# Covariant Systems

Recall a topological group $G$ is a group and a topological space such that the maps $(x,y)\mapsto xy$ and $x\mapsto x^{-1}$ are continuous.  Let $\Sigma$ be the $\sigma$-algebra generated by the compact subsets of $G$.  A measure $\mu$ on $\Sigma$ is left-invariant if $\mu(tU)=\mu(U)$ for all $t\in G$ and $U\in\Sigma$.  A left Haar measure on $G$ is a left-invariant Radon measure on $\Sigma$.

Theorem 1.  Let $G$ be a locally compact group.  Then there exists a unique left and unique right Haar measure on $G$ (up to multiplication by a constant).

See [2] for its construction.  Let $\mu$ be a left Haar measure on $G$.  We can define right translate Haar measures by $\mu_t(U)=\mu(Ut)$.  Clearly these are left invariant as they simply changes the input of the original left invariant Haar measure.

Proposition 2.  If $\mu$ is a left Haar measure, then there is a unique function $\Delta_G:G\to\mathbb{R},$ called the modular function of $G$, such that $\mu_t(U)=\Delta_G(t)\mu(U)$.

Let $\rho:G\to U(L(H))$ be a strongly continuous unitary representation of $G$ on a Hilbert space $H$.  That is, the group homomorphism is continuous with respect to the norm topology of $U(L(H))$, and $U(L(H))$ is the subset of $L(H)$ consisting of unitary linear operators on $H$.  Let $\mu$ be a left Haar measure on $G$ and $f\in L^1(G)$, then the operator

$\displaystyle\rho^*(f)=\int_G f(t)\rho(t)\,d\mu(t)$

in $L(H)$ is bounded and in fact defines a nondegenerate representation $\rho^*:L^1(G)\to L(H)$ as a Banach *-homomorphism with $f\mapsto\rho^*(f)$.  The product on $L^1(G)$ is convolution:

$\displaystyle (f*g)(t)=\int_G f(s)g(s^{-1}t)\,d\mu(s)$

and the involution is defined by

$f^*(t)=\Delta_G(t^{-1})\bar{f}(t^{-1}).$

Recall in the GNS construction we started with a positive functional $f$ on a C*-algebra $A$ and induced a representation $\rho_f:A\to L(H_f)$.  Here we start with one representation $\rho$ on $G$ and induce another, $\rho^*$, on $L^1(G)$.  We call $\rho^*$ the integrated form of $\rho$.  We can impose another norm (other than the default sup norm $\|f\|=\sup_{t\in G}|f(t)|$) on $L^1(G)$ defined by

$\|f\|=\sup_{\rho}\|\rho(f)\|$

where $\rho$ is a representation of $L^1(G)$.  The completion of $L^1(G)$ with respect to this norm is a C*-algebra called the group C*-algebra of $G$, which we denote by $C^*(G)$.  So every strongly continuous unitary representation of $G$ induces a nondegenerate representation of $L^1(G)$, and in fact, the converse is also true (that every nondegenerate representation of $L^1(G)$ is induced by a scu representation of $G$).  Moreover there is a bijection between the irreducible ones in each case.

Definition 3.  A covariant system is a triple $(G,A,\alpha)$ where $G$ is a locally compact group, $A$ is a C*-algebra, and $\alpha:G\to Aut(A)$ is a continuous representation (where $Aut(A)$ possesses the point-norm topology).  A covariant representation on a covariant system $(G,A,\alpha)$ is a pair $(\rho,\psi)$ of representations of $G$ and $A$ respectively on a Hilbert space $H$ such that $\rho$ is strongly continuous unitary, $\psi$ is nondegenerate, and

$\rho(t)\psi(a)\rho(t)^*=\psi(\alpha(t)(a))$

for all $t\in G$ and $a\in A$.

Definition 4.  We define the covariance algebra $L^1(G,A)$ of the covariant system $(G,A,\alpha)$ as the completion of $C_C(G,A)$ under the norm

$\displaystyle\|f\|_1=\int_G \|f(t)\|\,d\mu(t)$

where we define

$\displaystyle (f*g)(t)=\int_G f(s)\cdot\alpha(s)(g(s^{-1}t))\,d\mu(s)$

$\displaystyle f^*(t)=\Delta_G(t^{-1})\cdot\alpha(t)(f(t^{-1})^*).$

It is a Banach *-algebra.

Definition 5.  If $(\rho,\psi)$ is a covariant representation of $(G,A,\alpha)$, then there is a nondegenerate representation $\rho\times\psi:L^1(G,A)\to L(H)$ defined by

$\displaystyle(\rho\times\psi)(f)=\int_G \rho(f(t))\psi(t)\,d\mu(t)$

called the integrated form of $(\rho,\psi)$$L^1(G,A)$ together with a new norm

$\displaystyle\|f\|=\sup_{(\rho,\psi)}\|(\rho\times\psi)(f)\|$

is a C*-algebra called the crossed product of $(G,A,\alpha)$, which is denoted $C^*(G,A,\alpha)$.

[1]  Blackadar, Bruce.  Operator Algebras.  Encyclopedia of Mathematical Sciences.  Vol. 122.  Springer-Verlag.  2006.

[2]  Folland, Gerald.  Real Analysis: Modern Techniques and Their Applications.  2nd Edition.  John Wiley and Sons.  1999.