There are, roughly speaking, two kinds of algebras that can be functorially constructed from a group . The kind which is covariantly functorial is some variation on the group algebra , which is the free -module on with multiplication inherited from the multiplication on . The kind which is contravariantly functorial is some variation on the algebra of functions with pointwise multiplication.
When and when is respectively either a discrete group or a compact (Hausdorff) group, both of these algebras can naturally be endowed with the structure of a random algebra. In the case of , the corresponding state is a noncommutative refinement of Plancherel measure on the irreducible representations of , while in the case of , the corresponding state is by definition integration with respect to normalized Haar measure on .
In general, some nontrivial analysis is necessary to show that the normalized Haar measure exists, but for compact groups equipped with a faithful finite-dimensional unitary representation it is possible to at least describe integration against Haar measure for a dense subalgebra of the algebra of class functions on using representation theory. This construction will in some sense explain why the category of (finite-dimensional continuous unitary) representations of behaves like an inner product space (with being analogous to the inner product); what it actually behaves like is a random algebra, namely the random algebra of class functions on .