebra $C^{\ast }(G)$ and the character group $\hat{G}$.

• Any character $\psi \in \hat{G}$ gives rise to a multiplicative linear functional ${\phi }_{\psi }$ on $C_c(G)$ by setting ${\phi }_{\psi }(u_s)=\psi (s)$. This extends to a character on the full C*-algebra $C^{\ast }(G)$.
• Conversely, any character $\phi$ on $C^{\ast }(G)$ gives a group character $\psi$ on G via $\psi (s)=\phi (u_s)$.
• The topology of pointwise convergence on $\hat{G}$ corresponds to the weak-* topology on the character space of $C^{\ast }(G)$.
  $\square$

Examples of Dual Groups:

• If $G={\mathbb{Z}}_n$, its characters are of the form ${\chi }_k(j)=e^{2\pi ikj/n}$ for $k=0,\dots ,n-1$. Thus, ${\hat{\mathbb{Z}}}_n\cong {\mathbb{Z}}_n$. (In general, for any finite abelian group, $G\cong \hat{G}$).
• If $G=\mathbb{Z}$, any character $\chi$ is determined by $\chi (1)\in \text{T}$. Thus, $\hat{\mathbb{Z}}\cong \text{T}$.

Free Groups

Consider the free group on two generators, $G=F_2$.

• The group C*-algebra $C^{\ast }(F_2)$ is the universal C*-algebra generated by two unitaries $U,V$.
• This contrasts with $C^{\ast }({\mathbb{Z}}^2)$, which is the universal C*-algebra generated by two commuting unitaries $u,v$ (since ${\mathbb{Z}}^2$ is abelian).

Proposition

The C*-algebra $C^{\ast }(F_n)$ has a family of finite-dimensional representations $\{{\pi }_n\}$ such that their direct sum ${⨁}_{n\in \mathbb{N}}{\pi }_n$ is an isometric representation.

Proof Sketch
The proof relies on approximating the universal representation.

1. By the Gelfand-Naimark theorem, $C^{\ast }(F_2)$ has a faithful (isometric) universal representation $\pi :C^{\ast }(F_2)\to B(\mathcal{H})$ on a separable Hilbert space $\mathcal{H}$. Let $U=\pi ({\delta }_u)$ and $V=\pi ({\delta }_v)$ be the unitaries corresponding to the generators.
2. Let $\{e_k\}$ be an orthonormal basis for $\mathcal{H}$ and let $P_n$ be the projection onto the span of $\{e_1,\dots ,e_n\}$.
3. Define compressed operators $A_n=P_{n}U{|}_{P_n(\mathcal{H})}$ and $B_n=P_{n}V{|}_{P_n(\mathcal{H})}$. These are not necessarily unitary.
4. “Purify” these contractions to unitaries on a larger space. Define unitaries $U_n$ and $V_n$ on the space $P_n(\mathcal{H})\oplus P_n(\mathcal{H})$ using $A_n$ and $B_n$. For example: $V_n=\left(\begin{array}{cc}{B}_n& (I_n-B_n^{\ast }{B}_n{)}^{1/2}\\ (I_n-B_n{B}_n^{\ast }{)}^{1/2}& -B_n^{\ast }\end{array}\right)$
5. This defines a sequence of finite-dimensional representations ${\pi }_n:C^{\ast }(F_2)\to B(P_n(\mathcal{H})\oplus P_n(\mathcal{H}))$ by setting ${\pi }_n({\delta }_u)=U_n$ and ${\pi }_n({\delta }_v)=V_n$.
6. One can then show that for any element $f\in C_c(F_2)$, the norm $||{\pi }_n(f)||$ converges to $||\pi (f)||$. This implies that the direct sum representation is isometric: $||{⨁}_{n\ge 1}{\pi }_n(f)||={\sup }_{n\ge 1}||{\pi }_n(f)||=||\pi (f)||=||f|{|}_u$ This isometry extends from $C_c(F_2)$ to all of $C^{\ast }(F_2)$.
   $\square$