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 [2] 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 .

[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.