Recall a topological group is a group and a topological space such that the maps and are continuous. Let be the -algebra generated by the compact subsets of . A measure on is left-invariant if for all and . A left Haar measure on is a left-invariant Radon measure on .
Theorem 1. Let be a locally compact group. Then there exists a unique left and unique right Haar measure on (up to multiplication by a constant).
See  for its construction. Let be a left Haar measure on . We can define right translate Haar measures by . Clearly these are left invariant as they simply changes the input of the original left invariant Haar measure.
Proposition 2. If is a left Haar measure, then there is a unique function called the modular function of , such that .
Let be a strongly continuous unitary representation of on a Hilbert space . That is, the group homomorphism is continuous with respect to the norm topology of , and is the subset of consisting of unitary linear operators on . Let be a left Haar measure on and , then the operator
in is bounded and in fact defines a nondegenerate representation as a Banach *-homomorphism with . The product on is convolution:
and the involution is defined by
Recall in the GNS construction we started with a positive functional on a C*-algebra and induced a representation . Here we start with one representation on and induce another, , on . We call the integrated form of . We can impose another norm (other than the default sup norm ) on defined by
where is a representation of . The completion of with respect to this norm is a C*-algebra called the group C*-algebra of , which we denote by . So every strongly continuous unitary representation of induces a nondegenerate representation of , and in fact, the converse is also true (that every nondegenerate representation of is induced by a scu representation of ). Moreover there is a bijection between the irreducible ones in each case.
Definition 3. A covariant system is a triple where is a locally compact group, is a C*-algebra, and is a continuous representation (where possesses the point-norm topology). A covariant representation on a covariant system is a pair of representations of and respectively on a Hilbert space such that is strongly continuous unitary, is nondegenerate, and
for all and .
Definition 4. We define the covariance algebra of the covariant system as the completion of under the norm
where we define
It is a Banach *-algebra.
Definition 5. If is a covariant representation of , then there is a nondegenerate representation defined by
called the integrated form of . together with a new norm
is a C*-algebra called the crossed product of , which is denoted .
 Blackadar, Bruce. Operator Algebras. Encyclopedia of Mathematical Sciences. Vol. 122. Springer-Verlag. 2006.
 Folland, Gerald. Real Analysis: Modern Techniques and Their Applications. 2nd Edition. John Wiley and Sons. 1999.