More explicitly, it is a group equipped with a topology such that the multiplication and inversion maps are continuous.
A topological group carries two canonical uniformities: a right and left uniformity. The right uniformity consists of entourages where if ; here ranges over neighborhoods of the identity that are symmetric: . The left uniformity similarly consists of entourages where if . The uniform topology for either coincides with the topology of .
Obviously when is commutative, the left and right uniformities coincide. They also coincide if is compact Hausdorff, since in that case there is only one uniformity whose uniform topology reproduces the given topology.
Let , be topological groups, and equip each with their left uniformities. Let be a group homomorphism.
The following are equivalent:
The map is continuous at a point of ;
The map is continuous;
The map is uniformly continuous.
Suppose is continuous at . Since neighborhoods of a point are -translates of neighborhoods of the identity , continuity at means that for all neighborhoods of , there exists a neighborhood of such that
Since is a homomorphism, it follows immediately from cancellation that . Therefore, for every neighborhood of , there exists a neighborhood of such that
in other words such that . Hence is uniformly continuous with respect to the right uniformity. By similar reasoning, is uniformly continuous with respect to the right uniformity.
In physics, when a classical mechanical system is symmetric, i.e. invariant in a proper sense, with respect to the action of a topological group , then an unitary representation of is sometimes called a quantization of . See at geometric quantization and orbit method for more on this.
The reason that in the definition of a unitary representation, the strong operator topology on is used and not the norm topology, is that only few homomorphisms turn out to be continuous in the norm topology.
and this will generally not be continuous in the norm topology, but is always continuous in the strong topology.
The following monograph is not particulary about group representations, but some content of this page is based on it: