When an abstract group is represented by linear transformations on a vector space, the group of transformations generates an algebra. In the case of finite dimensional representations of a finite group, the algebra consists of linear combinations of elements in the range of the representation. It is therefore natural to associate an algebra with the abstract group in a similar way be defining the group algebra. The representation theory of the group is then equivalent to the representation theory of an algebra. Unfortunately there appears to be no single natural analogue of the concept of group algebra, to be used when studying infinite dimensional representations of topological groups. In the case of unitary representations of locally compact groups, the most popular choice for the role of "group algebra" has been L1(G). The crucial sense in which L' (G) generalizes the concept of the group algebra of a finite group is that the strongly continuous unitary representation theory of G is equivalent to the no-where trivial *-representation theory of L1(G).